# Ridges, Neural Networks, and the Radon Transform

Michael Unser<EMAIL_ADDRESS>
Biomedical Imaging Group
École polytechnique fédérale de Lausanne (EPFL)
CH-1015 Lausanne, Switzerland

###### Abstract

A ridge is a function that is characterized by a one-dimensional profile
(activation) and a multidimensional direction vector. Ridges appear in the
theory of neural networks as functional descriptors of the effect of a neuron,
with the direction vector being encoded in the linear weights. In this paper,
we investigate properties of the Radon transform in relation to ridges and to
the characterization of neural networks. We introduce a broad category of
hyper-spherical Banach subspaces (including the relevant subspace of measures)
over which the back-projection operator is invertible. We also give conditions
under which the back-projection operator is extendable to the full parent
space with its null space being identifiable as a Banach complement. Starting
from first principles, we then characterize the sampling functionals that are
in the range of the filtered Radon transform. Next, we extend the definition
of ridges for any distributional profile and determine their (filtered) Radon
transform in full generality. Finally, we apply our formalism to clarify and
simplify some of the results and proofs on the optimality of ReLU networks
that have appeared in the literature.

###### Contents

  1. 1 Introduction
  2. 2 Mathematical Preliminaries
    1. 2.1 Notations
    2. 2.2 Polynomial Spaces and Related Projectors
    3. 2.3 Radon Transform
      1. 2.3.1 Classical Integral Formulation
      2. 2.3.2 Distributional Extension
  3. 3 Radon-Compatible Banach Spaces
  4. 4 Radon-Domain Sampling Functionals
    1. 4.1 Abstract Characterization of Radon-Compatible Diracs
    2. 4.2 Constructive Description of Radon-Compatible Dirac
  5. 5 Ridges Revisited
    1. 5.1 Generalized Ridges
    2. 5.2 Connection with Prior Works
  6. 6 Variational Optimality of ReLU Networks
    1. 6.1 Learning with Radon-Domain Regularization
    2. 6.2 Discussion
  7. A Direct-Sum Topologies
    1. A.1 Projectors
    2. A.2 Direct Sums
  8. B Extreme Points

##  1 Introduction

A ridge is a multidimensional function ${\boldsymbol{x}}\mapsto
r({\boldsymbol{w}}^{\mathsf{T}}{\boldsymbol{x}})$ from
$\mathbb{R}^{d}\to\mathbb{R}$ that is characterized by a 1D profile
$r:\mathbb{R}\to\mathbb{R}$ and a weight vector
${\boldsymbol{w}}\in\mathbb{R}^{d}\backslash\\{{\boldsymbol{0}}\\}$ (Pinkus,
2015). Ridges are ubiquitous in mathematics and engineering. Most
significantly, the elementary unit (neuron) in a neural network is a function
of the form
$f_{k}({\boldsymbol{x}})=\sigma({\boldsymbol{w}}_{k}^{\mathsf{T}}{\boldsymbol{x}}-t_{k})$,
which is a ridge with a shifted profile $r=\sigma(\cdot-t_{k})$, where
$\sigma:\mathbb{R}\to\mathbb{R}$ is the activation function and where
$t_{k}\in\mathbb{R}$ (bias) and ${\boldsymbol{w}}_{k}\in\mathbb{R}^{d}$
(linear weights) are the trainable parameters of the $k$th neuron (Bishop,
2006). Variants of the universal-approximation theorem ensure that any
continuous function can be approximated as closely as desired by a weighted
sum of ridges with a fixed activation under mild conditions on $\sigma$
(Cybenko, 1989; Hornik et al., 1989; Barron, 1993).

Ridges are also intimately tied to the Radon transform (Logan and Shepp, 1975;
Madych, 1990) under the condition that the weight vector ${\boldsymbol{w}}$
has a unit norm, so that ${\boldsymbol{w}}\in\mathbb{S}^{d-1}$ where
$\mathbb{S}^{d-1}$ is the unit sphere in $\mathbb{R}^{d}$ whose generic
elements will be denoted by ${\boldsymbol{\xi}}$. This connection is exploited
in the ridgelet transform, which provides a wavelet-like representation of
functions where the basis elements are ridges (Murata, 1996; Rubin, 1998;
Candès, 1999; Candès and Donoho, 1999; Kostadinova et al., 2014). The
expansion of a function in terms of ridgelets is a precusor to sparse signal
approximation. There, the idea is to represent a function by a linear
combination of a small number of atoms taken within a dictionary (Elad, 2010;
Foucart and Rauhut, 2013). This paradigm, which is the basis for compressed
sensing (Donoho, 2006; Candès and Romberg, 2007), has been adapted to shallow
neural networks by considering a dictionary that consists of a continuum of
neurons. Mathematically, this can be implemented through the integral
representation (infinite-width neural network)

$\displaystyle f({\boldsymbol{x}})$
$\displaystyle=\int_{\mathbb{R}\times\mathbb{S}^{d-1}}\sigma({\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}-t){\rm
d}\mu(t,{\boldsymbol{\xi}}),$ (1)

where $\mu$ is a measure on $\mathbb{R}\times\mathbb{S}^{d-1}$ (hyper-
spherical domain). This model is fitted to data subject to a penalty on the
total-variation norm of $\mu$ (Bach, 2017). Remarkably, this infinite-
dimensional convex optimization problem results in sparse minimizers of the
form $\mu=\sum_{k=1}^{K}a_{k}\delta_{{\boldsymbol{z}}_{k}}$ with
${\boldsymbol{z}}_{k}=(t_{k},{\boldsymbol{\xi}}_{k})\in\mathbb{R}\times\mathbb{S}^{d-1}$,
which then map into standard two-layer neural networks (Bach, 2017).
Interestingly, we can relate (1) to the Radon transform by identifying $\mu$
as the (generalized) function $g_{\mu}$ (with ${\rm
d}\mu(t,{\boldsymbol{\xi}})=g_{\mu}(t,{\boldsymbol{\xi}}){\rm d}t{\rm
d}{\boldsymbol{\xi}}$) and by rewriting the integral as

$\displaystyle f({\boldsymbol{x}})$
$\displaystyle=\int_{\mathbb{S}^{d-1}}\left(\int_{\mathbb{R}}\sigma({\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}-t)g_{\mu}(t,{\boldsymbol{\xi}}){\rm
d}t\right){\rm d}{\boldsymbol{\xi}}={\mathrm{R}}^{\ast}{\mathrm{L}}_{\rm
rad}\\{g_{\mu}\\}({\boldsymbol{x}}),$ (2)

where ${\mathrm{R}}^{\ast}$ (the adjoint of the Radon transform) is the back-
projection operator of computer tomography (Natterer, 1984). Our “radial”
operator ${\mathrm{L}}_{\rm rad}:g_{\mu}\mapsto\sigma\ast_{t}g_{\mu}$ on the
right-hand side of (2) implements the Radon-domain convolution with $\sigma$
along the variable $t$.

While the synthesis approach to the learning problem proposed by Bach (2017)
is insightful, there is a strong incentive to make the connection with
regularization theory in direct analogy with the classical theory of learning
that relies on reproducing-kernel Hilbert spaces (Wahba, 1990; Poggio and
Girosi, 1990; Schölkopf et al., 1997, 2001; Alvarez et al., 2012; Unser,
2021). This is feasible provided that the linear relation between $f$ and
$g_{\mu}$ expressed by (2) be one-to-one. This requires that the operators
${\mathrm{L}}_{\rm rad}$ and ${\mathrm{R}}^{\ast}$ in (2) be both invertible.
Ongie et al. (2020) made an important step in that direction by showing that
ReLU networks are minimizers of a Radon-domain total-variation norm that
involves the Laplacian of $f$. Their optimality result was then generalized by
Parhi and Nowak (2021) who considered a broader class of differential
operators inspired by spline theory (Unser et al., 2017). The leading idea
there is that the operator ${\mathrm{L}}_{\rm rad}$ in (2) should implement
some variant of an $n$th-order integrator, with $\sigma={\rm ReLU}$ being the
solution for $n=2$. Such an ${\mathrm{L}}_{\rm rad}$ can be formally inverted
by applying an $n$th-order partial derivative (e.g., ${\mathrm{L}}^{-1}_{\rm
rad}=\partial^{n}_{t}$), which motivates the use of the latter as (filtered)
Radon-domain regularization operator.

The proposed spline-based approach to the inversion of (2) is elegant and
intuitively appealing. However, the formulation and resolution of the
corresponding optimization problem requires special care because the
underlying function spaces have a nontrivial kernel (null space) that needs to
be factored out. The latter statement applies not only to the regularization
operator (e.g., Laplacian and/or Radon-domain radial derivatives) but also to
${\mathrm{R}}^{\ast}$, which is an aspect that has been overlooked. While
there is a rich theory on the invertibility of the Radon transform (Helgason,
2011; Rubin, 1998; Boman and Lindskog, 2009; Ramm and Katsevich, 2020), there
are comparatively fewer—and not as strong—results on the invertibility of
${\mathrm{R}}^{\ast}$, the problem being that this operator has a huge null
space (Ludwig, 1966). The primary spaces on which ${\mathrm{R}}^{\ast}$ is
known to be injective, and hence invertible, are

  * •

${\mathcal{S}}(\mathbb{P}^{d})$ (the even part of Schwartz’ hyper-spherical—or
Radon-domain—test functions) (Solmon, 1987);

  * •

$L_{\infty,c}(\mathbb{P}^{d})$ (the bounded even functions of compact support)
(Ramm, 1996);

  * •

${\mathcal{S}}^{\prime}_{\rm Liz}(\mathbb{P}^{d})$ (the even Lizorkin
distributions) (Kostadinova et al., 2014).

The space ${\mathcal{S}}^{\prime}_{\rm Liz}(\mathbb{R}^{d})$ of Lizorkin
distributions, which is the topological dual of ${\mathcal{S}}_{\rm
Liz}(\mathbb{R}^{d})$ (the subspace of Schwartz functions that are orthogonal
to all polynomials), is especially attractive in that context. Indeed, the
Radon transform being an homeomorphism from ${\mathcal{S}}^{\prime}_{\rm
Liz}(\mathbb{R}^{d})$ onto ${\mathcal{S}}^{\prime}_{\rm Liz}(\mathbb{P}^{d})$,
the inversion process is straightforward (Kostadinova et al., 2014). Lizorkin
distributions also interact very nicely with the Laplace operator, which makes
then well suited to the investigation of fractional integrals (Samko et al.,
1993) and of wavelets (Saneva and Vindas, 2010). The Lizorkin framework,
however, has one basic limitation. The underlying objects—Lirzorkin
distributions—are abstract entities, with ${\mathcal{S}}^{\prime}_{\rm
Liz}(\mathbb{R}^{d})$ being isomorphic to the quotient space
${\mathcal{S}}^{\prime}(\mathbb{R}^{d})/{\mathcal{P}}$, where
${\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ and ${\mathcal{P}}$ are the spaces of
tempered distributions and polynomials, respectively. Thus, Lizorkin
distributions are generally identifiable only modulo some polynomial.
Fortunately, this is not a problem when dealing with ordinary functions $f\in
L_{p}(\mathbb{R}^{d})$ since $L_{p}(\mathbb{R}^{d})$ is continuously embedded
in ${\mathcal{S}}^{\prime}_{\rm Liz}(\mathbb{R}^{d})$ for any $p>1$ (Samko,
1982). This implies that the Lizorkin distribution
$f+{\mathcal{P}}\in{\mathcal{S}}^{\prime}_{\rm Liz}(\mathbb{R}^{d})$ has a
unique “concrete” representer $f\in L_{p}(\mathbb{R}^{d})$, which amounts to
simply setting the polynomial to zero. The situation, however, is not as
clearcut for functions and ridge profiles that exhibit polynomial growth at
infinity. To offer insights on the nature of the problem, let us consider
three distinct neuronal units $f_{1}(x)=(x-t_{k})_{+}$ (ReLU activation),
$f_{2}(x)=\tfrac{1}{2}|x-t_{k}|$, and $f_{3}(x)=x+(x-t_{k})_{+}$ (ReLU with
skip connection), which are all valid representers of the same Lizorkin
distribution $f=f_{i}+{\mathcal{P}}\in{\mathcal{S}}^{\prime}_{\rm
Liz}(\mathbb{R})$ for $i=1,2,3$ (since the $f_{i}$’s only differ by a first-
order polynomial). Suppose that a theoretical argument can be made concerning
the optimality of the announced $f\in{\mathcal{S}}^{\prime}_{\rm
Liz}(\mathbb{R})$. The practical difficulty then is to map this result into a
concrete architecture. Should the choice be one of the $f_{i}$, if any? The
least we can say is that the convenient rule of “setting the polynomial to
zero” is not applicable here because it is unclear what the underlying
polynomial truly is. This intrinsic ambiguity jeopardizes some of the
conclusions regarding the connection between ReLU neural networks, ridge
splines, and the Radon transform that have been reported in the literature
(Sonoda and Murata, 2017; Parhi and Nowak, 2021). We are in the opinion that
adjustments are needed.

In this paper, we revisit the topic and extend the existing formulation so
that it can handle arbitrary ridge profiles, without (polynomial) ambiguity.
Our four primary contributions are as follows.

  * •

A detailed investigation of the invertibility of the back-projection operator
for a broad family of Radon-compatible Banach subspaces
${\mathcal{X}}^{\prime}_{\rm Rad}\subset{\mathcal{X}}^{\prime}$, where
${\mathcal{X}}^{\prime}$ is the topological dual of some generic hyper-
spherical parent space ${\mathcal{X}}$.

  * •

A constructive characterization of the extreme points of the space of Radon-
compatible hyper-spherical measures.

  * •

The extension of ridges to distributional profiles
$r\in{\mathcal{S}}^{\prime}(\mathbb{R})$ and the determination of their Radon
transform.

  * •

The application of the formalism to the investigation of a functional-
optimization problem that results in solutions that are parameterized by ReLU
neural networks. Our contribution there is to clarify the analysis of Parhi
and Nowak (2021) and to provide a characterization of the full solution set.

The paper is organized as follows: We start with notations and mathematical
preliminaries in Section 2. In particular, we recall the main properties of
the classical Radon transform and its adjoint, and show how to extend them to
tempered distributions by duality. In Section 3, we develop a formulation that
leads to the identification of a generic family of Radon-compatible Banach
spaces over which the back-projection operator is invertible. Our results are
summarized in Theorem 7, which can be viewed as the Banach counterpart of the
classical result for tempered distributions (Ludwig, 1966). In Section 4, we
use our framework to characterize the sampling functionals (Radon-compatible
Diracs) that are in the range of the filtered Radon transform (Theorem 8). In
Section 5, we introduce a general definition of a ridge with an arbitrary
distributional profile and derive its (filtered) Radon transform. Finally, in
Section 6, we apply our formalism to the resolution of a multidimensional
supervised-learning problem with a 2nd-order Radon-domain regularization
formulated by Parhi and Nowak (2021), the outcome being Theorem 12 on the
optimality of ReLU networks.

##  2 Mathematical Preliminaries

###  2.1 Notations

We shall consider multidimensional functions $f$ on $\mathbb{R}^{d}$ that are
indexed by the variable ${\boldsymbol{x}}\in\mathbb{R}^{d}$. To describe their
partial derivatives, we use the multi-index
${\boldsymbol{k}}=(k_{1},\dots,k_{d})\in\mathbb{N}^{d}$ with the notational
conventions
${\boldsymbol{k}}!\stackrel{{\scriptstyle\vartriangle}}{{=}}\prod_{i=1}^{d}k_{i}!$,
$|{\boldsymbol{k}}|\stackrel{{\scriptstyle\vartriangle}}{{=}}k_{1}+\cdots+k_{d}$,
${\boldsymbol{x}}^{{\boldsymbol{k}}}\stackrel{{\scriptstyle\vartriangle}}{{=}}\prod_{i=1}^{d}x_{i}^{k_{i}}$
for any ${\boldsymbol{x}}\in\mathbb{R}^{d}$, and
$\partial^{\boldsymbol{k}}f({\boldsymbol{x}})\stackrel{{\scriptstyle\vartriangle}}{{=}}\frac{\partial^{|{\boldsymbol{k}}|}f(x_{1},\dots,x_{d})}{\partial^{k_{1}}_{x_{1}}\cdots\partial^{k_{d}}_{x_{d}}}$.
This allows us to write the multidimensional Taylor expansion around
${\boldsymbol{x}}_{0}$ of an analytical function
$f:\mathbb{R}^{d}\to\mathbb{R}$ explicitly as

$\displaystyle
f({\boldsymbol{x}})=\sum_{n=0}^{\infty}\sum_{|{\boldsymbol{k}}|=n}\frac{\partial^{\boldsymbol{k}}f({\boldsymbol{x}}_{0})({\boldsymbol{x}}-{\boldsymbol{x}}_{0})^{\boldsymbol{k}}}{{\boldsymbol{k}}!~},$
(3)

where the internal summation is over all multi-indices
${\boldsymbol{k}}=(k_{1},\dots,k_{d})$ such that $k_{1}+\cdots+k_{d}=n$.

Schwartz’ space of smooth and rapidly decreasing test functions
$\varphi:\mathbb{R}^{d}\to\mathbb{R}$ equipped with the usual Fréchet-Schwartz
topology is denoted by ${\mathcal{S}}(\mathbb{R}^{d})$. Its continuous dual is
the space ${\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ of tempered distributions.
In this setting, the Lebesgue spaces $L_{p}(\mathbb{R}^{d})$ for
$p\in[1,\infty)$ can be specified as the completion of
${\mathcal{S}}(\mathbb{R}^{d})$ equipped with the $L_{p}$-norm
$\|\cdot\|_{L_{p}}$; that is,
$L_{p}(\mathbb{R}^{d})=\overline{({\mathcal{S}}(\mathbb{R}^{d}),\|\cdot\|_{L_{p}})}$.
For the end point $p=\infty$, we have that
$\overline{({\mathcal{S}}(\mathbb{R}^{d}),\|\cdot\|_{L_{\infty}})}=C_{0}(\mathbb{R}^{d})$
with
$\|\varphi\|_{L_{\infty}}=\sup_{{\boldsymbol{x}}\in\mathbb{R}^{d}}|\varphi({\boldsymbol{x}})|$,
where $C_{0}(\mathbb{R}^{d})$ is the space of continuous functions that vanish
at infinity. Its continuous dual is the space
${\mathcal{M}}(\mathbb{R}^{d})=\\{f\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d}):\|f\|_{{\mathcal{M}}}<\infty\\}$
of bounded Radon measures with

$\|f\|_{{\mathcal{M}}}=\sup_{\varphi\in{\mathcal{S}}(\mathbb{R}^{d}):\|\varphi\|_{L_{\infty}}\leq
1}\langle f,\varphi\rangle.$

The latter is a superset of $L_{1}(\mathbb{R}^{d})$, which is isometrically
embedded in it, meaning that $\|f\|_{L_{1}}=\|f\|_{{\mathcal{M}}}$ for all
$f\in L_{1}(\mathbb{R}^{d})$.

The Fourier transform of a function $\varphi\in L_{1}(\mathbb{R}^{d})$ is
defined as

$\displaystyle\widehat{\varphi}({\boldsymbol{\omega}})\stackrel{{\scriptstyle\vartriangle}}{{=}}\mathcal{F}\\{\varphi\\}({\boldsymbol{\omega}})=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\varphi({\boldsymbol{x}})\mathrm{e}^{-\mathrm{j}\langle{\boldsymbol{\omega}},{\boldsymbol{x}}\rangle}{\rm
d}{\boldsymbol{x}}.$ (4)

Since the Fourier operator $\mathcal{F}$ continuously maps
${\mathcal{S}}(\mathbb{R}^{d})$ into itself, the transform can be extended by
duality to the whole space ${\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ of
tempered distribution. Specifically,
$\widehat{f}=\mathcal{F}\\{f\\}\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ is
the (unique) generalized Fourier transform of
$f\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ if and only if
$\langle\widehat{f},\varphi\rangle=\langle f,\widehat{\varphi}\rangle$ for all
$\varphi\in{\mathcal{S}}(\mathbb{R}^{d})$, where
$\widehat{\varphi}=\mathcal{F}\\{\varphi\\}$ is the “classical” Fourier
transform of $\varphi$ defined by (4).

###  2.2 Polynomial Spaces and Related Projectors

The regularization operators (e.g., the Laplacian) that are of interest to us
are isotropic and have a growth-restricted null space formed by the space
$\Pi_{n_{0}}$ of polynomials of degree $n_{0}$. Here, we choose to expand
these polynomials in the monomial/Taylor basis

$\displaystyle
m_{\boldsymbol{k}}({\boldsymbol{x}})\stackrel{{\scriptstyle\vartriangle}}{{=}}\frac{{\boldsymbol{x}}^{{\boldsymbol{k}}}}{{\boldsymbol{k}}!}$
(5)

with $|{\boldsymbol{k}}|\leq n_{0}$. Defining the $\ell_{2}$-norm of the
Taylor coefficients of the polynomial $p_{0}=\sum_{|{\boldsymbol{k}}|\leq
n_{0}}b_{\boldsymbol{k}}m_{{\boldsymbol{k}}}$ as

$\displaystyle\|p_{0}\|_{{\mathcal{P}}}\stackrel{{\scriptstyle\vartriangle}}{{=}}\|(b_{\boldsymbol{k}})_{|{\boldsymbol{k}}|\leq
n_{0}}\|_{2},$ (6)

we add a topological structure that results in the description

$\displaystyle{\mathcal{P}}_{n_{0}}=\\{p_{0}\in\Pi_{n_{0}}:\|p_{0}\|_{{\mathcal{P}}}<\infty\\}.$
(7)

The important point here is that (7) specifies a finite-dimensional Banach
subspace of ${\mathcal{S}}^{\prime}(\mathbb{R}^{d})$. Its continuous dual
${\mathcal{P}}^{\prime}_{n_{0}}$ is finite-dimensional as well, although it is
composed of “abstract” elements that are, in fact, equivalence classes in
${\mathcal{S}}^{\prime}(\mathbb{R}^{d})$. Yet, it possible to identify every
dual element $p_{0}^{\ast}\in{\mathcal{P}}^{\prime}_{n_{0}}$ concretely as a
function by selecting a particular dual basis
$\\{m_{{\boldsymbol{k}}}^{\ast}\\}_{|{\boldsymbol{k}}|\leq n_{0}}$ such that
$\langle
m^{\ast}_{{\boldsymbol{k}}},m_{{\boldsymbol{k}}^{\prime}}\rangle=\delta_{{\boldsymbol{k}}-{\boldsymbol{k}}^{\prime}}$
(Kroneker delta). Our specific choice is

$\displaystyle
m^{\ast}_{\boldsymbol{k}}=(-1)^{|{\boldsymbol{k}}|}\partial^{{\boldsymbol{k}}}\kappa_{\rm
iso}\in{\mathcal{S}}(\mathbb{R}^{d})$ (8)

with ${\boldsymbol{k}}\in\mathbb{N}^{d}$, where $\kappa_{\rm iso}$ is the
isotropic function described below.

###### Lemma 1

There exists an entire isotropic function $\kappa_{\rm
iso}\in{\mathcal{S}}(\mathbb{R}^{d})$ with $0\leq\widehat{\kappa}_{\rm
iso}({\boldsymbol{\omega}})\leq 1$ and $\widehat{\kappa}_{\rm
iso}({\boldsymbol{\omega}})=0$ for $\|{\boldsymbol{\omega}}\|\geq 1$ such that

$\displaystyle\int_{\mathbb{R}^{d}}\frac{{\boldsymbol{x}}^{{\boldsymbol{k}}}}{{\boldsymbol{k}}!}(-1)^{|{\boldsymbol{n}}|}\partial^{{\boldsymbol{n}}}\kappa_{\rm
iso}({\boldsymbol{x}}){\rm
d}{\boldsymbol{x}}=\;\delta_{{\boldsymbol{k}}-{\boldsymbol{n}}}$ (9)

for all ${\boldsymbol{k}},{\boldsymbol{n}}\in\mathbb{N}^{d}$.

Proof  We take $\kappa_{\rm iso}=\mathcal{F}^{-1}\\{\widehat{\kappa}_{\rm
rad}(\|\cdot\|)\\}$, where the radial profile $\widehat{\kappa}_{\rm
rad}:\mathbb{R}\to\mathbb{R}$ is such that $\widehat{\kappa}_{\rm
rad}\in{\mathcal{S}}(\mathbb{R})$, $\widehat{\kappa}_{\rm rad}(\omega)=1$ for
$0\leq|\omega|\leq R_{0}\leq\tfrac{1}{2}$, and $\widehat{\kappa}_{\rm
rad}(\omega)=0$ for $|\omega|\geq 1$. A particular construction with
$R_{0}=\tfrac{1}{2}$ is $\widehat{\kappa}_{\rm rad}={\rm rect}\ast\varphi$,
where $\varphi\in{\mathcal{S}}(\mathbb{R})$ is a symmetric, non-negative test
function (to avoid oscillations) with ${\rm
support}(\varphi)\subseteq[-\tfrac{1}{2},\tfrac{1}{2}]$ and
$\int_{\mathbb{R}}\varphi(x){\rm d}x=1$. Next, we observe that

$\displaystyle\langle
m_{{\boldsymbol{k}}},(-1)^{|{\boldsymbol{n}}|}\partial^{{\boldsymbol{n}}}\kappa_{\rm
iso}\rangle=\langle\partial^{{\boldsymbol{n}}}m_{{\boldsymbol{k}}},\kappa_{\rm
iso}\rangle=\begin{cases}\langle
m_{{\boldsymbol{k}}-{\boldsymbol{n}}},\kappa_{\rm
iso}\rangle,&{\boldsymbol{k}}-{\boldsymbol{n}}\geq{\boldsymbol{0}}\\\ \langle
0,\kappa_{\rm iso}\rangle=0,&\mbox{otherwise.}\end{cases}$ (10)

We evaluate the duality product for the case
${\boldsymbol{m}}=({\boldsymbol{k}}-{\boldsymbol{n}})\geq{\boldsymbol{0}}$ in
the Fourier domain as

$\displaystyle\langle m_{{\boldsymbol{m}}},\kappa_{\rm iso}\rangle$
$\displaystyle=\frac{1}{(2\pi)^{d}}\langle\mathcal{F}\\{m_{{\boldsymbol{m}}}\\},\widehat{\kappa}_{\rm
iso}\rangle$
$\displaystyle=\langle\frac{\mathrm{j}^{|{\boldsymbol{m}}|}}{{\boldsymbol{m}}!}\delta^{({\boldsymbol{m}})},\widehat{\kappa}_{\rm
iso}\rangle=\frac{-\mathrm{j}^{|{\boldsymbol{m}}|}}{{\boldsymbol{m}}!}\partial^{{\boldsymbol{m}}}\widehat{\kappa}_{\rm
iso}({\boldsymbol{0}})=\begin{cases}1,&{\boldsymbol{m}}={\boldsymbol{0}}\\\
0,&\mbox{otherwise},\end{cases}$ (11)

where we have used the relation
$\mathcal{F}\\{{\boldsymbol{x}}^{{\boldsymbol{m}}}\\}({\boldsymbol{\omega}})=(2\pi)^{d}\;\mathrm{j}^{|{\boldsymbol{m}}|}\delta^{({\boldsymbol{m}})}({\boldsymbol{\omega}})$.
Finally, since $\widehat{\kappa}_{\rm iso}$ is compactly supported, its
inverse Fourier transform $\kappa_{\rm iso}$ is an entire function of
exponential-type (by the Paley-Wiener theorem). This means that the function
${\boldsymbol{x}}\mapsto\kappa_{\rm iso}({\boldsymbol{x}})$ is analytic with a
convergent Taylor series of the form (3) for any
${\boldsymbol{x}},{\boldsymbol{x}}_{0}\in\mathbb{R}^{d}$.
This allows us to describe the dual space explicitly as

$\displaystyle{\mathcal{P}}^{\prime}_{n_{0}}=\\{p_{0}^{\ast}=\sum_{|{\boldsymbol{k}}|\leq
n_{0}}b^{\ast}_{\boldsymbol{k}}m^{\ast}_{{\boldsymbol{k}}}:\|p_{0}^{\ast}\|_{{\mathcal{P}}^{\prime}}\stackrel{{\scriptstyle\vartriangle}}{{=}}\|(b^{\ast}_{\boldsymbol{k}})\|_{2}<\infty\\}$
(12)

where each elements $p_{0}^{\ast}$ has a unique representation in terms of its
coefficients $(b^{\ast}_{\boldsymbol{k}})_{|{\boldsymbol{k}}|\leq n_{0}}$. We
can use the dual basis $\\{m_{\boldsymbol{k}}^{\ast}\\}$ to specify the
projection operator
$\mathrm{Proj}_{{\mathcal{P}}_{n_{0}}}:{\mathcal{S}}^{\prime}(\mathbb{R}^{d})\to{\mathcal{P}}_{n_{0}}$
as

$\displaystyle\mathrm{Proj}_{{\mathcal{P}}_{n_{0}}}\\{f\\}$
$\displaystyle=\sum_{|{\boldsymbol{k}}|\leq n_{0}}\langle
f,m_{\boldsymbol{k}}^{\ast}\rangle\;m_{\boldsymbol{k}},$ (13)

which is well-defined for any $f\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$
since $m_{\boldsymbol{k}}^{\ast}\in{\mathcal{S}}(\mathbb{R}^{d})$.

###  2.3 Radon Transform

The Radon transform integrates of a function of $\mathbb{R}^{d}$ over all
hyperplanes of dimension $(d-1)$. These hyperplanes are indexed over
$\mathbb{R}\times\mathbb{S}^{d-1}$, where
$\mathbb{S}^{d-1}=\\{{\boldsymbol{\xi}}\in\mathbb{R}^{d}:\|{\boldsymbol{\xi}}\|_{2}=1\\}$
is the unit sphere in $\mathbb{R}^{d}$. Specifically, the coordinates
${\boldsymbol{x}}$ of a hyperplane associated with an offset $t\in\mathbb{R}$
and a normal vector ${\boldsymbol{\xi}}\in\mathbb{S}^{d-1}$ satisfy

$\displaystyle{\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}=\xi_{1}x_{1}+\dots+\xi_{d}x_{d}=t.$
(14)

The transform is first described for ordinary (test) functions and then
extended to tempered distributions by duality.

####  2.3.1 Classical Integral Formulation

The Radon transform of the function $f\in L_{1}(\mathbb{R}^{d})$ is defined as

$\displaystyle{\mathrm{R}}\\{f\\}(t,{\boldsymbol{\xi}})$
$\displaystyle=\int_{\mathbb{R}^{d}}\delta(t-{\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}})f({\boldsymbol{x}}){\rm
d}{\boldsymbol{x}},\quad(t,{\boldsymbol{\xi}})\in\mathbb{R}\times\mathbb{S}^{d-1}.$
(15)

The adjoint of ${\mathrm{R}}$ is the back-projection operator
${\mathrm{R}}^{\ast}$. Its action on
$g:\mathbb{R}\times\mathbb{S}^{d-1}\to\mathbb{R}$ yields the function

$\displaystyle{\mathrm{R}}^{\ast}\\{g\\}({\boldsymbol{x}})=\int_{\mathbb{S}^{d-1}}g(\underbrace{{\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}}_{t},{\boldsymbol{\xi}}){\rm
d}{\boldsymbol{\xi}},\quad{\boldsymbol{x}}\in\mathbb{R}^{d}.$ (16)

Given the $d$-dimensional Fourier transform $\hat{f}=\mathcal{F}\\{f\\}$ of
$f\in L_{1}(\mathbb{R}^{d})$, one can calculate
${\mathrm{R}}\\{f\\}(\cdot,{\boldsymbol{\xi}}_{0})$ for any fixed
${\boldsymbol{\xi}}_{0}\in\mathbb{S}^{d-1}$ through

$\displaystyle{\mathrm{R}}\\{f\\}(t,{\boldsymbol{\xi}}_{0})=\frac{1}{2\pi}\int_{\mathbb{R}}\hat{f}(\omega{\boldsymbol{\xi}}_{0})\mathrm{e}^{\mathrm{j}\omega
t}{\rm d}\omega=\mathcal{F}^{-1}_{\omega\to
t}\\{\hat{f}(\omega{\boldsymbol{\xi}}_{0})\\}\\{t\\}.$ (17)

In other words, the restriction of $\hat{f}:\mathbb{R}^{d}\to\mathbb{C}$ along
the ray
$\\{{\boldsymbol{\omega}}=\omega{\boldsymbol{\xi}}_{0}:\omega\in\mathbb{R}\\}$
is equal to the 1D Fourier transform of
${\mathrm{R}}\\{f\\}(\cdot,{\boldsymbol{\xi}}_{0})$, a property that is
referred to as the Fourier slice theorem.

To describe the functional properties of the Radon transform, one needs the
(hyper)spherical (or Radon-domain) counterparts of the spaces described in
Section 2.1 where the Euclidean indexing with
${\boldsymbol{x}}\in\mathbb{R}^{d}$ is replaced by
$(t,{\boldsymbol{\xi}})\in\mathbb{R}\times\mathbb{S}^{d-1}$. The spherical
counterpart of ${\mathcal{S}}(\mathbb{R}^{d})$ is
${\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1})$. Correspondingly, an element
$g\in{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$ is a continuous
linear functional on ${\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1})$ whose
action on the test function $\phi(t,{\boldsymbol{\xi}})$ is represented by the
duality product $g:\phi\mapsto\langle g,\phi\rangle_{\rm Rad}$. When $g$ can
be identified as an ordinary function
$g:(t,{\boldsymbol{\xi}})\mapsto\mathbb{R}$, one has that

$\displaystyle\langle g,\phi\rangle_{\rm
Rad}=\int_{\mathbb{S}^{d-1}}\int_{\mathbb{R}}g(t,{\boldsymbol{\xi}})\phi(t,{\boldsymbol{\xi}}){\rm
d}t{\rm d}{\boldsymbol{\xi}},$ (18)

where ${\rm d}{\boldsymbol{\xi}}$ stands for a surface element on
$\mathbb{S}^{d-1}$ with $\|{\boldsymbol{\xi}}\|_{2}=1$. For instance, for
$d=2$, we parameterize $\mathbb{S}^{1}$ by setting
${\boldsymbol{\xi}}=(\cos\theta,\sin\theta)$ with ${\rm
d}{\boldsymbol{\xi}}={\rm d}\theta$ for $\theta\in[0,2\pi]$, which then yields

$\displaystyle\langle g,\phi\rangle_{\rm
Rad}=\int_{0}^{2\pi}\int_{\mathbb{R}}g(t,\theta)\phi(t,\theta){\rm d}t{\rm
d}\theta.$ (19)

Such explicit representations are also available in higher dimensions using
hyper-spherical polar coordinates. Of special importance to us is the
translated and rotated hyper-spherical Dirac distribution
$\delta_{(t_{0},{\boldsymbol{\xi}}_{0})}=\delta(\cdot-
t_{0})\delta(\cdot-{\boldsymbol{\xi}}_{0})\in{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$,
which is defined as $\langle\delta_{{\boldsymbol{z}}_{0}},\phi\rangle_{\rm
Rad}\stackrel{{\scriptstyle\vartriangle}}{{=}}\phi({\boldsymbol{z}}_{0})$ for
all $\phi\in{\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1})$ and any offset
${\boldsymbol{z}}_{0}=(t_{0},{\boldsymbol{\xi}}_{0})\in\mathbb{R}\times\mathbb{S}^{d-1}$.
This Dirac impulse, which is separable in the index variables $t$ and
${\boldsymbol{\xi}}$, is included in the Banach space
${\mathcal{M}}(\mathbb{R}\times\mathbb{S}^{d-1})$ (hyper-spherical Radon
measures) with the property that
$\|\delta_{{\boldsymbol{z}}_{0}}\|_{{\mathcal{M}}}=1$.

The key property for analysis is that the Radon transform is continuous on
${\mathcal{S}}$ and invertible (see (Ludwig, 1966; Helgason, 2011; Ramm and
Katsevich, 2020) for details).

######  Theorem 2 (Continuity and Invertibility of the Radon Transform on
${\mathcal{S}}(\mathbb{R}^{d})$)

The Radon operator ${\mathrm{R}}$ continuously maps
${\mathcal{S}}(\mathbb{R}^{d})\to{\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1})$.
Moreover, ${\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad}{\mathrm{R}}={\mathrm{K}}{\mathrm{R}}^{\ast}{\mathrm{R}}={\mathrm{R}}^{\ast}{\mathrm{R}}{\mathrm{K}}=\mathrm{Id}\mbox{
on~}{\mathcal{S}}(\mathbb{R}^{d})$, where
${\mathrm{K}}=({\mathrm{R}}^{\ast}{\mathrm{R}})^{-1}=c_{d}(-\Delta)^{\tfrac{d-1}{2}}$
with $c_{d}=\frac{1}{2(2\pi)^{d-1}}$ is the so-called “filtering” operator,
and is ${\mathrm{K}}_{\rm rad}$ its one-dimensional radial counterpart that
acts along the Radon-domain variable $t$. These filtering operators are
characterized by their frequency response
$\widehat{K}({\boldsymbol{\omega}})=c_{d}\|{\boldsymbol{\omega}}\|^{d-1}$ and
$\widehat{K}_{\rm rad}(\omega)=c_{d}|\omega|^{d-1}$.

The image of ${\mathcal{S}}(\mathbb{R}^{d})$ through the Radon transform is
the space ${\mathcal{S}}_{\rm
Rad}\stackrel{{\scriptstyle\vartriangle}}{{=}}{\mathrm{R}}\big{(}{\mathcal{S}}(\mathbb{R}^{d})\big{)}$:
a subset of the space of hyper-spherical test functions
${\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1})$ that can be characterized
explicitly (Gelfand and Shilov, 1966; Helgason, 2011; Ludwig, 1966). In
particular, a function $\phi={\mathrm{R}}\\{\varphi\\}\in{\mathcal{S}}_{\rm
Rad}$ must be symmetric and such as
$\int_{\mathbb{R}}{\mathrm{R}}\\{\varphi\\}(t,{\boldsymbol{\xi}}){\rm
d}t=\int_{\mathbb{R}^{d}}\varphi({\boldsymbol{x}}){\rm d}{\boldsymbol{x}}$ for
all ${\boldsymbol{\xi}}\in\mathbb{S}^{d-1}$ (see Theorem 9 for details). While
Theorem 2 implies that ${\mathrm{R}}$ is invertible on ${\mathcal{S}}_{\rm
Rad}$, there is also a stronger version of this property for
${\mathcal{S}}_{\rm Rad}$ equipped with the Schwartz-Fréchet topology
inherited from ${\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1})$ (Helgason,
2011, p. 60) (Hertle, 1983).

###### Theorem 3

The operator ${\mathrm{R}}:{\mathcal{S}}(\mathbb{R}^{d})\to{\mathcal{S}}_{\rm
Rad}$ is a continuous bijection, with a continuous inverse given by
${\mathrm{R}}^{-1}=({\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad}):{\mathcal{S}}_{\rm Rad}\to{\mathcal{S}}(\mathbb{R}^{d})$.

The bottom line is that the classical Radon transform is a homeomorphism
${\mathcal{S}}(\mathbb{R}^{d})\to{\mathcal{S}}_{\rm Rad}$.

####  2.3.2 Distributional Extension

To extend the framework to distributions, one proceeds by duality. For
instance, by invoking the property that ${\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad}{\mathrm{R}}=\mathrm{Id}$ on ${\mathcal{S}}(\mathbb{R}^{d})$ and noting
that ${\mathrm{K}}_{\rm rad}$ is self-adjoint, we make the manipulation

$\displaystyle\forall\varphi\in{\mathcal{S}}(\mathbb{R}^{d}),\quad\langle
f,\varphi\rangle$ $\displaystyle=\langle
f,{\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}{\mathrm{R}}\\{\varphi\\}\rangle$
$\displaystyle=\langle{\mathrm{K}}_{\rm
rad}{\mathrm{R}}\\{f\\},{\mathrm{R}}\\{\varphi\\}\rangle_{\rm
Rad}=\langle{\mathrm{K}}_{\rm rad}{\mathrm{R}}\\{f\\},\phi\rangle_{\rm Rad},$
(20)

with $\phi={\mathrm{R}}\\{\varphi\\}\in{\mathcal{S}}_{\rm Rad}$ and
$\varphi={\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}\\{\phi\\}$. Eq. (20) is
valid in the classical sense for $f\in L_{1}(\mathbb{R}^{d})$. Otherwise, it
is used as definition to extend the scope of the operator for
$f\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$.

###### Definition 4

The distribution
$g\in{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$ (respectively,
$g={\mathrm{R}}\\{f\\}\in\big{(}{\mathrm{K}}_{\rm
rad}{\mathrm{R}}\big{(}{\mathcal{S}})\big{)}^{\prime}$ to ensure unicity) is a
formal Radon transform of $f\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ if

$\displaystyle\forall\phi\in{\mathrm{K}}_{\rm
rad}{\mathrm{R}}\big{(}{\mathcal{S}}(\mathbb{R}^{d})\big{)}:\quad\langle
g,\phi\rangle_{\rm Rad}=\langle f,{\mathrm{R}}^{\ast}\\{\phi\\}\rangle.$ (21)

Likewise,
$\tilde{g}\in{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$
(respectively, $\tilde{g}={\mathrm{K}}_{\rm
rad}{\mathrm{R}}\\{f\\}\in{\mathcal{S}}_{\rm Rad}^{\prime}$ to ensure unicity)
is a formal filtered projection of
$f\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ if

$\displaystyle\forall\phi\in{\mathcal{S}}_{\rm
Rad}:\quad\langle\tilde{g},\phi\rangle_{\rm Rad}=\langle
f,{\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}\\{\phi\\}\rangle.$ (22)

Finally, the tempered distribution
$f={\mathrm{R}}^{\ast}\\{g\\}\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ is the
back-projection of
$g\in{{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})}$ if

$\displaystyle\forall\varphi\in{\mathcal{S}}(\mathbb{R}^{d}):\quad\langle{\mathrm{R}}^{\ast}\\{g\\},\varphi\rangle=\langle
g,{\mathrm{R}}\\{\varphi\\}\rangle_{\rm Rad}.$ (23)

While (23) yields a unique definition of the linear operator
${\mathrm{R}}^{\ast}:{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})\to{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$,
the establishement of a proper definition of the extended operators
${\mathrm{R}}$ and $({\mathrm{K}}_{\rm rad}{\mathrm{R}})$ is trickier because
there are always infinitely many distributions
$g,\tilde{g}\in{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$ that
satisfy (21) or (22). The root of the problem is that the Radon transform
${\mathrm{R}}:{\mathcal{S}}(\mathbb{R}^{d})\to{\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1})$
is not surjective: Since the test functions $\phi$ in (21) and (22) do not
span the whole space ${\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1})$, they
are unable to separate out the components that are in the null space of
${\mathrm{R}}^{\ast}$, which is huge111In addition to the odd distributions,
${\rm
ker}({\mathrm{R}}^{\ast})=\\{h\in{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1}):{\mathrm{R}}^{\ast}\\{h\\}=0\\}$
is formed of the set of rapidly decreasing distributions that are orthogonal
to functions of the form
$\varphi_{m}(t,{\boldsymbol{\xi}})=P_{m-1}(t)Y_{m}({\boldsymbol{\xi}})$ where
$P_{m-1}$ is a univariate polynomial of degree less than $m$ and $Y_{m}$ a
spherical harmonic of degree $m$ (Ludwig, 1966). . To illustrate this
ambiguity, we consider the Dirac ridge
$\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$
and refer to Definition (15) of the Radon transform to deduce that, for all
$\phi={\mathrm{R}}\\{\varphi\\}\in{\mathcal{S}}_{\rm Rad}$ with
$\varphi\in{\mathcal{S}}(\mathbb{R}^{d})$,

$\displaystyle\langle\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}\cdot-
t_{0}),{\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}\\{\phi\\}\rangle$
$\displaystyle=\langle\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}\cdot-
t_{0}),\overbrace{{\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad}{\mathrm{R}}}^{\mathrm{Id}}\\{\varphi\\}\rangle$
$\displaystyle=\int_{\mathbb{R}^{d}}\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})\varphi({\boldsymbol{x}}){\rm
d}{\boldsymbol{x}}={\mathrm{R}}\\{\varphi\\}(-{\boldsymbol{z}}_{0})=\langle\delta_{-{\boldsymbol{z}}_{0}},\phi\rangle_{\rm
Rad}$ (24)

where ${\boldsymbol{z}}_{0}=(t_{0},{\boldsymbol{\xi}}_{0})$, which shows that
the Dirac impulse $\delta_{-{\boldsymbol{z}}_{0}}$ is a formal filtered
projection of
$\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})$. Moreover,
since
$\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})=\delta(-{\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}+t_{0})$,
the same holds true for $\delta_{{\boldsymbol{z}}_{0}}$, as well as for
$\frac{1}{2}\big{(}\delta_{{\boldsymbol{z}}_{0}}+\delta_{-{\boldsymbol{z}}_{0}}\big{)}$.
In fact, there is an infinity of potential “formal” solutions, which is
consistent with the lack of injectivity of
${\mathrm{R}}^{\ast}:{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})\to{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$.

Classically, one obtains a unique characterization of $g={\mathrm{R}}\\{f\\}$
for $f\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ by restricting the scope of
(21) to $g\in\big{(}{\mathrm{K}}_{\rm
rad}{\mathrm{R}}({\mathcal{S}})\big{)}^{\prime}$ (Ludwig, 1966). Likewise, the
description of $\tilde{g}={\mathrm{K}}_{\rm rad}{\mathrm{R}}\\{f\\}$ in (22)
is unique if it is restricted to
$\tilde{g}\in\big{(}{\mathrm{R}}({\mathcal{S}})\big{)}^{\prime}={\mathcal{S}}^{\prime}_{\rm
Rad}$, which then yields a proper definition of $({\mathrm{K}}_{\rm
rad}{\mathrm{R}}):{\mathcal{S}}^{\prime}(\mathbb{R}^{d})\to{\mathcal{S}}^{\prime}_{\rm
Rad}$.

It turns out that this mechanism is equivalent to factoring out the null space
of
${\mathrm{R}}^{\ast}:{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})\to{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$.
Indeed, since the extended back-projection operator specified by (23) is
continuous
${\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})\to{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$,
its null space

$\displaystyle{\mathcal{N}}_{{\mathrm{R}}^{\ast}}=\\{g\in{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1}):{\mathrm{R}}^{\ast}\\{g\\}=0\quad\Leftrightarrow\quad\langle
g,\phi\rangle_{\rm Rad}=0,\forall\phi\in{\mathcal{S}}_{\rm Rad}\\}$ (25)

is a closed subspace of
${\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$. Accordingly, we
can identify ${\mathcal{S}}^{\prime}_{\rm Rad}$ as the abstract quotient space
${\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})/{\mathcal{N}}_{{\mathrm{R}}^{\ast}}$.
In other words, if we find a hyper-spherical distribution
$g_{0}\in{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$ such that
(22) is met for a given $f\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$, then,
strictly speaking, ${\mathrm{K}}_{\rm
rad}{\mathrm{R}}\\{f\\}\in{\mathcal{S}}^{\prime}_{\rm Rad}$ is the equivalence
class (or coset) given by

$\displaystyle{\mathrm{K}}_{\rm
rad}{\mathrm{R}}\\{f\\}=[g_{0}]=\\{g_{0}+h:h\in{\mathcal{N}}_{{\mathrm{R}}^{\ast}}\\}.$
(26)

The members of $[g_{0}]$ are interchangeable—we refer to them as “formal”
filtered projections of $f$ to remind us of this lack of unicity.

Based on those definitions, one obtains the classical result on the
invertibility of the (filtered) Radon transform on
${\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ (Ludwig, 1966), which is the dual of
Theorem 3.

######  Theorem 5 (Invertibility of the Radon Transform on
${\mathcal{S}}^{\prime}(\mathbb{R}^{d})$)

It holds that ${\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad}{\mathrm{R}}=\mathrm{Id}$ on ${\mathcal{S}}^{\prime}(\mathbb{R}^{d})$.
Moreover, the filtered Radon transform ${\mathrm{K}}_{\rm
rad}{\mathrm{R}}:{\mathcal{S}}^{\prime}(\mathbb{R}^{d})\to{\mathcal{S}}^{\prime}_{\rm
Rad}$ is bicontinuous and one-to-one, with its continuous inverse
$({\mathrm{K}}_{\rm rad}{\mathrm{R}})^{-1}$ being the back-projection
operation ${\mathrm{R}}^{\ast}:{\mathcal{S}}^{\prime}_{\rm
Rad}\to{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$.

The distributional extension of the Radon transform inherits most of the
properties of the “classical” operator defined in (15). Of special relevance
to us is the quasi-commutativity of ${\mathrm{R}}$ with convolution, also
known as the intertwining property. Specifically, let
$h,f\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ be two distributions whose
convolution $h\ast f$ is well defined in
${\mathcal{S}}^{\prime}(\mathbb{R}^{d})$. Then,

$\displaystyle{\mathrm{R}}\\{h\ast
f\\}={\mathrm{R}}\\{h\\}\ast_{t}{\mathrm{R}}\\{f\\}$ (27)

where the symbol “$\ast_{t}$” denotes a convolution along the radial variable
$t$; that is, $(u\ast_{t}g)(t,{\boldsymbol{\xi}})=\langle
u(\cdot,{\boldsymbol{\xi}}),g(t-\cdot,{\boldsymbol{\xi}})\rangle$. In
particular, when $h={\rm L}\\{\delta\\}$ is the (isotropic) impulse response
of an LSI operator whose frequency response
$\widehat{L}({\boldsymbol{\omega}})=\widehat{L}_{\rm
rad}(\|{\boldsymbol{\omega}}\|)$ is purely radial, we get that

$\displaystyle{\mathrm{R}}\\{h\ast f\\}={\mathrm{R}}{\rm L}\\{f\\}={\rm
L}_{\rm rad}{\mathrm{R}}\\{f\\},$ (28)

where ${\rm L}_{\rm rad}$ is the “$\ast_{t}$” convolution operator whose 1D
frequency response is $\widehat{L}_{\rm rad}(\omega)$. Likewise, by duality,
for $g\in{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$ we have
that

$\displaystyle{\rm L}{\mathrm{R}}^{\ast}\\{g\\}={\mathrm{R}}^{\ast}{\rm
L}_{\rm rad}\\{g\\}$ (29)

under the implicit assumption that ${\rm L}\\{{\mathrm{R}}^{\ast}g\\}$ and
${\rm L}_{\rm rad}\\{g\\}$ are well-defined distributions. By taking
inspiration from Theorem 2, we may then use Relations (28) and (29) for ${\rm
L}={\mathrm{K}}=({\mathrm{R}}^{\ast}{\mathrm{R}})^{-1}$ to show that
${\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad}{\mathrm{R}}\\{f\\}={\mathrm{R}}^{\ast}{\mathrm{R}}{\mathrm{K}}\\{f\\}={\mathrm{K}}{\mathrm{R}}^{\ast}{\mathrm{R}}\\{f\\}=f$
for a broad class of distributions. While the first form is valid for all
$f\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ (see Theorem 5), there is a
slight restriction with the second (resp., third), which requires that
${\mathrm{K}}\\{f\\}$ (resp., ${\mathrm{K}}\\{g\\}$ with
$g={\mathrm{R}}^{\ast}{\mathrm{R}}\\{f\\}\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$)
be well-defined in ${\mathcal{S}}^{\prime}(\mathbb{R}^{d})$. While the latter
condition is always met when $d$ is odd, it can fail in even dimensions for
distributions (e.g., polynomials) whose Fourier transform is singular at the
origin222For $d=2n$ even,
$\widehat{K}({\boldsymbol{\omega}})\propto\|{\boldsymbol{\omega}}\|^{2n-1}$
which is $C^{\infty}$ everywhere except at the origin, where it is only
$C^{2n-2}$, meaning that ${\mathrm{K}}$ can only properly handle (and
annihilate) polynomials up to degree $(2n-2)$..

The Fourier-slice theorem expressed by (17) also yields a unique (Fourier-
based) characterization of ${\mathrm{R}}\\{f\\}$, which remains valid for
$f\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ (Ramm and Katsevich, 2020). It is
especially helpful when the underlying function or distribution is isotropic.
An isotropic function $\rho_{\rm iso}:\mathbb{R}^{d}\to\mathbb{R}$ is
characterized by its radial profile $\rho:\mathbb{R}_{\geq 0}\to\mathbb{R}$,
with $\rho_{\rm iso}({\boldsymbol{x}})=\rho(\|{\boldsymbol{x}}\|)$. The
frequency-domain counterpart of this characterization is $\widehat{\rho}_{\rm
iso}({\boldsymbol{\omega}})=\widehat{\rho}_{\rm
rad}(\|{\boldsymbol{\omega}}\|)$ where the radial frequency profile can be
computed as

$\displaystyle\widehat{\rho}_{\rm
rad}(\omega)=\frac{(2\pi)^{d/2}}{|\omega|^{d/2-1}}\int_{0}^{+\infty}\rho(t)t^{d/2-1}J_{d/2-1}(\omega
t)t{\rm d}t,$ (30)

where $J_{\nu}$ is the Bessel function of the first kind of order $\nu$.

######  Proposition 6 (Radon Transform of Isotropic Distributions)

Let $\rho_{\rm iso}$ be an isotropic distribution whose radial frequency
profile is $\widehat{\rho}_{\rm rad}(\omega)$. Then,

$\displaystyle{\mathrm{R}}\\{\rho_{\rm
iso}(\cdot-{\boldsymbol{x}}_{0})\\}(t,{\boldsymbol{\xi}})$
$\displaystyle=\rho_{\rm
rad}(t-{\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}_{0})$ (31)
$\displaystyle{\mathrm{K}}_{\rm rad}{\mathrm{R}}\\{\rho_{\rm
iso}(\cdot-{\boldsymbol{x}}_{0})\\}(t,{\boldsymbol{\xi}}\\}$
$\displaystyle=\tilde{\rho}_{\rm
rad}(t-{\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}_{0})$ (32)
$\displaystyle{\mathrm{R}}\\{\partial^{{\boldsymbol{m}}}\rho_{\rm
iso}\\}(t,{\boldsymbol{\xi}})$
$\displaystyle={\boldsymbol{\xi}}^{{\boldsymbol{m}}}{\mathrm{D}}^{|{\boldsymbol{m}}|}\\{\rho_{\rm
rad}\\}(t)$ (33)

with $\rho_{\rm rad}(t)=\mathcal{F}^{-1}\\{\widehat{\rho}_{\rm
rad}(\omega)\\}(t)$ and $\tilde{\rho}_{\rm
rad}(t)=\tfrac{1}{2(2\pi)^{d-1}}\mathcal{F}^{-1}\\{|\omega|^{d-1}\widehat{\rho}_{\rm
rad}(\omega)\\}(t)$.

Proof  These identities are direct consequences of the Fourier-slice theorem.
For instance, by setting ${\boldsymbol{\omega}}=\omega{\boldsymbol{\xi}}$ in
the Fourier transform of $\partial^{{\boldsymbol{m}}}\rho_{\rm iso}$, we get
that

$\displaystyle\widehat{\partial^{{\boldsymbol{m}}}\rho_{\rm
iso}}(\omega{\boldsymbol{\xi}})=(\mathrm{j}\omega{\boldsymbol{\xi}})^{{\boldsymbol{m}}}\widehat{\rho}_{\rm
rad}(\omega)={\boldsymbol{\xi}}^{{\boldsymbol{m}}}(\mathrm{j}\omega)^{|{\boldsymbol{m}}|}\widehat{\rho}_{\rm
rad}(\omega)$ (34)

which, upon taking the inverse 1D Fourier transform, yields (33).

Let us note that both $\rho_{\rm rad}$ and $\tilde{\rho}_{\rm rad}$, as
inverse Fourier transform of a real-valued function, are symmetric, which is
consistent with the symmetry of the Radon transform and its filtered version.

##  3 Radon-Compatible Banach Spaces

The distribution formalism associated with Theorem 5 is extremely powerful. It
provides a definition of the (filtered) Radon transform of any tempered
distribution $f\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$, along with a way to
invert it via the back-projection operator
${\mathrm{R}}^{\ast}:{\mathcal{S}}^{\prime}_{\rm
Rad}\to{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$. While this is conceptually
very satisfying, it is only moderately helpful for our purpose because the
members of ${\mathcal{S}}^{\prime}_{\rm Rad}$ are “abstract” equivalence
classes of distributions. In fact, the investigation of functional-
optimization problems with Radon-domain regularization requires some Banach
counterpart of Theorems 3 and 5, where both ${\mathrm{R}}\\{f\\}$ and
${\mathrm{K}}_{\rm Rad}{\mathrm{R}}\\{f\\}$ have concrete representations as
hyper-spherical functions or measures. In particular, we are interested in
identifying specific Radon-domain Banach spaces—for instance, an appropriate
subspace of hyper-spherical measures—over which the back-projection operator
${\mathrm{R}}^{\ast}$ is guaranteed to be invertible.

To that end, we pick a “parent” hyper-spherical Banach space
${\mathcal{X}}=({\mathcal{X}},\|\cdot\|_{{\mathcal{X}}})$ such that
${\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1})\xhookrightarrow{\mbox{\tiny\rm
d.}}{\mathcal{X}}\xhookrightarrow{\mbox{\tiny\rm
d.}}{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$. This dense
embedding hypothesis has several implications.

  1. 1.

The Banach space ${\mathcal{X}}$ is the completion of
${\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1})$ in the
$\|\cdot\|_{{\mathcal{X}}}$ norm; i.e.,

$\displaystyle{\mathcal{X}}=\overline{\big{(}{\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1}),\|\cdot\|_{{\mathcal{X}}}\big{)}}.$
(35)

  2. 2.

The dual space
${\mathcal{X}}^{\prime}\xhookrightarrow{}{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$
is equipped with the norm

$\displaystyle\|g\|_{{\mathcal{X}}^{\prime}}=\sup_{\phi\in{\mathcal{X}}:\;\|\phi\|_{{\mathcal{X}}}\leq
1}\langle
g,\phi\rangle=\sup_{\phi\in{\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1}):\;\|\phi\|_{{\mathcal{X}}}\leq
1}\langle g,\phi\rangle,$ (36)

where the restriction of
$\phi\in{\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1})$ on the rightmost side
of (36) is justified by the denseness of
${\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1})$ in ${\mathcal{X}}$.

  3. 3.

The definition of $\|g\|_{{\mathcal{X}}^{\prime}}$ found in the rightmost side
of (36) is valid for any distribution
$g\in{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$ with
$\|g\|_{{\mathcal{X}}^{\prime}}=\infty$ for $g\notin{\mathcal{X}}^{\prime}$.
Accordingly, we can specify the topological dual of ${\mathcal{X}}^{\prime}$
as

$\displaystyle{\mathcal{X}}^{\prime}=\big{\\{}g\in{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1}):\|g\|_{{\mathcal{X}}^{\prime}}<\infty\big{\\}}.$
(37)

Prototypical examples where those properties are met are
$({\mathcal{X}},{\mathcal{X}}^{\prime})=\big{(}L_{p}(\mathbb{R}\times\mathbb{S}^{d-1}),L_{q}(\mathbb{R}\times\mathbb{S}^{d-1})\big{)}$
with $p\in[1,\infty)$ and $q=p/(p-1)$ (conjugate exponent), as well as
$({\mathcal{X}},{\mathcal{X}}^{\prime})=\big{(}C_{0}(\mathbb{R}\times\mathbb{S}^{d-1}),{\mathcal{M}}(\mathbb{R}\times\mathbb{S}^{d-1})\big{)}$
for $p=\infty$.

Likewise, by considering the dual pair $({\mathcal{S}}_{\rm
Rad},{\mathcal{S}}^{\prime}_{\rm Rad})$, we specify our Radon-compatible
Banach subspaces

$\displaystyle{\mathcal{X}}_{\rm Rad}$
$\displaystyle=\overline{({\mathcal{S}}_{\rm Rad},\|\cdot\|_{{\mathcal{X}}})}$
(38) $\displaystyle{\mathcal{X}}^{\prime}_{\rm Rad}$
$\displaystyle=\big{(}{\mathcal{X}}_{\rm
Rad}\big{)}^{\prime}=\big{\\{}g\in{\mathcal{S}}^{\prime}_{\rm
Rad}:\|g\|_{{\mathcal{X}}^{\prime}_{\rm Rad}}<\infty\big{\\}},$ (39)

where the underlying dual norms have a definition that is analogous to (36)
with ${\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1})$ and ${\mathcal{X}}$
being instanciated by ${\mathcal{S}}_{\rm Rad}$ and ${\mathcal{X}}_{\rm Rad}$.
We now show that ${\mathrm{R}}^{\ast}$ (resp., ${\mathrm{K}}_{\rm
rad}{\mathrm{R}}^{\ast}$) is invertible on ${\mathcal{X}}^{\prime}_{\rm Rad}$
(resp., on ${\mathcal{X}}_{\rm Rad}$), which is the main theoretical
contribution of this work.

Since ${\mathrm{R}}:{\mathcal{S}}(\mathbb{R}^{d})\to{\mathcal{S}}_{\rm Rad}$
is a homeomorphism and the $\|\cdot\|_{{\mathcal{X}}}$-norm is continuous in
the Fréchet topology of ${\mathcal{S}}_{\rm Rad}$ (resp., of
${\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1})$), we can consider the normed
space $({\mathcal{S}}(\mathbb{R}^{d}),\|\cdot\|_{{\mathcal{Y}}})$ with
$\|\varphi\|_{{\mathcal{Y}}}\stackrel{{\scriptstyle\vartriangle}}{{=}}\|{\mathrm{R}}\\{\varphi\\}\|_{{\mathcal{X}}}$
and identify the operator
${\mathrm{R}}:({\mathcal{S}}(\mathbb{R}^{d}),\|\cdot\|_{{\mathcal{Y}}})\to({\mathcal{S}}_{\rm
Rad},\|\cdot\|_{{\mathcal{X}}})$ as an isometry. We then invoke the BLT
theorem (Reed and Simon, 1980) to specify the unique extension
${\mathrm{R}}:{\mathcal{Y}}\to\overline{({\mathcal{S}}_{\rm
Rad},\|\cdot\|_{{\mathcal{X}}})}={\mathcal{X}}_{\rm Rad}$, where
${\mathcal{Y}}$ is a Banach space isometric to ${\mathcal{X}}_{\rm Rad}$. More
precisely, we have that

$\displaystyle{\mathcal{Y}}=\overline{({\mathcal{S}}(\mathbb{R}^{d}),\|\cdot\|_{{\mathcal{Y}}})}=\\{{\mathrm{R}}^{-1}\\{g\\}:g\in{\mathcal{X}}_{\rm
Rad}\\}$ (40)

with ${\mathrm{R}}^{-1}={\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}$ on
${\mathcal{S}}_{\rm Rad}$ and, by extension, on ${\mathcal{X}}_{\rm Rad}$.
This then leads to the following characterization.

######  Theorem 7 (Radon-compatible Banach spaces)

Let $({\mathcal{X}}_{\rm Rad},{\mathcal{X}}^{\prime}_{\rm Rad})$ be a dual
pair of hyper-spherical Banach spaces induced by a norm
$\|\cdot\|_{{\mathcal{X}}}$ and specified by (38) and (39). Then, the
following properties hold.

  1. 1.

The back-projection ${\mathrm{R}}^{\ast}:{\mathcal{X}}^{\prime}_{\rm
Rad}\to{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ is injective and
${\mathrm{K}}_{\rm rad}{\mathrm{R}}{\mathrm{R}}^{\ast}=\mathrm{Id}$ on
${\mathcal{X}}^{\prime}_{\rm Rad}$.

  2. 2.

The filtered back-projection ${\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad}:{\mathcal{X}}_{\rm Rad}\to{\mathcal{Y}}$ is an isometric bijection, with
${\mathrm{R}}{\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}=\mathrm{Id}$ on
${\mathcal{X}}_{\rm Rad}$.

  3. 3.

The corresponding “range” spaces
${\mathcal{Y}}={\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}({\mathcal{X}}_{\rm
Rad})$ and
${\mathcal{Y}}^{\prime}={\mathrm{R}}^{\ast}({\mathcal{X}}^{\prime}_{\rm Rad})$
from a dual Banach pair that is isomorphic to $({\mathcal{X}}_{\rm
Rad},{\mathcal{X}}^{\prime}_{\rm Rad})$.

Moreover, if there exists a complementary Banach space ${\mathcal{X}}^{\rm
c}_{\rm Rad}$ such that ${\mathcal{X}}={\mathcal{X}}_{\rm
Rad}\oplus{\mathcal{X}}^{\rm c}_{\rm Rad}$, then additional properties hold.

  1. 1.

The dual space is decomposable as
${\mathcal{X}}^{\prime}={\mathcal{X}}^{\prime}_{\rm
Rad}\oplus({\mathcal{X}}^{\rm c}_{\rm Rad})^{\prime}$.

  2. 2.

The dual complement $({\mathcal{X}}^{\rm c}_{\rm Rad})^{\prime}$ is the null
space of
${\mathrm{R}}^{\ast}:{\mathcal{X}}^{\prime}\to{\mathcal{Y}}^{\prime}$.

  3. 3.

The complement space ${\mathcal{X}}^{\rm c}_{\rm Rad}$ is the null space of
${\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}:{\mathcal{X}}\to{\mathcal{Y}}$.

  4. 4.

The operators ${\mathrm{P}}_{\rm
Rad}={\mathrm{R}}{\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad}:{\mathcal{X}}\to{\mathcal{X}}_{\rm Rad}$ and ${\mathrm{P}}^{\ast}_{\rm
Rad}={\mathrm{K}}_{\rm
rad}{\mathrm{R}}{\mathrm{R}}^{\ast}:{\mathcal{X}}^{\prime}\to{\mathcal{X}}_{\rm
Rad}$ form an adjoint pair of continuous projectors with ${\mathrm{P}}_{\rm
Rad}\big{(}{\mathcal{X}}\big{)}={\mathcal{X}}_{\rm Rad}$ and
${\mathrm{P}}^{\ast}_{\rm
Rad}\big{(}{\mathcal{X}}^{\prime}\big{)}={\mathcal{X}}_{\rm Rad}^{\prime}$.

Proof
First part: On the one hand, the existence of an extended isometry
${\mathrm{R}}:{\mathcal{Y}}\to{\mathcal{X}}_{\rm Rad}$ implies the continuity
of ${\mathrm{R}}^{\ast}:{\mathcal{X}}^{\prime}_{\rm
Rad}\to{\mathcal{Y}}^{\prime}$ with
$\|{\mathrm{R}}\|=\|{\mathrm{R}}^{\ast}\|=1$ and
${\mathrm{R}}^{\ast}({\mathcal{X}}^{\prime}_{\rm
Rad})\subseteq{\mathcal{Y}}^{\prime}$. On the other hand, Theorem 5 implies
that the back-projection operator
${\mathrm{R}}^{\ast}:{\mathcal{S}}^{\prime}_{\rm
Rad}\to{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ defined by (23) is injective
with a continuous inverse on ${\mathcal{S}}^{\prime}_{\rm Rad}$ that is given
by $({\mathrm{R}}^{\ast})^{-1}={\mathrm{K}}_{\rm rad}{\mathrm{R}}$. Let
${\mathcal{V}}={\mathrm{R}}^{\ast}\big{(}{\mathcal{X}}^{\prime}_{\rm
Rad}\big{)}$ be the range of the restricted operator
${\mathrm{R}}^{\ast}|_{{\mathcal{X}}^{\prime}_{\rm Rad}}$ with
${\mathcal{X}}^{\prime}_{\rm Rad}\xhookrightarrow{}{\mathcal{S}}_{\rm
Rad}^{\prime}$ (by construction). Then,
${\mathrm{R}}^{\ast}:{\mathcal{X}}^{\prime}_{\rm Rad}\to{\mathcal{V}}$ is
injective and invertible with $({\mathrm{R}}^{\ast})^{-1}={\mathrm{K}}_{\rm
rad}{\mathrm{R}}:{\mathcal{V}}\to{\mathcal{X}}^{\prime}_{\rm Rad}$. This
implies that
${\mathcal{V}}={\mathrm{R}}^{\ast}\big{(}{\mathcal{X}}^{\prime}_{\rm
Rad}\big{)}$ is a Banach space that is isomorphic to
${\mathcal{X}}^{\prime}_{\rm Rad}$ (see (Unser and Aziznejad, 2022,
Proposition 1)).

Injectivity of ${\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}$ on
${\mathcal{X}}_{\rm Rad}$: The continuity of ${\mathrm{K}}_{\rm
rad}{\mathrm{R}}:{\mathcal{V}}\to{\mathcal{X}}^{\prime}_{\rm Rad}$, together
with the isometric embedding of a Banach space in its bidual, implies the
continuity of ${\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}:{\mathcal{X}}_{\rm
Rad}\to{\mathcal{V}}^{\prime}$ (isometry). The condition
${\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}\\{\phi\\}=0$ can then be restated
as $\forall v\in{\mathcal{V}}:\langle v,{\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad}\\{\phi\\}\rangle=\langle{\mathrm{K}}_{\rm
rad}{\mathrm{R}}\\{v\\},\phi\rangle_{\rm Rad}=0$. Since
${\mathcal{X}}^{\prime}_{\rm Rad}={\mathrm{K}}_{\rm
rad}{\mathrm{R}}\big{(}{\mathcal{V}}\big{)}$, this is equivalent to say that
$\langle g,\phi\rangle_{\rm Rad}=0$ for all $g\in{\mathcal{X}}^{\prime}_{\rm
Rad}$, which leads to $\phi=0$ and proves that
${\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}:{\mathcal{X}}_{\rm
Rad}\to{\mathcal{V}}^{\prime}$ is injective. Consequently,
$\widetilde{{\mathcal{Y}}}={\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad}({\mathcal{X}}_{\rm Rad})\xhookrightarrow{\mbox{\tiny\rm
iso.}}{\mathcal{V}}^{\prime}$ is a Banach space that is isomorphic to
${\mathcal{X}}_{\rm Rad}$, while there exists an inverse operator
$({\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad})^{-1}$ such that
${\mathcal{X}}_{\rm Rad}=({\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad})^{-1}\big{(}\widetilde{{\mathcal{Y}}}\big{)}$. By invoking the continuity
of ${\mathrm{R}}:{\mathcal{V}}^{\prime}\to{\mathcal{X}}^{\prime\prime}_{\rm
Rad}$, we then manipulate the duality product as

$\displaystyle\langle g,\phi\rangle_{\rm Rad}=\langle
w,\tilde{y}\rangle=\langle{\mathrm{R}}^{\ast}\\{g\\},\tilde{y}\rangle$
$\displaystyle=\langle
g,{\mathrm{R}}\\{\tilde{y}\\}\rangle_{{\mathcal{X}}^{\prime}_{\rm
Rad}\times{\mathcal{X}}^{\prime\prime}_{\rm Rad}}$ $\displaystyle=\langle
g,{\mathrm{R}}{\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}\\{\phi\\}\rangle_{\rm
Rad},$ (41)

for all $(g,\phi)\in{\mathcal{X}}^{\prime}_{\rm Rad}\times{\mathcal{X}}_{\rm
Rad}$ with $w={\mathrm{R}}^{\ast}\\{g\\}\in{\mathcal{Y}}^{\prime}$ and
$\tilde{y}={\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}\\{\phi\\}$. This identity
proves that ${\mathrm{R}}{\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad}=\mathrm{Id}$ on ${\mathcal{X}}_{\rm Rad}$. Thus,
${\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}:{\mathcal{X}}_{\rm
Rad}\to\widetilde{{\mathcal{Y}}}$ is an isometric bijection with
${\mathcal{X}}_{\rm
Rad}={\mathrm{R}}(\widetilde{{\mathcal{Y}}})={\mathrm{R}}({\mathcal{Y}})$,
which also leads to the identification
$\widetilde{{\mathcal{Y}}}={\mathcal{Y}}$.

Since $({\mathcal{X}}_{\rm Rad},{\mathcal{Y}})$ and
$({\mathcal{X}}^{\prime}_{\rm Rad},{\mathcal{V}})$ are homeomorphic pairs of
Banach spaces with ${\mathcal{V}}\subseteq{\mathcal{Y}}^{\prime}$, we readily
deduce that ${\mathcal{V}}={\mathcal{Y}}^{\prime}$, with the dual pair
$({\mathcal{Y}},{\mathcal{Y}}^{\prime})$ ultimately being isometrically
isomorphic to $({\mathcal{X}}_{\rm Rad},{\mathcal{X}}^{\prime}_{\rm Rad})$.
The corresponding functional picture can then be summarized as follows.

The dual spaces $({\mathcal{X}}^{\prime}_{\rm Rad},{\mathcal{Y}}^{\prime})$:
For every $w\in{\mathcal{Y}}^{\prime}$, there exists a unique
$g={\mathrm{K}}_{\rm rad}{\mathrm{R}}\\{w\\}\in{\mathcal{X}}^{\prime}_{\rm
Rad}$ such that $w={\mathrm{R}}^{\ast}\\{g\\}$ with
$\|w\|_{{\mathcal{Y}}^{\prime}}=\|g\|_{{\mathcal{X}}^{\prime}_{\rm Rad}}$, and
vice versa (i.e., ${\mathcal{X}}^{\prime}_{\rm Rad}={\mathrm{K}}_{\rm
rad}{\mathrm{R}}({\mathcal{Y}}^{\prime})$). Another way to put it is that
${\mathrm{K}}_{\rm rad}{\mathrm{R}}{\mathrm{R}}^{\ast}=\mathrm{Id}$ on
${\mathcal{X}}^{\prime}_{\rm Rad}$, which is the statement in Item 1.
Likewise, we have that ${\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad}{\mathrm{R}}=\mathrm{Id}$ on ${\mathcal{Y}}^{\prime}$.

The predual spaces $({\mathcal{X}}_{\rm Rad},{\mathcal{Y}})$: For every
$y\in{\mathcal{Y}}$, there exists a unique
$\phi={\mathrm{R}}\\{y\\}\in{\mathcal{X}}_{\rm Rad}$ such that
$y={\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}\\{\phi\\}$ with
$\|y\|_{{\mathcal{Y}}}=\|\phi\|_{{\mathcal{X}}_{\rm Rad}}$, and vice versa
(i.e., ${\mathcal{X}}_{\rm Rad}={\mathrm{R}}({\mathcal{Y}})$). This also
implies that ${\mathrm{R}}{\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad}=\mathrm{Id}$ on ${\mathcal{X}}_{\rm Rad}$, as stated in Item 2. Likewise,
we have that ${\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad}{\mathrm{R}}=\mathrm{Id}$ on ${\mathcal{Y}}$.

Second part: Item 1’ follows from the generic property that
$({\mathcal{X}}_{1}\oplus{\mathcal{X}}_{2})^{\prime}={\mathcal{X}}^{\prime}_{1}\oplus{\mathcal{X}}^{\prime}_{2}$,
where the Banach spaces ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$ can be
arbitrary. In particular, Item 1’ tells us that ${\mathcal{X}}^{\prime}_{\rm
Rad},({\mathcal{X}}^{\rm c}_{\rm
Rad})^{\prime}\subseteq{\mathcal{X}}^{\prime}$ with the embedding being
continuous. Moreover, it implies that ${\mathcal{X}}_{\rm Rad}$ is the
annihilator of ${\mathcal{X}}_{\rm Rad}^{\prime}$ in ${\mathcal{X}}^{\prime}$,
with

$\displaystyle({\mathcal{X}}^{\rm c}_{\rm Rad})^{\prime}$
$\displaystyle=\\{g\in{\mathcal{X}}^{\prime}:\langle g,\phi\rangle_{\rm
Rad}=0,\forall\phi\in{\mathcal{X}}_{\rm Rad}\\}$
$\displaystyle=\\{g\in{\mathcal{X}}^{\prime}:\langle g,\phi\rangle_{\rm
Rad}=0,\forall\phi\in{\mathcal{S}}_{\rm Rad}\\},$ (42)

where the substitution of ${\mathcal{X}}_{\rm Rad}$ with ${\mathcal{S}}_{\rm
Rad}$ in (42) is legitimate because ${\mathcal{S}}_{\rm Rad}$ is a dense
subspace of ${\mathcal{X}}_{\rm Rad}$. Since
${\mathcal{X}}^{\prime}\subset{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$,
this shows that $({\mathcal{X}}^{\rm c}_{\rm
Rad})^{\prime}\subseteq{\mathcal{N}}_{{\mathrm{R}}^{\ast}}$ (Item 2’). It also
yields Item 3’ by duality. The null-space property implies that
${\mathrm{P}}^{\ast}_{\rm Rad}\big{(}({\mathcal{X}}^{\rm c}_{\rm
Rad})^{\prime}\big{)}=\\{0\\}$ and ${\mathrm{P}}_{\rm
Rad}\big{(}{\mathcal{X}}^{\rm c}_{\rm Rad}\big{)}=\\{0\\}$. The final element
is the identity in Item 1 (resp., Item 2), which ensures that
${\mathrm{P}}^{\ast}_{\rm
Rad}:{\mathcal{X}}^{\prime}\to{\mathcal{X}}^{\prime}_{\rm Rad}$ (resp.,
${\mathrm{P}}_{\rm Rad}:{\mathcal{X}}\to{\mathcal{X}}_{\rm Rad}$) is the
canonical projector on ${\mathcal{X}}^{\prime}_{\rm Rad}$ (resp., on
${\mathcal{X}}_{\rm Rad}$). The existence (unicity) and continuity of the
latter is guaranteed for any pair of complemented Banach spaces (see Appendix
A).

The dual direct-sum decomposition in Theorem 7 has the following corollary:
The Banach complement ${\mathcal{X}}^{\rm c}_{\rm Rad}$ (resp.,
${\mathcal{X}}_{\rm Rad}$) is the annihilator of ${\mathcal{X}}^{\prime}_{\rm
Rad}$ in ${\mathcal{X}}$ (resp., of $({\mathcal{X}}^{\rm c}_{\rm
Rad})^{\prime}$ in ${\mathcal{X}}^{\prime}$), and vice versa. In particular,
$({\mathcal{X}}^{\rm c}_{\rm
Rad})^{\prime}\xhookrightarrow{}{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$
is specified by (42) (as a set), which clearly shows that $({\mathcal{X}}^{\rm
c}_{\rm Rad})^{\prime}\subseteq{\mathcal{N}}_{{\mathrm{R}}^{\ast}}$ (see
(25)).

The existence of the projection operator ${\mathrm{P}}^{\ast}_{\rm Rad}$ in
the second part of Theorem 7 is very handy because it enables us to convert
any “formal” filtered projection $\tilde{g}={\mathrm{K}}_{\rm
rad}{\mathrm{R}}\\{f\\}\in{\mathcal{X}}^{\prime}\xhookrightarrow{}{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$
(see (22) in Definition 4) into a concrete representer
${\mathrm{P}}^{\ast}_{\rm Rad}\\{\tilde{g}\\}\in{\mathcal{X}}^{\prime}_{\rm
Rad}$, which has a unique, unambiguous interpretation.

Since the members of ${\mathcal{S}}_{\rm Rad}$ must be even (see Theorem 9),
we readily deduce that ${\mathcal{X}}_{\rm Rad}\subseteq{\mathcal{X}}_{\rm
even}=\overline{\big{(}{\mathcal{S}}_{\rm
even}(\mathbb{R}\times\mathbb{S}^{d-1}),\|\cdot\|_{{\mathcal{X}}}\big{)}}$. In
particular, when the inclusion is a set equality and when
${\mathcal{X}}={\mathcal{X}}_{\rm even}\oplus{\mathcal{X}}_{\rm odd}$, then
${\mathcal{X}}^{\prime}_{\rm Rad}={\mathcal{X}}^{\prime}_{\rm even}$ and
${\mathrm{P}}^{\ast}_{\rm Rad}={\mathrm{P}}_{\rm Rad}={\mathrm{P}}_{\rm
even}$, which is the self-adjoint projector that extracts the even part of a
function.

##  4 Radon-Domain Sampling Functionals

The canonical sampling functional that acts on continuous functions expressed
in hyper-spherical coordinates is the shifted hyper-spherical Dirac
distribution
$\delta_{{\boldsymbol{z}}_{0}}\in{\mathcal{M}}(\mathbb{R}\times\mathbb{S}^{d-1})$
with
${\boldsymbol{z}}_{0}=(t_{0},{\boldsymbol{\xi}}_{0})\in\mathbb{R}\times\mathbb{S}^{d-1}$
and $\|\delta_{{\boldsymbol{z}}_{0}}\|_{{\mathcal{M}}}=1$, which also happens
to be a “formal” filtered projection of
$\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})$.

We now show how we can describe
$[\delta_{{\boldsymbol{z}}_{0}}]\in{\mathcal{S}}^{\prime}_{\rm Rad}$ by its
unique representer $e_{{\boldsymbol{z}}_{0}}:g\mapsto g({\boldsymbol{z}}_{0})$
in ${\mathcal{M}}_{\rm Rad}=(C_{0,{\rm Rad}})^{\prime}$. We start with a
theoretical investigation where $e_{{\boldsymbol{z}}_{0}}$ is characterized
indirectly through its functional properties. We then provide an explicit
construction that allows us to identity $e_{{\boldsymbol{z}}_{0}}$ as the
limit of a normalized Radon-compatible distribution whose unit mass get
concentrated at ${\boldsymbol{z}}=\pm{\boldsymbol{z}}_{0}$.

###  4.1 Abstract Characterization of Radon-Compatible Diracs

Since ${\mathcal{S}}_{\rm Rad}\subseteq{\mathcal{S}}_{\rm
even}(\mathbb{R}\times\mathbb{S}^{d-1})$ (see Theorem 9), we have that
$C_{0,\rm Rad}\subseteq C_{0,\rm even}(\mathbb{R}\times\mathbb{S}^{d-1})$,
which implies that all functions $g\in C_{0,\rm Rad}$ are continuous and even.
This ensures that the evaluation functional
$e_{{\boldsymbol{z}}_{0}}:g\mapsto\langle
e_{{\boldsymbol{z}}_{0}},f\rangle=g({\boldsymbol{z}}_{0})$ is well-defined for
any ${\boldsymbol{z}}_{0}\in\mathbb{R}\times\mathbb{S}^{d-1}$. We now prove
that $e_{{\boldsymbol{z}}_{0}}$ is a continuous linear functional on $C_{0,\rm
Rad}$—that is, $e_{{\boldsymbol{z}}_{0}}\in(C_{0,\rm
Rad})^{\prime}={\mathcal{M}}_{\rm Rad}$ (the space of Radon-compatible
measures)—and establish its basic properties, which are compabible with those
of the Dirac distribution. By the same token, we get a characterization of the
extreme points of the unit ball in ${\mathcal{M}}_{\rm Rad}$.

######  Theorem 8 (Properties of $e_{{\boldsymbol{z}}_{0}}$)

The Radon-domain functionals $e_{{\boldsymbol{z}}_{0}}:C_{0,{\rm
Rad}}\to\mathbb{R}$ with
${\boldsymbol{z}}_{0}=(t_{0},{\boldsymbol{\xi}}_{0})\in\mathbb{R}\times\mathbb{S}^{d-1}$
have the following properties.

  1. 1.

Definition (sampling at ${\boldsymbol{z}}_{0}$)

$\displaystyle\forall\phi\in C_{0,\rm Rad}:\quad\langle
e_{(t_{0},{\boldsymbol{\xi}}_{0})},\phi\rangle_{\rm
Rad}=\phi(t_{0},{\boldsymbol{\xi}}_{0}).$ (43)

  2. 2.

Symmetry: $e_{{\boldsymbol{z}}_{0}}=e_{-{\boldsymbol{z}}_{0}}$.

  3. 3.

Continuity: $\displaystyle\|e_{{\boldsymbol{z}}_{0}}\|=\sup_{\phi\in C_{0,{\rm
Rad}}:\ \|\phi\|_{L_{\infty}}\leq 1}\langle
e_{{\boldsymbol{z}}_{0}},\phi\rangle=1$, which implies that
$e_{{\boldsymbol{z}}_{0}}\in{\mathcal{M}}_{\rm Rad}.$

  4. 4.

Let $\\{{\boldsymbol{z}}_{k}\\}$ be any finite set of distinct points. Then,
$\|\sum_{k}a_{k}e_{{\boldsymbol{z}}_{k}}\|_{{\mathcal{M}}_{\rm
Rad}}=\sum_{k}|a_{k}|$.

  5. 5.

${\mathrm{R}}^{\ast}\\{e_{(t_{0},{\boldsymbol{\xi}}_{0})}\\}({\boldsymbol{x}})=\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})\
\Leftrightarrow\ e_{(t_{0},{\boldsymbol{\xi}}_{0})}={\mathrm{K}}_{\rm
rad}{\mathrm{R}}\\{\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})\\}$
in ${\mathcal{M}}_{\rm Rad}$.

  6. 6.

If $e\in{\rm Ext}B_{{\mathcal{M}}_{\rm Rad}}$, then $e=\pm
e_{{\boldsymbol{z}}_{k}}$ for some
${\boldsymbol{z}}_{k}\in\mathbb{R}\times\mathbb{S}^{d-1}$.

Proof  Item 1 is a paraphrasing of the definition of the evaluation
functional, while Item 2 follows from the property that
$\phi({\boldsymbol{z}}_{0})=\phi(-{\boldsymbol{z}}_{0})$ for all $\phi\in
C_{0,{\rm Rad}}$.

Continuity: From the definition of the ${\mathcal{M}}$-norm and the denseness
of ${\mathcal{S}}_{{\rm Rad}}$ in $C_{0,{\rm Rad}}$, we immediately deduce
that

$\displaystyle\|e_{{\boldsymbol{z}}_{0}}\|_{{\mathcal{M}}_{\rm Rad}}$
$\displaystyle=\sup_{\phi\in{\mathcal{S}}_{{\rm Rad}}:\
\|\phi\|_{L_{\infty}}\leq
1}\phi({\boldsymbol{z}}_{0})=\sup_{\phi\in{\mathcal{S}}_{{\rm Rad}}:\
\|\phi\|_{L_{\infty}}\leq
1}\langle\delta_{{\boldsymbol{z}}_{0}},\phi\rangle\leq\|\delta_{{\boldsymbol{z}}_{0}}\|_{{\mathcal{M}}}=1$

for all ${\boldsymbol{z}}_{0}\in\mathbb{R}\times\mathbb{S}^{d-1}$. To prove
that $\|e_{{\boldsymbol{z}}_{0}}\|_{{\mathcal{M}}_{\rm Rad}}=1$, it therefore
suffices to exhibit a function $g_{{\boldsymbol{x}}_{0}}\in{\mathcal{S}}_{{\rm
Rad}}$ with $\|g_{{\boldsymbol{x}}_{0}}\|_{L_{\infty}}=1$ such that $\langle
g_{{\boldsymbol{x}}_{0}},\delta_{{\boldsymbol{z}}_{0}}\rangle=1$. We take
$g_{{\boldsymbol{x}}_{0}}(t,{\boldsymbol{\xi}})=\mathrm{e}^{-\tfrac{1}{2}(t\textendash{\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}_{0})^{2}}$
with any choice of ${\boldsymbol{x}}_{0}\in\mathbb{R}^{d}$ such that
${\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}_{0}=t_{0}$: With the help
of Proposition 6, we easily verify that $g_{{\boldsymbol{x}}_{0}}$ is the
Radon transform of a standardized Gaussian centered on ${\boldsymbol{x}}_{0}$,
which ensures that $g_{{\boldsymbol{x}}_{0}}\in{\mathcal{S}}_{\rm Rad}$.

Now, let $f=\sum_{k=1}^{K}a_{k}e_{{\boldsymbol{z}}_{k}}$, which is such that
$\|f\|_{{\mathcal{M}}_{\rm
Rad}}\leq\sum_{k=1}^{K}|a_{k}|=\|{\boldsymbol{a}}\|_{\ell_{1}}$, by the
triangle inequality. In addition, since the ${\boldsymbol{z}}_{k}$ are
distinct, there exits some $\epsilon_{0}>0$ such that
$\|{\boldsymbol{z}}_{k}-{\boldsymbol{z}}_{k^{\prime}}\|<\epsilon_{0}$ for all
$k\neq k^{\prime}$. Again, we shall prove that $\|f\|_{{\mathcal{M}}_{\rm
Rad}}=\|{\boldsymbol{a}}\|_{\ell_{1}}$ by constructing a conjugate function
$f^{\ast}\in{\mathcal{S}}_{\rm rad}$ with $\|f^{\ast}\|_{L_{\infty}}=1$. This
requires the use of more sophisticaded atoms whose localization is adjustable;
namely, the functions

$\phi_{\epsilon,{\boldsymbol{x}}_{k}}=\frac{1}{d_{\epsilon}(0,{\boldsymbol{e}}_{1})}d_{\epsilon}(t-{\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}_{k},{\bf{U}}_{k}{\boldsymbol{\xi}})\in{\mathcal{S}}_{\rm
rad}$

where $d_{\epsilon}(t,{\boldsymbol{\xi}})$ is specified by (47) and where
${\boldsymbol{x}}_{k}\in\mathbb{R}^{d}$ and the rotation matrix
${\bf{U}}_{k}\in\mathbb{R}^{d\times d}$ are chosen such that
${\boldsymbol{\xi}}_{k}^{\mathsf{T}}{\boldsymbol{x}}_{k}=t_{k}$ and
${\boldsymbol{e}}_{1}={\bf{U}}_{k}{\boldsymbol{\xi}}_{k}$. The function
$\phi_{\epsilon,{\boldsymbol{x}}_{k}}$ with $\epsilon\in(0,1)$ is symmetric,
non-negative and bounded by $1$; it achieves its maximum at
${\boldsymbol{z}}={\boldsymbol{z}}_{k}$ and is decreasing towards zero as
${\boldsymbol{z}}$ moves away from $\pm{\boldsymbol{z}}_{k}$ with the speed of
decay becoming arbitrarily fast as $\epsilon\to 0$ (see Figure 1).
Consequently, one can always find some $\epsilon>0$ such that
$|\phi_{\epsilon,{\boldsymbol{x}}_{k}}({\boldsymbol{z}})|<1/K$ for all
${\boldsymbol{z}}\in\mathbb{R}\times\mathbb{S}^{d-1}$ with
$\|{\boldsymbol{z}}\pm{\boldsymbol{z}}_{k}\|>\epsilon_{0}$. Then,
$f^{\ast}=\sum_{k=1}^{K}{\rm
sign}(a_{k})\phi_{\epsilon,{\boldsymbol{x}}_{k}}\in{\mathcal{S}}_{\rm Rad}$ is
such that $\|f^{\ast}\|_{L_{\infty}}=1$ and $\langle
f^{\ast},f\rangle=\sum_{k=1}|a_{k}|$. The latter implies that
$\|f\|_{{\mathcal{M}}_{\rm Rad}}\geq\|{\boldsymbol{a}}\|_{\ell_{1}}$, which
proves the claim.

Filtered Radon transform: Theorem 7 with
${\mathcal{X}}=C_{0}(\mathbb{R}\times\mathbb{S}^{d-1})$ ensures that the
adjoint pair of operators ${\mathrm{K}}_{\rm rad}{\mathrm{R}}^{\ast}:C_{0,\rm
Rad}\to{\mathcal{Y}}$ and ${\mathrm{K}}_{\rm
rad}{\mathrm{R}}:{\mathcal{Y}}^{\prime}\to{\mathcal{M}}_{\rm Rad}$ are
continuous (isometries). This Banach setting also allows us to specify the
corresponding back-projection operator ${\mathrm{R}}^{\ast}=({\mathrm{K}}_{\rm
rad}{\mathrm{R}})^{-1}:{\mathcal{M}}_{\rm Rad}\to{\mathcal{Y}}^{\prime}$ by
extending the scope of Definition (23) for
$\varphi\in{\mathcal{Y}}={\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}(C_{0,\rm
Rad})$. We then use the same manipulations as in (24) with $\phi\in C_{0,\rm
Rad}$ (resp., $\varphi\in{\mathcal{Y}}$) to prove that: (i)
$\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})={\mathrm{R}}^{\ast}\\{e_{t_{0},{\boldsymbol{\xi}}_{0}}\\}\in{\mathcal{Y}}^{\prime}$,
and (ii) ${\mathrm{K}}_{\rm
rad}{\mathrm{R}}\\{\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})\\}=e_{t_{0},{\boldsymbol{\xi}}_{0}}\in{\mathcal{M}}_{\rm
Rad}$.

The abstract interpretation of Items 1 and 2 is that the evaluation
functionals on $C_{0,\rm Rad}$ are spanned (with a double covering) by
$e_{{\boldsymbol{z}}}$ with
${\boldsymbol{z}}\in{\mathcal{Z}}\stackrel{{\scriptstyle\vartriangle}}{{=}}\mathbb{R}\times\mathbb{S}^{d-1}$.
Since $C_{0,\rm Rad}$ is a closed subspace of $C_{0}({\mathcal{Z}})$, we can
invoke Lemma 15 in the Appendix, which tells us that the extreme points of the
unit ball in ${\mathcal{M}}_{\rm Rad}=\big{(}C_{0,\rm Rad}\big{)}^{\prime}$
are all of the form $\pm e_{{\boldsymbol{z}}}$ for some
${\boldsymbol{z}}\in{\mathcal{Z}}$.

###  4.2 Constructive Description of Radon-Compatible Dirac

In complement to the abstract characterization of $e_{{\boldsymbol{z}}_{0}}$
in Theorem 8, we now describe the underlying distribution concretely. We
consider the $d$-dimensional Gaussian density function

$\displaystyle g_{\epsilon}({\boldsymbol{x}})$
$\displaystyle=\frac{\epsilon^{d-1}}{(2\pi)^{d/2}}\exp\left(-\frac{1}{2}\big{(}\frac{x^{2}_{1}}{\epsilon^{2}}+\epsilon^{2}(x_{2}^{2}+\cdots+x_{d}^{2})\big{)}\right)\in{\mathcal{S}}(\mathbb{R}^{d})$
(44)

whose Fourier transform is

$\displaystyle\hat{g}_{\epsilon}({\boldsymbol{{\boldsymbol{\omega}}}})=\exp\left(-\frac{1}{2}\big{(}\epsilon^{2}\omega_{1}^{2}+\frac{\omega_{2}^{2}+\cdots+\omega_{d}^{2}}{\epsilon^{2}}\big{)}\right).$
(45)

The parameter $\epsilon<1$ controls the degree of ellipticity. When $\epsilon$
is small, $g_{\epsilon}({\boldsymbol{x}})$ gets narrow along the $x_{1}$ axis,
while it spreads out along the other directions. Setting
${\boldsymbol{\omega}}=\omega{\boldsymbol{\xi}}$, we rewrite (45) as

$\displaystyle\hat{g}_{\epsilon}(\omega{\boldsymbol{\xi}})$
$\displaystyle=\exp\left(-\frac{\omega^{2}}{2}\sigma_{\epsilon}^{2}({\boldsymbol{\xi}})\right)\mbox{
with
}\sigma_{\epsilon}^{2}({\boldsymbol{\xi}})=\epsilon^{2}\xi_{1}^{2}+\frac{\xi_{2}^{2}+\cdots+\xi_{d}^{2}}{\epsilon^{2}}.$
(46)

From (17), we obtain the Radon transform of $g_{\epsilon}$ as

$\displaystyle
d_{\epsilon}(t,{\boldsymbol{\xi}})={\mathrm{R}}\\{g_{\epsilon}\\}(t,{\boldsymbol{\xi}})$
$\displaystyle=\frac{1}{\sqrt{2\pi\sigma_{\epsilon}^{2}({\boldsymbol{\xi}})}}\exp\left(-\frac{t^{2}}{2\sigma_{\epsilon}^{2}({\boldsymbol{\xi}})}\right)\in{\mathcal{S}}_{\rm
Rad},$ (47)

which is a radial Gaussian with a spherical dependence on the variance. For
$0<\epsilon<1$, $d_{\epsilon}(t,{\boldsymbol{\xi}})$ attains its maximum at
$(t,{\boldsymbol{\xi}})=(0,\pm{\boldsymbol{e}}_{1})$. As $\epsilon$ gets
smaller, the maximum increases while the distribution becomes peakier and more
and more localized around ${\boldsymbol{z}}=(0,{\boldsymbol{e}}_{1})$.
However, the integral of the function is preserved since
$\int_{\mathbb{S}^{d-1}}\int_{\mathbb{R}}d_{\epsilon}(t,{\boldsymbol{\xi}}){\rm
d}t{\rm d}{\boldsymbol{\xi}}=\int_{\mathbb{S}^{d-1}}{\rm
d}{\boldsymbol{\xi}}=\frac{2\pi^{d/2}}{\Gamma(d/2)}$ for any $\epsilon>0$.
This allows us to identify our Radon-compatible sampling functional as

$\displaystyle
e_{(t_{0},{\boldsymbol{\xi}}_{0})}(t,{\boldsymbol{\xi}})=\lim_{\epsilon\to
0+}d_{\epsilon}(t-{\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}_{0},{\bf{U}}_{0}{\boldsymbol{\xi}}),$
(48)

where ${\bf{U}}_{0}\in\mathbb{R}^{d\times d}$ is a rotation matrix such that
${\boldsymbol{e}}_{1}={\bf{U}}_{0}{\boldsymbol{\xi}}_{0}$ and
${\boldsymbol{x}}_{0}\in\mathbb{R}^{d}$ is such that
${\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}_{0}=t_{0}$. Examples of such
functions for $d=2$, ${\boldsymbol{\xi}}_{0}={\boldsymbol{e}}_{1}=(1,0)$, and
${\boldsymbol{x}}_{0}=(2,2)$ are shown in Figure 1.

Figure 1: Localization effect of the parameter $\epsilon$ for the
approximation of $e_{(2,{\boldsymbol{e}}_{1})}$ for $d=2$, displayed as the
sinogram of the functions
$d_{\epsilon}(t-{\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}_{0},\theta)$
with ${\boldsymbol{\xi}}=(\cos\theta,\sin\theta)$,
${\boldsymbol{x}}_{0}=(2,2)$, and
$\epsilon=1,\tfrac{1}{2},\tfrac{1}{4},\tfrac{1}{8}$.

While this construction reminds us of the description of a Dirac as the limit
of a Gaussian distribution whose standard deviation tends to zero, there is
one important difference: unlike a conventional Gaussian, the functions
$d_{\epsilon}$ on the left hand side of (48) all satisfy the range conditions
of the Radon transform, which are stated in Theorem 9 below.

##  5 Ridges Revisited

A 1D profile along the direction ${\boldsymbol{e}}_{1}=(1,0,\dots,0)$ is a
generalized function of the form
$r_{{\boldsymbol{e}}_{1}}({\boldsymbol{x}})=r(x_{1})\times 1$ with
$r\in{\mathcal{S}}^{\prime}(\mathbb{R})$. Since the latter is separable, its
generalized Fourier transform is

$\displaystyle\mathcal{F}\\{r_{{\boldsymbol{e}}_{1}}\\}({\boldsymbol{\omega}})=\hat{r}(\omega_{1})\prod_{k=2}^{d}2\pi\delta(\omega_{k})=\hat{r}(\omega_{1})(2\pi)^{d-1}\delta(\omega_{2},\dots,\omega_{d}),$
(49)

which is a weighted Dirac mass localized along the $\omega_{1}$ axis. An
equivalent formulation of (49) that involves test functions is

$\displaystyle\forall\varphi\in{\mathcal{S}}(\mathbb{R}^{d}):\langle
r_{{\boldsymbol{e}}_{1}},\varphi\rangle=\frac{1}{2\pi}\int_{\mathbb{R}}\hat{r}(\omega)\hat{\varphi}(\omega{\boldsymbol{e}}_{1}){\rm
d}\omega$ (50)

where $\hat{\varphi}\in{\mathcal{S}}(\mathbb{R}^{d})$ is the $d$-dimensional
Fourier transform of $\varphi$. The argument remains valid when we rotate the
coordinate system, which allows us to consider more general ridges of the form
$r_{{\boldsymbol{\xi}}_{0}}({\boldsymbol{x}})\stackrel{{\scriptstyle\vartriangle}}{{=}}r({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}})$;
that is, 1D profiles along the direction
${\boldsymbol{\xi}}_{0}\in\mathbb{S}{d-1}$.

###  5.1 Generalized Ridges

By identifying $\hat{\varphi}(\omega{\boldsymbol{e}}_{1})$ in (50) as the 1D
Fourier transform of ${\mathrm{R}}\\{\varphi(\cdot,{\boldsymbol{e}}_{1})\\}$
and by substituting ${\boldsymbol{e}}_{1}$ by ${\boldsymbol{\xi}}_{0}$, we
obtain the general signal-domain relation:

$\displaystyle\forall\varphi\in{\mathcal{S}}(\mathbb{R}^{d}):\quad\langle
r_{{\boldsymbol{\xi}}_{0}},\varphi\rangle=\langle
r,{\mathrm{R}}\\{\varphi\\}(\cdot,{\boldsymbol{\xi}}_{0})\rangle,$ (51)

which will be referred to as the ridge identity. Under the assumption that $r$
is a locally integrable function, we can establish (51) by making the change
of coordinates ${\boldsymbol{y}}={\bf{U}}{\boldsymbol{x}}$, where
${\bf{U}}\in\mathbb{R}^{d\times d}$ is any rotation matrix such that
$y_{1}={\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}$. We then rewrite
the integral explicitly as

$\displaystyle\int_{\mathbb{R}^{d}}\varphi({\boldsymbol{x}})r_{{\boldsymbol{\xi}}_{0}}({\boldsymbol{x}}){\rm
d}{\boldsymbol{x}}$
$\displaystyle=\int_{\mathbb{R}^{d}}\varphi({\boldsymbol{y}})r(y_{1}){\rm
d}y_{1}\dots\rm dy_{d}$
$\displaystyle=\int_{\mathbb{R}}\underbrace{\left(\int_{\mathbb{R}^{d-1}}\varphi({\boldsymbol{y}}){\rm
d}y_{2}\dots\rm
dy_{d}\right)}_{{\mathrm{R}}\varphi(y_{1},{\boldsymbol{\xi}}_{0})}r(y_{1}){\rm
d}y_{1}.$

Otherwise, when $r\in{\mathcal{S}}^{\prime}(\mathbb{R})$ has no pointwise
interpretation, we simply use (51) as definition for the ridge distribution
$r_{{\boldsymbol{\xi}}_{0}}\in{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$, which
is legitimate since
${\mathrm{R}}\\{\varphi\\}(\cdot,{\boldsymbol{\xi}}_{0})\in{\mathcal{S}}(\mathbb{R})$.

Note that the special case of (51) with $r(t)=\mathrm{e}^{-\mathrm{j}\omega
t}$ and
$r_{\boldsymbol{\xi}}({\boldsymbol{x}})=\mathrm{e}^{-\mathrm{j}\omega{\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}}=\mathrm{e}^{-\mathrm{j}{\boldsymbol{\omega}}^{\mathsf{T}}{\boldsymbol{x}}}$
yields the Fourier-slice theorem (17).

The ridge identity (51) also helps us delineate the range of the Radon
transform. Take $r(t)=t^{k}$ and define

$\Phi_{k}({\boldsymbol{\xi}})\stackrel{{\scriptstyle\vartriangle}}{{=}}\int_{\mathbb{R}}{\mathrm{R}}\\{\varphi\\}(t,{\boldsymbol{\xi}})\;t^{k}{\rm
d}t=\int_{\mathbb{R}^{d}}\varphi({\boldsymbol{x}})({\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}})^{k}{\rm
d}{\boldsymbol{x}}=\sum_{|{\boldsymbol{k}}|=k}a_{{\boldsymbol{k}}}{\boldsymbol{\xi}}^{\boldsymbol{k}}$

with
$a_{{\boldsymbol{k}}}=k!\int_{\mathbb{R}^{d}}\frac{{\boldsymbol{x}}^{\boldsymbol{k}}}{{\boldsymbol{k}}!}\varphi({\boldsymbol{x}}){\rm
d}{\boldsymbol{x}}$.

######  Theorem 9 (Gelfand and Shilov (1966); Helgason (2011); Ludwig (1966))

A hyper-spherical test function
$\phi\in{\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d-1})$ is a valid Radon
transform in the sense that $\phi\in{\mathcal{S}}_{\rm
Rad}=\\{{\mathrm{R}}\\{\varphi\\}:\varphi\in{\mathcal{S}}(\mathbb{R}^{d})\\}$
if and only if

  1. 1.

Evenness: $\phi(t,{\boldsymbol{\xi}})=\phi(-t,-{\boldsymbol{\xi}})$.

  2. 2.

$\Phi_{k}({\boldsymbol{\xi}})=\int_{\mathbb{R}}\phi(t,{\boldsymbol{\xi}})t^{k}{\rm
d}t$ is a homogeneous polynomial in ${\boldsymbol{\xi}}\in\mathbb{S}^{d-1}$
for any $k\in\mathbb{N}$.

In particular, for $k=0$, we must have that
$\int_{\mathbb{R}}\phi(t,{\boldsymbol{\xi}}){\rm d}t=a_{{\boldsymbol{0}}}={\rm
Constant}$ for all ${\boldsymbol{\xi}}\in\mathbb{S}^{d-1}$.

The most basic version of a ridge is
$\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})$ with
$r=\delta(\cdot-t_{0})$, which is a Dirac ridge along ${\boldsymbol{\xi}}_{0}$
with offset $t_{0}$. Since the Fourier transform of such ridges is entirely
localized along the ray
$\\{{\boldsymbol{\omega}}=\omega{\boldsymbol{\xi}}_{0}:\omega\in\mathbb{R}\\}$,
we can expect their Radon transform to vanish away from
$\pm{\boldsymbol{\xi}}_{0}$. The latter can be readily identified as follows,
where the square bracket notation $[g]$ with
$g\in{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$ reminds us that
the members of ${\mathcal{S}}^{\prime}_{\rm Rad}$ (resp., of
${\mathrm{K}}_{\rm
rad}{\mathrm{R}}\big{(}{\mathcal{S}}(\mathbb{R}^{d})\big{)}^{\prime}$) are
equivalence classes of distributions.

######  Proposition 10 (Radon transform of ridge distributions)

Let
$(t_{0},{\boldsymbol{\xi}}_{0})={\boldsymbol{z}}_{0}\in\mathbb{R}\times\mathbb{S}^{d-1}$
and $r\in{\mathcal{S}}^{\prime}(\mathbb{R})$. Then,

$\displaystyle{\mathrm{K}}_{\rm
rad}{\mathrm{R}}\\{\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})\\}(t,{\boldsymbol{\xi}})$
$\displaystyle=[\delta(\cdot-
t_{0})\delta(\cdot-{\boldsymbol{\xi}}_{0})]\quad\in{\mathcal{S}}^{\prime}_{\rm
Rad}$
$\displaystyle{\mathrm{R}}\\{\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}})\\}(t,{\boldsymbol{\xi}})$
$\displaystyle=[q_{d}(t)\delta({\boldsymbol{\xi}}-{\boldsymbol{\xi}}_{0})]\quad\in{\mathrm{K}}_{\rm
rad}{\mathrm{R}}\big{(}{\mathcal{S}}(\mathbb{R}^{d})\big{)}^{\prime}$
$\displaystyle{\mathrm{K}}_{\rm
rad}{\mathrm{R}}\\{r({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}})\\}(t,{\boldsymbol{\xi}})$
$\displaystyle=[r(t)\delta({\boldsymbol{\xi}}-{\boldsymbol{\xi}}_{0})]\quad\in{\mathcal{S}}^{\prime}_{\rm
Rad}$
$\displaystyle{\mathrm{R}}\\{r({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}})\\}(t,{\boldsymbol{\xi}})$
$\displaystyle=[(q_{d}\ast
r)(t)\delta({\boldsymbol{\xi}}-{\boldsymbol{\xi}}_{0})]\quad\in{\mathrm{K}}_{\rm
rad}{\mathrm{R}}\big{(}{\mathcal{S}}(\mathbb{R}^{d})\big{)}^{\prime}$

where $q_{d}(t)=2(2\pi)^{d-1}\mathcal{F}^{-1}\\{1/|\omega|^{d-1}\\}(t)$ is the
1D impulse response of the Radon-domain inverse filtering operator
${\mathrm{K}}_{\rm rad}^{-1}$.

Proof  The fact that
$\delta_{{\boldsymbol{z}}_{0}}=\delta(t-t_{0})\delta({\boldsymbol{\xi}}-{\boldsymbol{\xi}}_{0})$
is a formal filtered projection of
$\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})$ has
already been mentioned in the text—it is a direct consequence of Definition
(15).

To derive the third identity, we observe that, for all
$\varphi\in{\mathcal{S}}(\mathbb{R}^{d})$, one has that

$\displaystyle\langle
r(\cdot)\delta(\cdot-{\boldsymbol{\xi}}_{0}),{\mathrm{R}}\\{\varphi\\}\rangle_{\rm
Rad}$
$\displaystyle=\int_{\mathbb{R}}r(t){\mathrm{R}}\\{\varphi\\}(t,{\boldsymbol{\xi}}_{0}){\rm
d}t=\langle r,{\mathrm{R}}\\{\varphi\\}(\cdot,{\boldsymbol{\xi}}_{0})\rangle$
$\displaystyle=\int_{\mathbb{R}^{d}}r({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}})\varphi({\boldsymbol{x}}){\rm
d}{\boldsymbol{x}}=\langle
r({\boldsymbol{\xi}}_{0}^{\mathsf{T}}\cdot),{\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad}\\{{\mathrm{R}}\varphi\\}\rangle,$

where we have made use of (51). By setting
$\phi={\mathrm{R}}\\{\varphi\\}\in{\mathcal{S}}_{\rm rad}$, we then refer to
(22) to deduce that $r(t)\delta({\boldsymbol{\xi}}-{\boldsymbol{\xi}}_{0})$ is
a formal filtered projection of
$r({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}})$. This then also
yields the second and fourth identities by substituting $r$ with $q_{d}(\cdot-
t_{0})$ and $q_{d}\ast r$, respectively. The additional element there is
$r({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}})={\mathrm{K}}{\mathrm{K}}^{-1}\\{r({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}})\\}({\boldsymbol{x}})={\mathrm{K}}\\{\tilde{r}({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}})\\}({\boldsymbol{x}})$
with $\tilde{r}(t)=(q_{d}\ast r)(t)$, which is readily verified in the Fourier
domain.

An equivalent form of the first identity in Proposition 10 is

$\displaystyle\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})$
$\displaystyle={\mathrm{R}}^{\ast}\\{\delta_{(t_{0},{\boldsymbol{\xi}}_{0})}\\}({\boldsymbol{x}}),$
(52)

which results from ${\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm
rad}{\mathrm{R}}=\mathrm{Id}$ on ${\mathcal{S}}^{\prime}(\mathbb{R}^{d})$.
Likewise, when the back-projection in (52) is followed by an isotropic
operator ${\rm L}$ whose radial frequency response is $\widehat{L}_{\rm
rad}:\mathbb{R}\to\mathbb{R}$, we use the intertwining property to show that

$\displaystyle{\rm
L}{\mathrm{R}}^{\ast}\\{\delta_{(t_{0},{\boldsymbol{\xi}}_{0})}\\}({\boldsymbol{x}})={\mathrm{R}}^{\ast}\\{{\rm
L}_{\rm
rad}\delta_{(t_{0},{\boldsymbol{\xi}}_{0})}\\}({\boldsymbol{x}})=r({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})$
(53)

where $r(t)=\mathcal{F}^{-1}\\{\widehat{L}_{\rm rad}\\}(t)$.

###  5.2 Connection with Prior Works

Most authors who use the Radon transform in connection with neural networks do
not distinguish “formal” from “range-compatible” Radon transforms of
distributions (Candès, 1999; Sonoda and Murata, 2017; Ongie et al., 2020;
Parhi and Nowak, 2021). They bypass the difficulty by focusing their analysis
on some appropriate subspace of ${\mathcal{S}}^{\prime}_{\rm Rad}$ over which
the backpropagation operator ${\mathrm{R}}^{\ast}$ is known to be invertible,
for instance ${\mathcal{M}}_{\rm
even}(\mathbb{R}\times\mathbb{S}^{d-1})={\mathcal{M}}(\mathbb{P}^{d})$ and/or
${\mathcal{S}}^{\prime}_{\rm Liz,even}$. To specify their norm for Radon-
domain measures, Ongie et al. (2020) consider the subspace of test functions
${\mathcal{S}}_{\rm
even}=\\{\phi\in{\mathcal{S}}(\mathbb{R}\times\mathbb{S}^{d}):\phi(t,{\boldsymbol{\xi}})=\phi(-t,-{\boldsymbol{\xi}})\\}$
for which the validity of the inversion formula ${\mathrm{K}}_{\rm
rad}{\mathrm{R}}{\mathrm{R}}^{\ast}=\mathrm{Id}$ has been established by
Solmon (1987). The caveat is that the functions
$f\in{\mathrm{R}}^{\ast}({\mathcal{S}}_{\rm even})$ that are in the range of
${\mathrm{R}}^{\ast}$ can decay as badly as $O(1/\|{\boldsymbol{x}}\|)$ (Ramm
and Katsevich, 2020, Corollary 3.1.1, p. 73), meaning that their “classical”
Radon transform specified by (15) can be ill-defined. Parhi and Nowak (2021)
follow a different path and identify ${\mathcal{M}}_{\rm
even}(\mathbb{R}\times\mathbb{S}^{d-1})$ as a subspace of the space of even
Lizorkin distributions ${\mathcal{S}}^{\prime}_{\rm Liz,even}$, which is the
topological dual of ${\mathcal{S}}_{\rm Liz,even}=\\{\phi\in{\mathcal{S}}_{\rm
even}:\int_{\mathbb{R}}\phi(t,{\boldsymbol{\xi}})t^{k}=0,\forall
k\in\mathbb{N},{\boldsymbol{\xi}}\in\mathbb{S}^{d-1}\\}$. Implicit in the
calculation of Example 1 in (Ongie et al., 2020) is the property that

$\displaystyle{\mathrm{K}}_{\rm
rad}{\mathrm{R}}\\{\delta({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})\\}(t,{\boldsymbol{\xi}})=\frac{1}{2}\Big{(}\delta(t-t_{0})\delta({\boldsymbol{\xi}}-{\boldsymbol{\xi}}_{0})+\delta(t+t_{0})\delta({\boldsymbol{\xi}}+{\boldsymbol{\xi}}_{0})\Big{)},$
(54)

which needs to be related to the “abstract” description of
$e_{{\boldsymbol{z}}_{0}}$ in Theorem 8. As it turns out, the two forms are
equivalent. The abstract version, combined with the properties in Theorem 8,
conveys the same information as (54). In complement is the concrete
representation of $e_{(t_{0},{\boldsymbol{\xi}}_{0})}(t,{\boldsymbol{\xi}})$
given by (48), which gives us a sense of how and why the Radon-domain mass
concentrates around the points $\pm{\boldsymbol{z}}_{0}$ as the Gaussian
“blob” $g_{\epsilon}$ gets thinner along the primary axis and elongates in the
perpendicular directions.

Even though the test functions used in the listed works are different from
ours with ${\mathcal{S}}_{\rm Liz}\subset{\mathcal{S}}_{\rm
Rad}\subset{\mathcal{S}}_{\rm even}$, the approaches are reconciled by
invoking the property that

$\displaystyle C_{0,{\rm even}}=\overline{({\mathcal{S}}_{\rm
even},\|\cdot\|_{L_{\infty}})}=\overline{({\mathcal{S}}_{\rm
Liz},\|\cdot\|_{L_{\infty}})}=\overline{({\mathcal{S}}_{\rm
Rad},\|\cdot\|_{L_{\infty}})}=C_{0,{\rm Rad}}$ (55)

where the domain of all underlying spaces is
$(\mathbb{R}\times\mathbb{S}^{d-1})$ (Neumayer and Unser, 2022, Lemma 1).
While Proposition 10 gives the filtered Radon transform of ridges with the
greatest possible level of generality, the caveat is that ${\mathrm{K}}_{\rm
rad}{\mathrm{R}}\\{r_{{\boldsymbol{\xi}}_{0}}\\}\in{\mathcal{S}}^{\prime}_{\rm
Rad}$ is an abstract equivalence class. As complement, we are providing the
“concrete” version of the main result for the case where the profile is a
measure.

######  Corollary 11 (Filtered Radon Transform of Ridge Measures)

Let
$r_{{\boldsymbol{\xi}}_{0}}=r({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}})$
be the ridge with profile $r\in{\mathcal{S}}^{\prime}(\mathbb{R})$ and
direction ${\boldsymbol{\xi}}_{0}\in\mathbb{S}^{d-1}$. If
$r\in{\mathcal{M}}(\mathbb{R})$, then the equality

$\displaystyle{\mathrm{K}}_{\rm
rad}{\mathrm{R}}\\{r_{{\boldsymbol{\xi}}_{0}}\\}(t,{\boldsymbol{\xi}})$
$\displaystyle=\frac{1}{2}\big{(}r(t)\delta({\boldsymbol{\xi}}-{\boldsymbol{\xi}}_{0})+r(-t)\delta({\boldsymbol{\xi}}+{\boldsymbol{\xi}}_{0})\big{)}$
(56)

holds in ${\mathcal{M}}_{\rm Rad}={\mathcal{M}}_{\rm
even}(\mathbb{R}\times\mathbb{S}^{d-1})$.

Indeed, we know that $r(t)\delta({\boldsymbol{\xi}}-{\boldsymbol{\xi}}_{0})$
is a formal filtered Radon transform of $r_{{\boldsymbol{\xi}}_{0}}$ and that
it is included in ${\mathcal{M}}(\mathbb{R}\times\mathbb{S}^{d-1})$ if
$r\in{\mathcal{M}}(\mathbb{R})$. Moreover, Theorem 7 with
${\mathcal{X}}=C_{0}(\mathbb{R}\times\mathbb{S}^{d-1})$, together with (55),
implies that ${\mathcal{M}}_{\rm Rad}=(C_{0,{\rm
even}})^{\prime}={\mathcal{M}}_{\rm even}$, whose Banach complement in
${\mathcal{M}}(\mathbb{R}\times\mathbb{S}^{d-1})$ is ${\mathcal{M}}_{\rm
odd}$. This ensures the validity of the second part of Theorem 7 with
${\mathrm{P}}^{\ast}_{\rm Rad}={\mathrm{P}}_{\rm even}$, where
${\mathrm{P}}_{\rm
even}\\{g\\}(t,{\boldsymbol{\xi}})=\tfrac{1}{2}\big{(}g(t,{\boldsymbol{\xi}})+g(-t,-{\boldsymbol{\xi}})\big{)}$.
Accordingly, we can identify ${\mathrm{P}}_{\rm
even}\\{r(t)\delta({\boldsymbol{\xi}}-{\boldsymbol{\xi}}_{0})\\}$ as the
unique representer in ${\mathcal{M}}_{\rm Rad}$ of ${\mathrm{K}}_{\rm
rad}{\mathrm{R}}\\{r_{{\boldsymbol{\xi}}_{0}}\\}=[r(t)\delta({\boldsymbol{\xi}}-{\boldsymbol{\xi}}_{0})]\in{\mathcal{S}}_{\rm
Rad}^{\prime}$.

The argument also suggests that (56) is likely to be extendable to broader
families of distributions. The condition for its validity is that
$r(t)\delta({\boldsymbol{\xi}}-{\boldsymbol{\xi}}_{0})$ be included in a space
${\mathcal{X}}^{\prime}=\big{(}{\mathcal{X}}_{\rm Rad}\oplus{\mathcal{X}}^{\rm
c}_{\rm Rad}\big{)}^{\prime}$ with ${\mathcal{X}}_{\rm
Rad}=\overline{({\mathcal{S}}_{\rm
Rad},\|\cdot\|_{{\mathcal{X}}})}=\overline{({\mathcal{S}}_{\rm
even},\|\cdot\|_{{\mathcal{X}}})}$, so that the underlying projector can be
readily identified as ${\mathrm{P}}^{\ast}_{\rm Rad}={\mathrm{P}}_{\rm
even}:{\mathcal{X}}^{\prime}\to{\mathcal{X}}^{\prime}_{\rm Rad}$.

##  6 Variational Optimality of ReLU Networks

As application of the proposed formalism, we shall now link ReLU neural
networks with functional optimization, revisiting the energy-minimization
property uncovered in (Ongie et al., 2020) as well as the general variational-
learning problem investigated in (Parhi and Nowak, 2021).

###  6.1 Learning with Radon-Domain Regularization

In order to state the relevant optimization problem, we consider the
regularization operator

$\displaystyle\Delta_{\rm R}={\mathrm{K}}_{\rm
rad}{\mathrm{R}}\Delta:{\mathcal{M}}_{{\mathrm{\Delta}}_{\rm
R}}(\mathbb{R}^{d})\to{\mathcal{M}}(\mathbb{R}\times\mathbb{S}^{d-1})$ (57)

that was first proposed by Ongie et al. (2020, Lemma 2 p. 6), where $\Delta$
is the $d$-dimensional Laplace operator. The ${\mathcal{M}}$-norm (a.k.a.
total variation) of this “Radonized” Laplacian is then used as regularization.
The corresponding native space ${\mathcal{M}}_{{\mathrm{\Delta}}_{\rm
R}}(\mathbb{R}^{d})$ is the largest Banach space over which this seminorm is
well-defined under the constraint that the null space of $\Delta_{\rm R}$ be
limited to polynomials of degree $1$.

######  Theorem 12 (Optimality of Shallow ReLU Networks)

Let $E:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ be a strictly convex loss
function, $({\boldsymbol{x}}_{m},y_{m})\in\mathbb{R}^{d}\times\mathbb{R}$ with
$m=1,\dots,M$ a given set of distinct data points, and $\lambda>0$ some fixed
regularization parameter. Then, for $M>d+1$, the solution set

$\displaystyle S=\left\\{\arg\min_{f\in{\mathcal{M}}_{\Delta_{\rm
R}}(\mathbb{R}^{d})}\left(\sum_{m=1}^{M}E(y_{m},f({\boldsymbol{x}}_{m}))+\lambda\|\Delta_{\rm
R}f\|_{{\mathcal{M}}}\right)\right\\},$ (58)

of the functional optimization problem is nonempty and weak* compact. It is
the weak* closure of the convex hull of its extreme points, which all take the
form

$\displaystyle f_{\rm
ridge}({\boldsymbol{x}})=b+{\boldsymbol{b}}^{\top}{\boldsymbol{x}}+\sum_{k=1}^{K_{0}}a_{k}{\rm
ReLU}({\boldsymbol{\xi}}_{k}^{\mathsf{T}}{\boldsymbol{x}}-\tau_{k})$ (59)

with $(b,{\boldsymbol{b}})\in\mathbb{R}\times\mathbb{R}^{d}$, and a number
$K_{0}<M$ of adaptive ridges with weight, direction, and offset parameters
$(a_{k},{\boldsymbol{\xi}}_{k},\tau_{k})\in\mathbb{R}\times\mathbb{S}^{d-1}\times\mathbb{R}$.
The corresponding regularization cost, which is common to all solutions, is
$\|\Delta_{\rm R}f_{\rm ridge}\|_{{\mathcal{M}}}=\sum_{k=1}^{K_{0}}|a_{k}|$.

The key here is that the search space ${\mathcal{M}}_{\Delta_{\rm
R}}(\mathbb{R}^{d})$ is isometrically isomorphic to ${\mathcal{M}}_{\rm
Rad}\times{\mathcal{P}}_{1}$, where ${\mathcal{M}}_{\rm Rad}=\big{(}C_{0,\rm
Rad}\big{)}^{\prime}$ (see Section 3 with
$\|\cdot\|_{{\mathcal{X}}}=\|\cdot\|_{L_{\infty}}$ and ${\mathcal{X}}_{\rm
Rad}=C_{0,{\rm Rad}}$) and ${\mathcal{P}}_{1}$ is the space of polynomials of
degree $1$ specified by (7) with $n_{0}=1$. An equivalent representation of
the latter space of affine functions on $\mathbb{R}^{d}$ is
${\mathcal{P}}_{1}=\\{b+{\boldsymbol{b}}^{\mathsf{T}}{\boldsymbol{x}}\\}$ with
$b=b_{\boldsymbol{0}}$ and
${\boldsymbol{b}}=(b_{{\boldsymbol{e}}_{i}})_{i=1}^{d}$, which matches the
leading term in (59).

The crucial element for this construction is the pseudoinverse operator

$\displaystyle\Delta_{\rm
R}^{\dagger}\stackrel{{\scriptstyle\vartriangle}}{{=}}(\mathrm{Id}-\mathrm{Proj}_{{\mathcal{P}}_{1}})\Delta^{-1}{\mathrm{R}}^{\ast}:{\mathcal{M}}_{\rm
Rad}\to{\mathcal{M}}_{\Delta_{\rm R}}(\mathbb{R}^{d}),$ (60)

where
$\mathrm{Proj}_{{\mathcal{P}}_{1}}:{\mathcal{S}}^{\prime}(\mathbb{R}^{d})\to{\mathcal{P}}_{1}$
is the projector defined by (13) with $n_{0}=1$. The native space is then
given by

$\displaystyle{\mathcal{M}}_{{\mathrm{\Delta}}_{\rm R}}(\mathbb{R}^{d})$
$\displaystyle=\Delta_{\rm R}^{\dagger}\big{(}{\mathcal{M}}_{\rm
Rad}\big{)}\oplus{\mathcal{P}}_{1}$ $\displaystyle=\\{\Delta_{\rm
R}^{\dagger}\\{w\\}+p_{0}:(w,p_{0})\in{\mathcal{M}}_{\rm
Rad}\times{\mathcal{P}}_{1}\\}$ (61)

equipped with the composite norm induced by ${\mathcal{M}}_{\rm
Rad}\times{\mathcal{P}}_{1}$. Equivalently, given the generic form of $f$ in
(61), it is possible to retrieve the components $(w,p_{0})$ with the help of
suitable linear maps. Specifically, letting $f=\Delta_{\rm
R}^{\dagger}\\{w\\}+p_{0}$, one verifies that

$\displaystyle\Delta_{\rm R}\\{f\\}$ $\displaystyle=\Delta_{\rm
R}\\{\Delta_{\rm R}^{\dagger}w\\}+\Delta_{\rm R}\\{p_{0}\\}$
$\displaystyle={\mathrm{K}}_{\rm
rad}{\mathrm{R}}\Delta(\mathrm{Id}-\mathrm{Proj}_{{\mathcal{P}}_{1}})\Delta^{-1}{\mathrm{R}}^{\ast}\\{w\\}+0$
$\displaystyle={\mathrm{K}}_{\rm
rad}{\mathrm{R}}\Delta\Delta^{-1}{\mathrm{R}}^{\ast}\\{w\\}-{\mathrm{K}}_{\rm
rad}{\mathrm{R}}\underbrace{\Delta\\{\mathrm{Proj}_{{\mathcal{P}}_{1}}\Delta^{-1}{\mathrm{R}}^{\ast}w\\}}_{0}$
$\displaystyle={\mathrm{K}}_{\rm
rad}{\mathrm{R}}{\mathrm{R}}^{\ast}\\{w\\}={\mathrm{P}}^{\ast}_{\rm
Rad}\\{w\\}=w\in{\mathcal{M}}_{\rm Rad}$

and

$\displaystyle\mathrm{Proj}_{{\mathcal{P}}_{1}}\\{f\\}$
$\displaystyle=\mathrm{Proj}_{{\mathcal{P}}_{1}}\\{\Delta_{\rm
R}^{\dagger}w\\}+\mathrm{Proj}_{{\mathcal{P}}_{1}}\\{p_{0}\\}$
$\displaystyle=\mathrm{Proj}_{{\mathcal{P}}_{1}}(\mathrm{Id}-\mathrm{Proj}_{{\mathcal{P}}_{1}})\Delta^{-1}{\mathrm{R}}^{\ast}\\{w\\}+p_{0}=0+p_{0}\in{\mathcal{P}}_{1},$

where the annihilation of the $w$-component follows from the idempotence of
the projector.

Proof [Proof of Theorem 12] Theorem 7 with
${\mathcal{X}}=C_{0}(\mathbb{R}\times\mathbb{S}^{d-1})$ ensures that the back-
projection operator ${\mathrm{R}}^{\ast}$ is invertible on
${\mathcal{X}}^{\prime}_{\rm Rad}={\mathcal{M}}_{\rm Rad}=(C_{0,\rm
Rad})^{\prime}$. This is the fundamental ingredient that makes
${\mathcal{M}}_{\Delta_{\rm R}}(\mathbb{R}^{d})$ isometrically isomorphic to
${\mathcal{M}}_{\rm Rad}\times{\mathcal{P}}_{1}$ via the reversible mapping
$w=\Delta_{\rm R}\\{f\\}\in{\mathcal{M}}_{\rm Rad}$ and
$p_{0}=\mathrm{Proj}_{{\mathcal{P}}_{1}}\\{f\\}\in{\mathcal{P}}_{1}$. This
equivalent representation of $f$ enables us to derive the result as a
corollary of the third case of the abstract representer theorem for direct
sums in (Unser and Aziznejad, 2022, Theorem 3). This representer theorem gives
the generic form of the extreme points of the solution set $S$ as $f_{\rm
extreme}=p_{0}+\sum_{k=1}^{K_{0}}a_{k}e_{k}$, where
$p_{0}\in{\mathcal{P}}_{1}$ is a null-space component and where the $e_{k}$
are extreme points of the unit ball $B_{{\mathcal{U}}^{\prime}}$ of the
primary-component space ${\mathcal{U}}^{\prime}=\Delta_{\rm
R}^{\dagger}\big{(}{\mathcal{M}}_{\rm Rad}\big{)}$. Based on the form of the
extreme points of ${\mathcal{M}}_{\rm Rad}$ given by Theorem 8 and the
property that ${\rm Ext}B_{{\mathcal{U}}^{\prime}}=\Delta_{\rm
R}^{\dagger}\big{(}{\rm Ext}B_{{\mathcal{M}}_{\rm Rad}}\big{)}$ (since
$\Delta_{\rm R}^{\dagger}$ is an isometry), we deduce that any $e_{k}\in{\rm
Ext}B_{{\mathcal{U}}^{\prime}}$ can be written as

$\displaystyle e_{k}$ $\displaystyle=\pm\Delta_{\rm
R}^{\dagger}\\{e_{(t_{k},{\boldsymbol{\xi}}_{k})}\\}=(\mathrm{Id}-\mathrm{Proj}_{{\mathcal{P}}_{1}})\Delta^{-1}{\mathrm{R}}^{\ast}\\{e_{(t_{k},{\boldsymbol{\xi}}_{k})}\\}$
$\displaystyle=\pm(\mathrm{Id}-\mathrm{Proj}_{{\mathcal{P}}_{1}})\\{\tfrac{1}{2}|{\boldsymbol{\xi}}_{k}^{\mathsf{T}}\cdot-
t_{k}|\\}$
$\displaystyle=\pm\tfrac{1}{2}|{\boldsymbol{\xi}}_{k}^{\mathsf{T}}\cdot-
t_{k}|\pm p_{k}\quad\mbox{ with }\quad p_{k}\in{\mathcal{P}}_{1},$

where we have used (53) with ${\rm L}=\Delta^{-1}$ and $\widehat{L}_{\rm
rad}(\omega)=1/\omega^{2}$ to evaluate the back-projection. Since
$\tfrac{1}{2}|t|=(t)_{+}-\tfrac{1}{2}t$, we can also write $e_{k}$ as

$\displaystyle e_{k}$ $\displaystyle=\pm({\boldsymbol{\xi}}_{k}^{\top}\cdot-
t_{k})_{+}\pm\tilde{p}_{k},$

where
$\tilde{p}_{k}=p_{k}-\tfrac{1}{2}\left({\boldsymbol{\xi}}_{k}^{\top}\cdot+t_{k}\right)\in{\mathcal{P}}_{1}$.
We then obtain (59) by forming the linear combination of extreme points and by
grouping all polynomial components in a single term
$b+{\boldsymbol{b}}^{\mathsf{T}}{\boldsymbol{x}}$. Since $\Delta_{\rm
R}\\{f_{\rm
ridge}\\}=\sum_{k=1}^{K_{0}}a_{k}e_{(t_{k},{\boldsymbol{\xi}}_{k})}$, we also
deduce that $\|\Delta_{\rm R}\\{f_{\rm
ridge}\\}\|_{{\mathcal{M}}}=\sum_{k=1}^{K_{0}}|a_{k}|$ by invoking Theorem 8.

###  6.2 Discussion

Our formulation of Theorem 12 owes a lot to the pioneering works of Ongie et
al. (2020) and Parhi and Nowak (2021) (PN). Our is merely a refinement of the
results published by these authors together with a clarification of the
underlying mathematics. The interesting outcome is that the solution of the
functional-optimization problem in (58) can be implemented by a 2-layer ReLU
network.

In their work which pulls together ideas from Unser et al. (2017) and Ongie et
al. (2020), PN restrict the domain of the test functions to the so-called
Lizorkin functions

$\displaystyle{\mathcal{S}}_{\rm
Liz}(\mathbb{R}^{d})=\\{\varphi\in{\mathcal{S}}(\mathbb{R}^{d}):\int_{\mathbb{R}^{d}}{\boldsymbol{x}}^{{\boldsymbol{m}}}\varphi({\boldsymbol{x}}){\rm
d}{\boldsymbol{x}}=0,\forall{\boldsymbol{m}}\in\mathbb{N}^{d}\\},$ (62)

which are orthogonal to the polynomials. This choice is motivated by the
property that the Radon transform is a homeomorphism
${\mathrm{R}}:{\mathcal{S}}_{\rm Liz}(\mathbb{R}^{d})\to{\mathcal{S}}_{\rm
Liz,even}$, where ${\mathcal{S}}_{\rm Liz,even}$ denotes the even part of the
Radon-domain Lizorkin space ${\mathcal{S}}_{\rm
Liz}(\mathbb{R}\times\mathbb{S}^{d-1})$ with
${\mathrm{R}}^{\ast}{\mathrm{R}}{\mathrm{K}}_{\rm rad}=\mathrm{Id}$ on
${\mathcal{S}}_{\rm Liz,even}$, as well as on ${\mathcal{S}}^{\prime}_{\rm
Liz,even}$, by duality (Helgason, 2011; Kostadinova et al., 2014).

While the adoption of this formalism leads to a well-defined functional-
optimization problem, PN’s derivation/interpretation of Lemmas 17, 18, and 21
is flawed because they implicitly assume that there is a systematic, one-to-
one association between a “concrete” spline ridge
$\rho_{m}({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})$ with
$\rho_{m}(t)=\tfrac{1}{2}{\rm sign}(t)\frac{t^{m-1}}{m!}$ and some abstract
Lizorkin distribution
$\rho_{m}({\boldsymbol{\xi}}_{0}^{\mathsf{T}}{\boldsymbol{x}}-t_{0})+{\mathcal{P}}\in{\mathcal{S}}_{\rm
Liz}^{\prime}(\mathbb{R}^{d})$, which is unlikely to be the case for the
reasons exposed in the introduction. We have recent evidence that such an
association can be made, but that it requires a specific polynomial correction
that depends on the shift $t_{0}$ (Neumayer and Unser, 2022). That said, it
remains that the main results and conclusions reported by Parhi and Nowak
(2021) are qualitatively correct and in agreement with Theorem 12 (up to the
mentioned technicalities). Also, the mathematical arguments proposed by these
authors can easily be corrected/upgraded by extending their space of test
functions to ${\mathcal{S}}_{\rm Rad}$ and by using our results in Theorems 7
and 8. The same holds true for PN’s higher-order generalizations (ridge
splines). In fact, PN’s definition of the native space of the $m$th-order
ridge splines is equivalent to that of ${\mathcal{M}}_{{\mathrm{\Delta}}_{\rm
R}}(\mathbb{R}^{d})$ for $m=2$.

Now, the one aspect where Theorem 12 improves upon Ongie et al. (2020, Theorem
1) and Parhi and Nowak (2021, Theorem 1 with $m=1$) is that it contains the
characterization of the full solution set. The theorem tells us that all
extreme points of the optimization problem in (58) have the same parametric
form (59), which is a much stronger statement than the existence of one such
neural-network-like solution. Ideally, one would like to identify the sparsest
solution within the solution set, in other words, the one with the fewest
neurons. While there is a direct algorithm that will find the sparsest
solution for $d=1$ (Debarre et al., 2022), it is not known yet if this can be
generalized to a larger number of dimensions.

We also like to point out that the functional-learning problem in (58) is
invariant to similarity transformations of the data points
${\boldsymbol{x}}_{m}$. This is to say that any such transformation of the
data characterized by a scale $s$ can be accounted for via a proper rescaling
of the regularization parameter $\lambda\to s\lambda$.

###### Proposition 13

The regularization functional in (58) is translation-, scale- and rotation-
invariant in the sense that

$\displaystyle\|\Delta_{\rm
R}\\{f(s{\bf{U}}\cdot-{\bf{b}})\\}\|_{{\mathcal{M}}}=s\|\Delta_{\rm
R}\\{f\\}\|_{{\mathcal{M}}}$ (63)

for any $f\in{\mathcal{M}}_{\Delta_{\rm R}}(\mathbb{R}^{d})$, and any scaling
factor $s\in\mathbb{R}^{+}$, offset ${\bf{b}}\in\mathbb{R}^{d}$, and rotation
matrix ${\bf{U}}\in\mathbb{R}^{d\times d}$ with
${\bf{U}}^{\mathsf{T}}{\bf{U}}={\bf{I}}$.

There has been concern about the well-posedness of the generative model used
in (Parhi and Nowak, 2021) and summarized by $f=\Delta_{\rm
R}^{\dagger}\\{w\\}$ with $w\in{\mathcal{M}}_{\rm Rad}={\mathcal{M}}_{\rm
even}$ in the present formulation. To make the link with (Bartolucci et al.,
2021), it is instructive to describe our inverse operator $\Delta_{\rm
R}^{\dagger}=(\mathrm{Id}-\mathrm{Proj}_{{\mathcal{P}}_{1}})\Delta^{-1}{\mathrm{R}}^{\ast}:{\mathcal{M}}_{\rm
Rad}\to{\mathcal{M}}_{\Delta_{\rm R}}(\mathbb{R}^{d})$ explicitly in terms of
the generic integral equation

$\displaystyle\Delta_{\rm
R}^{\dagger}\\{w\\}({\boldsymbol{x}})=\int_{\mathbb{R}}\int_{\mathbb{S}^{d-1}}h({\boldsymbol{x}};t,{\boldsymbol{\xi}})w(t,{\boldsymbol{\xi}}){\rm
d}{\boldsymbol{\xi}}{\rm d}t,$ (64)

with a “kernel”
$h:\mathbb{R}^{d}\times(\mathbb{R}\times\mathbb{S}^{d-1})\to\mathbb{R}$ that
is given by

$\displaystyle h({\boldsymbol{x}};t,{\boldsymbol{\xi}})$
$\displaystyle=\tfrac{1}{2}|{\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}-t|-\sum_{|{\boldsymbol{k}}|\leq
1}\frac{{\boldsymbol{x}}^{\boldsymbol{k}}}{{\boldsymbol{k}}!}q_{\boldsymbol{k}}(t,{\boldsymbol{\xi}}),$
(65)

where
$q_{\boldsymbol{k}}(t,{\boldsymbol{\xi}})=\langle\tfrac{1}{2}|{\boldsymbol{\xi}}^{\mathsf{T}}\cdot-t|,m_{\boldsymbol{k}}^{\ast}\rangle$
and where the dual basis $(m^{\ast}_{\boldsymbol{k}})_{|{\boldsymbol{k}}|\leq
1}$ is specified by (8). This form is compatible with (Parhi and Nowak, 2021,
Lemma 21) and has been interpreted as an infinite-width neural network. By
evaluating the underlying duality products explicitly and by regrouping the
first-order correction terms, we obtain the simplified formula

$\displaystyle h({\boldsymbol{x}};t,{\boldsymbol{\xi}})$
$\displaystyle=\tfrac{1}{2}|t-{\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}|-(\kappa_{\rm
rad}\ast\tfrac{1}{2}|\cdot|)(t)+({\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}})(\kappa_{\rm
rad}\ast\tfrac{1}{2}{\rm sign})(t),$ (66)

where the radial convolution with $\kappa_{\rm rad}$ has a mollifying effect
on the two correction terms. If we fix $t,{\boldsymbol{\xi}}$ in (66), then
${\boldsymbol{x}}\mapsto h({\boldsymbol{x}};t,{\boldsymbol{\xi}})$ grows like
$O(|{\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}|)$ with the correction
contributing a first-order polynomial. The effect of the correction is more
remarkable along the radial variable $t$, in that it neutralizes the growth of
the leading term
$\tfrac{1}{2}|t-{\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}|=O(|t|)$: This
is because the profile $t\mapsto h({\boldsymbol{x}};t,{\boldsymbol{\xi}})$ is
bounded with a maximum proportional to
$|{\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}|$ around the origin,
continuous (due to the convolution of ${\rm sign}$ with $\kappa_{\rm rad}$),
and vanishing towards $0$ as $t\to\pm\infty$. This can be translated into the
following properties, which are the mathematical bases for the present
construction.

  1. 1.

For any ${\boldsymbol{x}}_{0}\in\mathbb{R}^{d}$,
$h({\boldsymbol{x}}_{0};\cdot,\cdot)=\Delta_{\rm
R}^{\dagger\ast}\\{\delta(\cdot-{\boldsymbol{x}}_{0})\\}\in
C_{0}(\mathbb{R}\times\mathbb{S}^{d-1})$, which is equivalent to the weak*
continuity of the sampling functionals
$\delta(\cdot-{\boldsymbol{x}}_{0}):f\mapsto f({\boldsymbol{x}}_{0})$; that
is, $\delta(\cdot-{\boldsymbol{x}}_{0})\in C_{0,\Delta_{\rm
R}}(\mathbb{R}^{d})=\Delta_{\rm R}^{\ast}\big{(}C_{0,\rm
Rad}\big{)}\oplus{\mathcal{P}}_{1}^{\prime}$ with
${\mathcal{P}}^{\prime}_{1}={\rm
span}\\{m^{\ast}_{\boldsymbol{k}}\\}_{|{\boldsymbol{k}}|\leq 1}$, where
$C_{0,\Delta_{\rm R}}$ is the predual of ${\mathcal{M}}_{\Delta_{\rm R}}$.

  2. 2.

Stability: the integral operator (64) satisfies the bound

$\displaystyle\big{|}\Delta_{\rm
R}^{\dagger}\\{w\\}({\boldsymbol{x}})\big{|}\leq\|h({\boldsymbol{x}};\cdot,\cdot)\|_{L_{\infty}}\;\|w\|_{{\mathcal{M}}}\leq
C(1+\|{\boldsymbol{x}}\|)\;\|w\|_{{\mathcal{M}}(\mathbb{R}\times\mathbb{S}^{d-1})},$
(67)

which ensures that $\Delta_{\rm R}^{\dagger}\\{w\\}({\boldsymbol{x}})$ is
well-defined (and continuous) for any ${\boldsymbol{x}}\in\mathbb{R}^{d}$ and
$w\in{\mathcal{M}}_{\rm even}(\mathbb{R}\times\mathbb{S}^{d-1})$.

Note that one can also take
$w\in{\mathcal{M}}(\mathbb{R}\times\mathbb{S}^{d-1})$ in (64) since the null
space of $\Delta_{\rm
R}^{\dagger}:{\mathcal{M}}(\mathbb{R}\times\mathbb{S}^{d-1})\to{\mathcal{M}}_{\Delta_{\rm
R}}(\mathbb{R}^{d})$ (and of
${\mathrm{R}}^{\ast}:{\mathcal{M}}(\mathbb{R}\times\mathbb{S}^{d-1})\to{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$)
is precisely the complementary space ${\mathcal{M}}_{\rm
Rad^{\perp}}={\mathcal{M}}_{\rm odd}(\mathbb{R}\times\mathbb{S}^{d-1})$ (see
Theorem 7, Item 6).

An alternative to (66) is to consider a “tempered” kernel of the form
$h_{\beta}({\boldsymbol{x}};t,{\boldsymbol{\xi}})=\frac{1}{2}|{\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}-t|\,\beta(t)$,
where $\beta(t)>0$ is a weighting function (e.g.,
$\beta(t)=\frac{1}{1+|t|^{2+\epsilon}})$ that compensates the linear growth of
the first factor333The first factor can also be replaced by
$({\boldsymbol{\xi}}^{\mathsf{T}}{\boldsymbol{x}}-t)_{+}$ modulo some
adjustments in $\beta$, as shown by the authors. (Bartolucci et al., 2021). In
effect, this mechanism, whose stability is intrinsically guaranteed, reduces
the size of the native space. Interestingly, the corresponding optimization
problem admits the same form of solution—a neural network with one hidden ReLU
layer (Bartolucci et al., 2021)—with the caveat that the underlying
regularization is no longer translation-invariant. In this modified scenario,
the optimal cost is $\|\Delta_{\rm R}f_{\rm
ridge}\|_{{\mathcal{M}}_{1/\beta}}=\sum_{n=1}^{K_{0}}|a_{k}|\frac{1}{|\beta(\tau_{k})|}$,
which tends to favor smaller biases $\tau_{k}$. This also means that one then
looses the invariance to similarity transformations of the data (Proposition
13).

##  A Direct-Sum Topologies

There are two standard ways to define direct sums: explicitly, via the use of
projectors; or abstractly, via the use of quotient spaces. The two methods are
equivalent whenever one can explicitly identify the underlying quotient space
as a (concrete) complemented subspace of the original space.

###  A.1 Projectors

Let ${\mathcal{X}}$ be a topological vector space. A continuous linear
operator ${\mathrm{P}}:{\mathcal{X}}\to{\mathcal{X}}$ with the property that
${\mathrm{P}}={\mathrm{P}}\circ{\mathrm{P}}={\mathrm{P}}^{2}$ (idempotence) on
${\mathcal{X}}$ is called a projection operator (Dunford and Schwartz, 1988, p
480). In particular, when ${\mathcal{X}}$ is a Banach space or a Fréchet
space, the range ${\mathcal{U}}={\mathrm{P}}({\mathcal{X}})$ of ${\mathrm{P}}$
is necessarily a closed subspace of ${\mathcal{X}}$. In that case,
${\mathrm{P}}=\mathrm{Proj}_{{\mathcal{U}}}$ is a projector from
${\mathcal{X}}$ onto ${\mathcal{U}}$, and
${\mathcal{X}}={\mathcal{U}}\oplus{\mathcal{V}}$ where ${\mathcal{V}}={\rm
ker}{\mathrm{P}}$ is the null space of ${\mathrm{P}}$ or, equivalently, the
range of the complementary projector
$\mathrm{Proj}_{\mathcal{V}}=(\mathrm{Id}-{\mathrm{P}})$.

More generally, when ${\mathcal{X}}$ is a topological vector space, the space
${\mathrm{P}}({\mathcal{X}})$ equipped with the topology induced by
${\mathcal{X}}$ is a topological space as well, with the same properties as
the original space ${\mathcal{X}}$ (e.g., completeness). Likewise, if
$({\mathcal{X}},{\mathcal{X}}^{\prime})$ is a dual pair of topological spaces,
then so is
$\Big{(}{\mathrm{P}}({\mathcal{X}}),{\mathrm{P}}^{\ast}({\mathcal{X}}^{\prime})\big{)}$,
where ${\mathrm{P}}^{\ast}:{\mathcal{X}}^{\prime}\to{\mathcal{X}}^{\prime}$ is
the dual projection operator.

###  A.2 Direct Sums

1) Direct-sum decomposition of a vector space ${\mathcal{X}}$: Let
${\mathcal{U}}$ and ${\mathcal{V}}$ be two (complementary) closed subspaces of
${\mathcal{X}}$. The notation ${\mathcal{X}}={\mathcal{U}}\oplus{\mathcal{V}}$
indicates that every element $x\in{\mathcal{X}}$ has a unique decomposition as
$x=u+v$ with $(u,v)\in{\mathcal{U}}\times{\mathcal{V}}$. The underlying
projection operators are

$\displaystyle x=u+v$
$\displaystyle\mapsto\mathrm{Proj}_{{\mathcal{U}}}\\{x\\}=u$ $\displaystyle x$
$\displaystyle\mapsto\mathrm{Proj}_{{\mathcal{V}}}\\{x\\}=(\mathrm{Id}-\mathrm{Proj}_{{\mathcal{U}}})\\{x\\}=v.$

To summarize, given a (closed) subspace ${\mathcal{U}}$ of a normed vector
space ${\mathcal{X}}$, the search of a complement ${\mathcal{V}}$ for
${\mathcal{U}}$ in ${\mathcal{X}}$ is equivalent to the search of a
(continuous) projection operator ${\mathrm{P}}$ on ${\mathcal{X}}$ (with
${\mathrm{P}}^{2}={\mathrm{P}}$) whose range is ${\mathcal{U}}$. Then,
${\mathcal{V}}=\mathrm{Proj}_{{\mathcal{V}}}({\mathcal{X}})$ with
$\mathrm{Proj}_{{\mathcal{V}}}=(\mathrm{Id}-{\mathrm{P}})$.
2) Annihilator: Let ${\mathcal{U}}$ be a closed subset of ${\mathcal{X}}$. One
then defines ${\mathcal{U}}^{\perp}$ as the annihilator of ${\mathcal{U}}$ in
${\mathcal{X}}^{\prime}$, which is the subset

${\mathcal{U}}^{\perp}=\\{f\in{\mathcal{X}}^{\prime}:\langle
f,u\rangle=0\mbox{ for
all~}u\in{\mathcal{U}}\\}\subseteq{\mathcal{X}}^{\prime}.$

3) Dual space: The dual of the direct sum
${\mathcal{X}}={\mathcal{U}}\oplus{\mathcal{V}}$ is
${\mathcal{X}}^{\prime}={\mathcal{U}}^{\prime}\oplus{\mathcal{V}}^{\prime}$,
where
${\mathcal{U}}^{\prime}={\mathrm{P}}^{\ast}({\mathcal{X}}^{\prime})={\mathcal{V}}^{\perp}$
and ${\mathcal{U}}^{\prime}={\mathcal{U}}^{\perp}$.
4) Quotient space: Under the assumption that ${\mathcal{V}}$ is a closed
subset of ${\mathcal{X}}$, one defines the quotient space
${\mathcal{X}}/{\mathcal{V}}$ whose elements are equivalence classes (or
cosets) denoted by $[x]=x+{\mathcal{V}}$. The corresponding quotient map
$q:{\mathcal{X}}\to{\mathcal{X}}/{\mathcal{V}}$ is linear and its kernel (null
space) is ${\mathcal{V}}$. When ${\mathcal{X}}$ is a Banach space, the
quotient norm is

$\|[x]\|_{{\mathcal{X}}/{\mathcal{V}}}=\inf_{v\in{\mathcal{V}}}\|x+v\|_{{\mathcal{X}}},$

which measures the distance from $x$ to ${\mathcal{V}}$. The quotient space
${\mathcal{X}}/{\mathcal{V}}$ equipped with the quotient norm is a Banach
space as well. Moreover, there is a natural isomorphism between
$({\mathcal{X}}/{\mathcal{V}})^{\prime}$ (the dual of the quotient of
${\mathcal{X}}$ by ${\mathcal{V}}$) and ${\mathcal{V}}^{\perp}$ (the
annihilator of ${\mathcal{V}}$ in ${\mathcal{X}}^{\prime}$), so that
$({\mathcal{X}}/{\mathcal{V}})^{\prime}\xhookrightarrow{}{\mathcal{X}}^{\prime}$.

Also of relevance is the property that the bounded operators on
${\mathcal{X}}$ that annihilate the elements of ${\mathcal{V}}$ “factor
through” ${\mathcal{X}}/{\mathcal{V}}$. Let
${\mathrm{T}}:{\mathcal{X}}\to{\mathcal{Y}}$ with ${\mathcal{V}}\subseteq{\rm
ker}{\mathrm{T}}$. Then, there exists a unique linear operator
${\mathrm{T}}_{q}:{\mathcal{X}}/{\mathcal{V}}\to{\mathcal{Y}}$ such that
${\mathrm{T}}_{q}q(x)={\mathrm{T}}(x)$ and
$\|{\mathrm{T}}_{q}\|=\|{\mathrm{T}}\|$.

The kernel of any bounded operator
${\mathrm{T}}:{\mathcal{X}}\to{\mathcal{Y}}$ is a closed subspace of
${\mathcal{X}}$ (Markin, 2020, Proposition 4.2, p. 172). Hence, the quotient
space ${\mathcal{X}}/{\rm ker}({\mathrm{T}})$ is a vector space that is
isomorphic to ${\mathrm{T}}({\mathcal{X}})$.

##  B Extreme Points

######  Definition 14 (Extreme Points)

Let $C$ be a convex set of a Banach space ${\mathcal{X}}$. The extreme points
of $C$ are the points $x\in C$ such that, if there exist $x_{1},x_{2}\in C$
and $\theta\in(0,1)$ such that $x=\theta x_{1}+(1-\theta)x_{2}$, then it
necessarily holds that $x_{1}=x=x_{2}$. The set of these extreme points is
denoted by ${\rm Ext}(C)$.

We now present a classical result that gives the explicit form of the extreme
points of the dual ${\mathcal{X}}^{\prime}$ of any closed subspace
${\mathcal{X}}\subseteq C({\mathcal{Z}})$, where $C({\mathcal{Z}})$ is the
space of continuous functions $z\mapsto f(z)$ on some compact Hausdorff space
${\mathcal{Z}}$ equipped with the norm $\|f\|=\sup_{z\in{\mathcal{Z}}}|f(z)|$.

######  Lemma 15 ((Dunford and Schwartz, 1988, p. 441))

Let ${\mathcal{X}}$ be a closed linear manifold of the Banach space
$C({\mathcal{Z}})$ of all real continuous functions on the compact Hausdorff
space ${\mathcal{Z}}$. For each $z\in{\mathcal{Z}}$, let the evaluation
functional $e_{z}\in{\mathcal{X}}^{\prime}$ be defined by

$\displaystyle\langle e_{z},f\rangle=f(z),\quad f\in{\mathcal{X}}.$ (68)

Then, every extreme point of the closed unit ball in ${\mathcal{X}}^{\prime}$,

$\displaystyle
B_{{\mathcal{X}}^{\prime}}=\\{x^{\ast}\in{\mathcal{X}}^{\prime}:\|x^{\ast}\|_{{\mathcal{X}}^{\prime}}=\sup_{f\in{\mathcal{X}}:|f(z)|\leq
1}\langle x^{\ast},f\rangle\leq 1\\},$ (69)

is of the form $\pm e_{z}$ with $z\in{\mathcal{Z}}$. If
${\mathcal{X}}=C({\mathcal{Z}})$, then the converse is true as well; that is,
${\rm Ext}B_{{\mathcal{M}}({\mathcal{Z}})}=\\{\pm e_{z}:z\in{\mathcal{Z}}\\}$
with ${\mathcal{M}}({\mathcal{Z}})=\big{(}C({\mathcal{Z}})\big{)}^{\prime}$.

Lemma 15 generalizes to $C_{0}({\mathcal{Z}})$, where ${\mathcal{Z}}$ is a
locally compact Hausdorff space, which covers the case that is of interest to
us: ${\mathcal{Z}}=\mathbb{R}\times\mathbb{S}^{d-1}$.

Proof  Let $E$ be the set of all points in ${\mathcal{X}}^{\prime}$ of the
form $\pm e_{z}$ with $z\in{\mathcal{Z}}$. The space ${\mathcal{X}}^{\prime}$
is equipped with its weak$\ast$ (or ${\mathcal{X}}$) topology for the Krein-
Milman theorem to apply. As $\|e_{z}\|_{{\mathcal{X}}^{\prime}}\leq 1$,
$E\subseteq B_{{\mathcal{X}}^{\prime}}$. Since $B_{{\mathcal{X}}^{\prime}}$ is
convex, weak*-compact and, hence, weak*-closed, the inclusion also holds for
the closed convex hull, with ${\rm cch}E\subseteq B_{{\mathcal{X}}^{\prime}}$.
Next, we invoke a variant of the Hahn-Banach theorem (Rudin, 1991, Theorem
3.5, p. 59). For any $x^{\ast}\notin{\rm cch}E$, there exists a linear
functional $f\in{\mathcal{X}}$ that separates
$x^{\ast}\in{\mathcal{X}}^{\prime}$ from the closed convex set ${\rm cch}E$.
This means that there are a constant $c>0$ and some $\epsilon>0$ such that

$\pm f(z)\leq c-\epsilon<c\leq\langle x^{\ast},f\rangle$

for all $z\in{\mathcal{Z}}$. Hence, $\|f\|\leq(c-\epsilon)$ which, when
combined with $\|x^{\ast}\|_{{\mathcal{X}}^{\prime}}\|f\|\geq c$, gives that
$\|x^{\ast}\|_{{\mathcal{X}}^{\prime}}>1$. Thus, ${\rm cch}E\supseteq
B_{{\mathcal{X}}^{\prime}}$, from which we conclude that ${\rm
cch}E=B_{{\mathcal{X}}^{\prime}}$. Finally, since $E$ is compact, the extreme
points of ${\rm cch}E$ necessarily lie in $E$ (see (Rudin, 1991, Milman’s
theorem, p. 76)).

For the converse implication, we invoke the Riesz-representation theorem,
which allows us to represent any unit-norm functional on
$C_{0}({\mathcal{Z}})$ by a real-valued measure
$\mu\in{\mathcal{M}}({\mathcal{Z}})$ of total variation $1$. If the support of
$\mu$ consists of one point, it is a signed multiple of a Dirac mass.
Otherwise, ${\rm supp}\mu$ contains two distinct points $z_{1}\neq z_{2}$. Let
$U,V\subset{\mathcal{Z}}$ be disjoint neighborhoods of $z_{1}$ and $z_{2}$. By
the definition of the support, $|\mu|(U)>0$ and $|\mu|(V)>0$. Define
$t=|\mu|(U)$, which is such that $0<t<1$. Now let $\lambda=t^{-1}\mu|_{U}$ and
$\nu=(1-t)^{-1}\mu|_{U^{c}}$. Then, both $\lambda$ and $\nu$ are unit-norm
functionals and $\mu=t\lambda+(1-t)\nu$, which proves that $\mu$ is not
extreme.

### Acknowlegdments

The research was partially supported by the Swiss National Science Foundation
under Grant 200020-184646. The author is thankful to Sebastian Neumayer and
Shayan Aziznejad for very helpful discussions.

## References

  * Alvarez et al. (2012) Mauricio A Alvarez, Lorenzo Rosasco, and Neil D Lawrence.  Kernels for vector-valued functions: A review.  _Foundations and Trends in Machine Learning_ , 4(3):195–266, 2012.
  * Bach (2017) Francis Bach.  Breaking the curse of dimensionality with convex neural networks.  _Journal of Machine Learning Research_ , 18:1–53, 2017\.
  * Barron (1993) Andrew R Barron.  Universal approximation bounds for superpositions of a sigmoidal function.  _IEEE Transactions on Information Theory_ , 39(3):930–945, May 1993.  doi: 10.1109/18.256500.
  * Bartolucci et al. (2021) Francesca Bartolucci, Ernesto De Vito, Lorenzo Rosasco, and Stefano Vigogna.  Understanding neural networks with reproducing kernel Banach spaces.  _arXiv:2109.09710_ , 2021.
  * Bishop (2006) Christopher M Bishop.  _Pattern Recognition and Machine Learning_.  Springer, 2006.
  * Boman and Lindskog (2009) Jan Boman and Filip Lindskog.  Support theorems for the Radon transform and Cramér-Wold theorems.  _Journal of Theoretical Probability_ , 22(3):683–710, March 2009.  doi: https://doi.org/10.1007/s10959-008-0151-0.
  * Candès (1999) Emmanuel J Candès.  Harmonic analysis of neural networks.  _Applied and Computational Harmonic Analysis_ , 6(2):197–218, 1999.
  * Candès and Donoho (1999) Emmanuel J Candès and David L Donoho.  Ridgelets: A key to higher-dimensional intermittency?  _Phil. Trans. R. Soc. Lond. A._ , pages 2495–2509, 1999.
  * Candès and Romberg (2007) Emmanuel J Candès and Justin Romberg.  Sparsity and incoherence in compressive sampling.  _Inverse Problems_ , 23(3):969–985, 2007.
  * Cybenko (1989) George Cybenko.  Approximation by superpositions of a sigmoidal function.  _Mathematics of Control, Signals and Systems_ , 2(4):303–314, 1989.
  * Debarre et al. (2022) Thomas Debarre, Quentin Denoyelle, Michael Unser, and Julien Fageot.  Sparsest piecewise-linear representation of data.  _Journal of Computational and Applied Mathematics_ , 406:in press, 2022.  doi: https://doi.org/10.1016/j.cam.2021.114044.
  * Donoho (2006) David L Donoho.  Compressed sensing.  _IEEE Transactions on Information Theory_ , 52(4):1289–1306, 2006.
  * Dunford and Schwartz (1988) Nelson Dunford and Jacob T Schwartz.  _Linear Operators, Part 1: General Theory_.  John Wiley and Sons, 1988.
  * Elad (2010) Michael Elad.  _Sparse and Redundant Representations. From Theory to Applications in Signal and Image Processing_.  Springer, 2010.
  * Foucart and Rauhut (2013) Simon Foucart and Holger Rauhut.  _A Mathematical Introduction to Compressive Sensing_.  Springer, 2013.
  * Gelfand and Shilov (1966) Isreal M Gelfand and Georgy Shilov.  _Generalized Functions. Vol. 5. Integral Geometry and Representation Theory_.  Academic Press, New York, USA, 1966.
  * Helgason (2011) Sigurdur Helgason.  _Integral Geometry and Radon Transforms_.  Springer, 2011.
  * Hertle (1983) Alexander Hertle.  Continuity of the Radon transform and its inverse on Euclidean space.  _Mathematische Zeitschrift_ , 184(2):165–192, 1983.
  * Hornik et al. (1989) Kurt Hornik, Maxwell Stinchcombe, and Halbert White.  Multilayer feedforward networks are universal approximators.  _Neural networks_ , 2(5):359–366, 1989.
  * Kostadinova et al. (2014) Sanja Kostadinova, Stevan Pilipović, Katerina Saneva, and Jasson Vindas.  The ridgelet transform of distributions.  _Integral Transforms and Special Functions_ , 25(5):344–358, 2014.
  * Logan and Shepp (1975) Benjamin F Logan and Larry A Shepp.  Optimal reconstruction of a function from its projections.  _Duke Mathematical Journal_ , 42(4):645–659, 1975\.
  * Ludwig (1966) Donald Ludwig.  The Radon transform on Euclidean space.  _Communications on Pure and Applied Mathematics_ , 19(1):49–81, September 1966.  doi: 10.1002/cpa.3160190105.
  * Madych (1990) Wolodymyr R Madych.  Summability and approximate reconstruction from Radon transform data.  _Contemporary Mathematics_ , 113:189–219, 1990.
  * Markin (2020) Marat V Markin.  _Elementary Operator Theory_.  De Gruyter, 2020.
  * Murata (1996) Noboru Murata.  An integral representation of functions using three-layered networks and their approximation bounds.  _Neural Networks_ , 9(6):947–956, 1996.
  * Natterer (1984) Frank Natterer.  _The Mathematics of Computed Tomography_.  John Willey & Sons Ltd, 1984.
  * Neumayer and Unser (2022) Sebastian Neumayer and Michael Unser.  Explicit representations for Banach subspaces of Lizorkin distributions.  _Preprint_ , 2022.
  * Ongie et al. (2020) Greg Ongie, Rebecca Willett, Daniel Soudry, and Nathan Srebro.  A function space view of bounded norm infinite width ReLU nets: The multivariate case.  _International Conference on Representation Learning (ICLR)_ , 2020\.
  * Parhi and Nowak (2021) Rahul Parhi and Robert D Nowak.  Banach space representer theorems for neural networks and ridge splines.  _Journal of Machine Learning Research_ , 22(43):1–40, 2021.
  * Pinkus (2015) Allan Pinkus.  _Ridge Functions_.  Cambridge Tracts in Mathematics. Cambridge University Press, 2015.
  * Poggio and Girosi (1990) Tomaso Poggio and Federico Girosi.  Regularization algorithms for learning that are equivalent to multilayer networks.  _Science_ , 247(4945):978–982, 1990.
  * Ramm (1996) Alexander G Ramm.  Inversion formula and singularities of the solution for the backprojection operator in tomography.  _Proceedings of the American Mathematical Society_ , 124(2):567–577, 1996.  doi: 10.1090/s0002-9939-96-03155-3.
  * Ramm and Katsevich (2020) Alexander G Ramm and Alexander I Katsevich.  _The Radon transform and local tomography_.  CRC Press, 2020.
  * Reed and Simon (1980) Michael Reed and Barry Simon.  _Methods of Modern Mathematical Physics. Vol. 1: Functional Analysis_.  Academic Press, San Diego, 1980.
  * Rubin (1998) Boris Rubin.  The Calderón reproducing formula, windowed x-ray transforms, and Radon transforms in $L^{p}$-spaces.  _Journal of Fourier Analysis and Applications_ , 4(2):175–197, 1998.
  * Rudin (1991) Walter Rudin.  _Functional Analysis_.  McGraw-Hill, New York, 2nd edition, 1991.  McGraw-Hill Series in Higher Mathematics.
  * Samko (1982) Stefan G Samko.  Denseness of the Lizorkin-type spaces $\Phi_{V}$ in ${L_{p}(\mathbb{R}^{n})}$.  _Mathematical notes of the Academy of Sciences of the USSR_ , 31(6):432–437, 1982.
  * Samko et al. (1993) Stefan G Samko, Anatoly A Kilbas, and Oleg I Marichev.  _Fractional Integrals and Derivatives: Theory and Applications_.  Gordon and Breach Science Publishers, 1993.
  * Saneva and Vindas (2010) Katerina Saneva and Jasson Vindas.  Wavelet expansions and asymptotic behavior of distributions.  _Journal of Mathematical Analysis and Applications_ , 370(2):543–554, October 2010.  doi: https://doi.org/10.1016/j.jmaa.2010.04.041.