11institutetext: Ronald R. Coifman 22institutetext: Department of Mathematics,
Program in Applied Mathematics, Yale University, New Haven, CT 06510, USA,
22email<EMAIL_ADDRESS>33institutetext: Jacques Peyrière
44institutetext: Institut de Mathématiques d’Orsay, CNRS, Université Paris-
Saclay, 91405 Orsay, France, 44email: jacques.peyriere@universite-paris-
saclay.fr

#  On Complex Analytic tools, and the Holomorphic Rotation methods

Ronald R. Coifman  Jacques Peyrière  and Guido Weiss

##  1 introduction

This paper in honor of Guido Weiss was written posthumously, jointly with him,
as we had, all of his initial notes and ideas related to the program described
below.

Our task, here, is to recount ideas, explorations, and visions that Guido his
collaborators and students, developed over the last 60 years. To point out the
connection of ideas between the original views of the interplay between
complex and real analysis as envisioned by Zygmund and his students Calderón,
Guido Weiss, Eli Stein,and many others, 70 years ago, and the current
approaches introducing nonlinear multi layered analysis for the organization
and processing of complicated oscillatory functions.

It was Zygmund’s view that harmonic analysis provides the infrastructure
linking most areas of analysis, from complex analysis to partial differential
equations, to probability, number theory, and geometry.

In particular he pushed forward the idea that the remarkable tools of complex
analysis, which include; contour integration, conformal mappings,
factorization, tools which were used to provide miraculous proofs in real
analysis, should be deciphered and converted to real variable tools. Together
with Calderón, they bucked the trend for abstraction, prevalent at the time,
and formed a school pushing forward this interplay between real and complex
analysis. A principal bridge was provided by real variable methods, multiscale
analysis, Littlewood Paley theory, and related Calderon representation
formulas. Our aim, here, is to elaborate on the ”magic” of complex analysis
and indicate potential applications in Higher dimensions. An old idea of
Calderón and Zygmund, the so called ”rotation method”, enabled the reductions
of the study of $L^{p}$ estimates for multi dimensional singular integrals to
a superposition,over all directions,of Hilbert transforms. Thereby allowing
the use of one complex variable methods. A related idea was the invention of
systems of Harmonic functions satisfying generalised Cauchy Riemann equations,
such as the Riesz systems, exploiting their special properties. CW

Our goal is to extend these ideas to enable remarkable nonlinear complex
analytic tools for the adapted analysis of functions in one variable, to apply
in higher dimensions.

Guido has been pushing the idea that factorization theorems like Blaschke
products are a key to a variety of nonlinear analytic methods CW1 . Our goal
here is to demonstrate this point, deriving amazing approximation theorems, in
one variable, and opening doors to higher dimensional applications.
Application in which each harmonic function is the average of special
holomorphic functions in planes and constant in orthogonal directions.

We start by describing recent developments in nonlinear complex analysis,
exploiting the tools of factorization and composition. In particular we will
sketch methods extending conventional Fourier analysis, exploiting both phase
and amplitudes of holomorphic functions. The ”miracles of nonlinear complex
analysis”, such as factorization and composition of functions lead to new
versions of holomorphic wavelets, and relate them to multiscale dynamical
systems.

Our story interlaces the role of the phase of signals with their
analytic/geometric properties. The Blaschke factors are a key ingredient, in
building analytic tools, starting with the Malmquist-Takenaka orthonormal
bases of the Hardy space $\mathsf{H}^{2}({\mathbb{T}})$, continuing with
”best” adapted bases obtained through phase unwinding, and describing
relations to composition of Blaschke products and their dynamics (on the disc
and upper half plane). Specifically we construct multiscale orthonormal
holomorphic wavelet bases, generalized scaled holomorphic orthogonal bases, to
dynamical systems, obtained by composing Blaschke products.

We also, remark, that the phase of a Blaschke product is a one layer neural
net with ($\arctan$ as an activation sigmoid) and that the composition is a
”Deep Neural Net” whose ”depth” is the number of compositions. Our results
provide a wealth of related libraries of orthogonal bases.

We sketch these ideas in various ”vignette” subsections and refer for more
details on analytic methods CP , related to the Blaschke based nonlinear phase
unwinding decompositions coifman ; CSW ; nahon . We also consider orthogonal
decompositions of invariant subspaces of Hardy spaces. In particular we
constructed a multiscale decomposition, described below, of the Hardy space of
the upper half-plane.

Such a decomposition can be carried in the unit disk by conformal mapping. A
somewhat different multiscale decomposition of the space
$\mathsf{H}^{2}({\mathbb{T}})$ has been constructed by using Malmquist-
Takenaka bases associated with Blaschke products whose zeroes are
$\displaystyle(1-2^{-n})\mathrm{e}^{2\mathrm{i}\pi j/2^{n}}$ where $n\geq 1$
and $0\leq j<2^{n}$ feichtinger . Here we provide a variety of multiscale
decompositions by considering iterations of Blaschke products.

In the next chapter we will show how with help of an extended Radon transform
we can introduce a method of rotations to enable us to lift the one
dimensional tools to higher dimensions. In particular the various orthogonal
bases of holomorphic functions in one dimension, give rise to orthogonal bases
of Harmonic functions in the higher dimensional upper half space.

##  2 Preliminaries and notation

For $p\geq 1$, $\mathsf{H}^{p}({\mathbb{T}})$ stands for the space of analytic
functions $f$ on the unit disk ${\mathbb{D}}$ such that

$\sup_{0<r<1}\int_{0}^{2\pi}|f(r\mathrm{e}^{\mathrm{i}\theta})|^{p}\frac{\mathrm{d}\theta}{2\pi}<+\infty.$

Such functions have boundary values almost everywhere, and the Hardy space
$\mathsf{H}^{p}({\mathbb{T}})$ can be identified with the set of $L^{p}$
functions on the torus ${\mathbb{T}}=\partial{\mathbb{D}}$ whose Fourier
coefficients of negative order vanish. We will alternate between analysis on
the disk, and the parallel theory for analytic functions on the upper half
plane ${\mathbb{H}}=\\{x+\mathrm{i}y\ :\ y>0\\}$. The space of analytic
functions $f$ on ${\mathbb{H}}$ such that

$\sup_{y>0}\|f(\cdot+\mathrm{i}y)\|_{L^{p}({\mathbb{R}})}<+\infty$

is denoted by $\mathsf{H}^{p}({\mathbb{R}})$. These functions have boundary
values in $L^{p}({\mathbb{R}})$ when $p\geq 1$. The space
$\mathsf{H}^{p}({\mathbb{R}})$ is identified to the space of $L^{p}$ functions
whose Fourier transform vanishes on the negative half line $(-\infty,0)$.

##  3  Analysis on The upper half plane

We present some known results CP , without proof. In this section one simply
writes $\mathsf{H}^{2}$ instead of $\mathsf{H}^{2}({\mathbb{R}})$.

### Malmquist-Takenaka bases

Let $(a_{j})_{1\leq j}$ be a sequence (finite or not)) of complex numbers with
positive imaginary parts and such that

$\displaystyle\sum_{j\geq 0}\frac{\Im a_{j}}{1+|a_{j}|^{2}}<+\infty.$ (1)

The corresponding Blaschke product is

${\mathsf{B}}(x)=\prod_{j\geq
0}\frac{\left|1+a_{j}^{2}\right|}{1+a_{j}^{2}}\,\frac{x-a_{j}}{x-\overline{a}_{j}},$

where, $0/0$, which appears if $a_{j}=\mathrm{i}$, should be understood as 1.
The factors $\displaystyle\frac{\left|1+a_{j}^{2}\right|}{1+a_{j}^{2}}$ insure
the convergence of this product when there are infinitely many zeroes. But, in
some situations, it is more convenient to use other convergence factors as we
shall see below.

Whether the series (1) is convergent or not, one defines (for $n\geq 0$) the
functions

$\phi_{n}(x)=\frac{1}{\sqrt{\pi}}\left(\prod_{0\leq
j<n}\frac{x-a_{j}}{x-\overline{a}_{j}}\right)\,\frac{1}{x-\overline{a}_{n}}.$

Then these functions form an orthonormal system in $\mathsf{H}^{2}$. If the
series (1) diverges, it is a Malmquist-Takenaka orthonormal basis of
$\mathsf{H}^{2}$, otherwise it is a basis of the orthogonal complement of
${\mathsf{B}}\,\mathsf{H}^{2}$ in $\mathsf{H}^{2}$.

We remark that roughly a hundred years ago these bases were constructed
takenaka ; malmquist  through a Gram Schmidt orthogonalization of the list of
rational functions with poles in the lower half plane .

Observe that for a rational function with a pole of order M at $a$ the
corresponding M basis functions have the form

$\phi_{n}(x)=\mathrm{e}^{\mathrm{i}{n}\theta(x)}\frac{1}{x-\overline{a}_{n}}\qquad(n=1..M).$

These are localized ”Fourier like” basis functions around the real part of $a$
scaled by the imaginary part.

###  Example of a multiscale Wavelet decomposition

The infinite Blaschke products

$G_{n}(x)=\prod_{j\leq
n}\frac{j-\mathrm{i}}{j+\mathrm{i}}\,\frac{x-j-\mathrm{i}}{x-j+\mathrm{i}}\text{\quad
and\quad}G(x)=\prod_{j\in{\mathbb{Z}}}\frac{j-\mathrm{i}}{j+\mathrm{i}}\,\frac{x-j-\mathrm{i}}{x-j+\mathrm{i}}$

can be expressed in terms of known functions:

$G_{n}(x)=\frac{\Gamma(-\mathrm{i}-n)}{\Gamma(\mathrm{i}-n)}\,\frac{\Gamma(x-n+\mathrm{i})}{\Gamma(x-n-\mathrm{i})}\text{\quad
and\quad}G(x)=\frac{\sin\pi(\mathrm{i}-x)}{\sin\pi(\mathrm{i}+x)}.$ (2)

### An orthonormal system

Consider the function
$\phi(x)=\displaystyle\frac{\Gamma(x-1+\mathrm{i})}{\sqrt{\pi}\Gamma(x-\mathrm{i})}$.
It is easily checked that

$\phi(x-n)=\frac{\Gamma(\mathrm{i}-n)}{\Gamma(-\mathrm{i}-n)}\,\frac{G_{n}(x)}{\sqrt{\pi}\bigl{(}x-(n+1)+\mathrm{i}\bigr{)}}.$

Set $\phi_{n}(x)=\phi(x-n)$. For fixed $m$, the functions $\phi_{n}/G_{m}$,
for $n\geq m$, form a Malmquist-Takenaka basis of $(G/G_{m})\mathsf{H}^{2}$.
In other terms, the functions $\phi_{n}$, for $n\geq m$, form an orthonormal
basis of $G_{m}\mathsf{H}^{2}\ominus G\mathsf{H}^{2}$. This means that the
functions $\phi_{n}$ (for $n\in{\mathbb{Z}}$) form a Malmquist-Takenaka basis
of the orthogonal complement of $G\mathsf{H}^{2}$ in $\mathsf{H}^{2}$.

#### Multiscale decomposition

As $|1-G(2^{n}x)|\leq C2^{n}$ all the products
$\displaystyle{\mathscr{B}}_{n}(x)=\prod_{j<n}G(2^{j}x)$ are convergent and
$\displaystyle\lim_{n\to-\infty}{\mathscr{B}}_{n}=1$ uniformly.

Let ${\mathscr{B}}={\mathscr{B}}_{0}$. Obviously,
${\mathscr{B}}_{n}(x)={\mathscr{B}}(2^{n}x)$. Consider the following subspaces
of $\mathsf{H}^{2}$: ${\mathsf{E}}_{n}={\mathscr{B}}_{n}\mathsf{H}^{2}$. This
is a decreasing sequence. The space
$\displaystyle{\mathsf{E}}_{+\infty}=\bigcap_{n\in{\mathbb{Z}}}{\mathsf{E}}_{n}$
is equal to $\\{0\\}$ since a function orthogonal to this space would have too
many zeros, and the space
$\displaystyle{\mathsf{E}}_{-\infty}=\mathrm{closure~{}of}\bigcup_{n\in{\mathbb{Z}}}{\mathsf{E}}_{n}$
is equal to $\mathsf{H}^{2}$ since ${\mathscr{B}}_{n}$ converges uniformly to
1 when $n$ goes to $-\infty$.

For all $n$ and $j$, let

$\phi_{n,j}(x)=2^{n/2}\phi(2^{n}x-j){\mathscr{B}}(2^{n}x).$

Then, for all $n$, $(\phi_{n,j})_{j\in{\mathbb{Z}}}$ is an orthonormal basis
of ${\mathsf{E}}_{n}\ominus{\mathsf{E}}_{n+1}$. We conclude that
$(\phi_{n,j})_{n,j\in{\mathbb{Z}}}$ is an orthonormal basis of
$\mathsf{H}^{2}$.

##  4  Adapted MT bases, ”phase unwinding”

We now find a ”best” adapted Malmquist Takenaka basis to analyze or unwind the
oscillations of a given function.

The idea is to peel off the oscillation of a function by dividing by its
Blaschke product defined by the zeroes of the function, this procedure is
iterated to yield an expansion in an orthogonal collection of functions or
Blaschke products which of course are naturally embedded in a MT basis, once
the zeroes are ordered.

### The unwinding series.

There is a natural way to iterate the Blaschke factorization, it is inspired
by the power series expansion of a holomorphic function on the disk. If $G$
has no zeroes inside $\mathbb{D}$, its Blaschke factorization is the trivial
one $G=1\cdot G$, however, the function $G(z)-G(0)$ certainly has at least one
root inside the unit disk $\mathbb{D}$ and will therefore yield some
nontrivial Blaschke factorization $G(z)-G(0)={\mathsf{B}}_{1}G_{1}$. We write

$\displaystyle F(z)$ $\displaystyle=$ $\displaystyle{\mathsf{B}}(z)\cdot
G(z)={\mathsf{B}}(z)\cdot\bigl{(}G(0)+(G(z)-G(0)\bigr{)}$ $\displaystyle=$
$\displaystyle{\mathsf{B}}(z)\cdot\bigl{(}G(0)+{\mathsf{B}}_{1}(z)G_{1}(z)\bigr{)}=G(0){\mathsf{B}}z+{\mathsf{B}}(z){\mathsf{B}}_{1}(z)G_{1}(z).$

An iterative application gives rise to the unwinding series

$F=a_{1}{\mathsf{B}}_{1}+a_{2}{\mathsf{B}}_{1}{\mathsf{B}}_{2}+a_{3}{\mathsf{B}}_{1}{\mathsf{B}}_{2}{\mathsf{B}}_{3}+a_{4}{\mathsf{B}}_{1}{\mathsf{B}}_{2}{\mathsf{B}}_{3}{\mathsf{B}}_{4}+\dots$

This orthogonal expansion first appeared in the PhD thesis of Michel Nahon
nahon  and independently by T. Qian in qtao ; qw  Detailed approximations in
smoothness spaces were derived by S. Steinerberger in coifman . Given a
general function $F$ it is not numerically feasible to actually compute the
roots of the function; a crucial insight in nahon  is that this is not
necessary – one can numerically obtain the Blaschke product in a stable way by
using a method that was first mentioned in a paper of Guido and Mary Weiss ww
and has been investigated with respect to stability by Nahon nahon  Using the
boundedness of the Hilbert transform one can prove easily convergence in
$L^{p},1<p<\infty$.

###  The fast algorithm of Guido and Mary Weiss ww

Our starting point is the theorem that any Hardy function can be decomposed as

$F={\mathsf{B}}\cdot G,$

where $B$ is a Blaschke product, that is a function of the form

${\mathsf{B}}(z)=z^{m}\prod_{i\in
I}{\frac{\overline{a_{i}}}{|a_{i}|}\frac{z-a_{i}}{1-\overline{a_{i}}z}},$

where $m\in\mathbb{N}_{0}$ and $a_{1},a_{2},\dots\in\mathbb{D}$ are zeroes
inside the unit disk $\mathbb{D}$ and $G$ has no roots in $\mathbb{D}$. For
$|z|=1$ we have $|{\mathsf{B}}(z)|=1$ which motivates the analogy

${\mathsf{B}}\sim\mbox{frequency and}~{}G\sim\mbox{amplitude}$

for the function restricted to the boundary. However, the function $G$ need
not be constant: it can be any function that never vanishes inside the unit
disk. If $F$ has roots inside the unit disk, then the Blaschke factorization
$F={\mathsf{B}}\cdot G$ is going to be nontrivial (meaning
${\mathsf{B}}\not\equiv 1$ and $G\not\equiv F$). $G$ should be ’simpler’ than
$F$ because the winding number around the origin decreases.

In fact since $|F|=|G|$ and $\ln(G)$ is analytic in the disk we have formally
that $G=\exp(\ln|F|+\mathrm{i}(\ln|F|)^{\sim}))=\exp(\mathscr{H}(\ln|F|))$
where $\mathscr{H}$ is the projection onto the Hardy space. and
${\mathsf{B}}=F/G$. G can be computed easily using the FFT nahon .

###  A remarkable unwinding

The following is an explicit unwinding of a singular inner function in the
upper half plane illustrating this exponentially fast approximation of
$\exp\frac{2\mathrm{i}\pi}{x}$:

$\exp\frac{2\mathrm{i}\pi}{x}=\mathrm{e}^{-2\pi}+\bigl{(}1-\mathrm{e}^{-4\pi}\bigr{)}\sum_{n\geq
0}(-1)^{n}\mathrm{e}^{-2n\pi}B(x)^{n+1},$

where $B$ is a Blaschke product whose zeros are
$\\{1/(j+\mathrm{i})\\}_{j\in{\mathbb{Z}}}$.

##  5 Geometric function theory: the role of compositions of Blaschke
products.

#### Iteration of Blaschke products

We claim that by building Blaschke product through composition we open up rich
dynamical structures, and libraries of corresponding Malmquist Takenaka bases.

We are interested in iteration of finite Blaschke products

${\mathsf{B}}(z)=\mathrm{e}^{\mathrm{i}\theta}z^{\nu}\prod_{j=1}^{\mu}\frac{z+a_{j}}{1+\overline{a}_{j}z},$

where $\mu$ and $\nu$ are nonnegative integers and the $a_{j}$ are complex
numbers of modulus less than 1.

It is well known that ${\mathbb{T}}$ and ${\mathbb{D}}$ are globally invariant
under ${\mathsf{B}}$, as well as the complement of $\overline{\mathbb{D}}$ in
the Riemann sphere.

A careful discussion can be found in CP . Here is the main result.

###### Theorem 5.1

Let ${\mathsf{B}}$ be a finite Blaschke product with a fixed point $\alpha$
inside the unit disk. Then there exists a sequence
$\alpha,a_{1},a_{2},\dots,a_{j},\dots$ of complex numbers in the unit disk and
an increasing sequence $(\nu_{j})_{j\geq 1}$ of positive integers such that
$a_{1},a_{2},\dots,a_{\nu_{n}}$ are the zeros, counted according to their
multiplicity, of ${\mathsf{B}}_{n}$ (the nth iterate of ${\mathsf{B}}$).
Moreover $\displaystyle\sum_{j\geq 1}(1-|a_{j}|)=+\infty$. Also,
${\mathsf{B}}_{n}$ converges towards $\alpha$ unformly on compact subsets of
the open unit disk.

###  Dynamic Multiscale analysis through composition of Blaschke products

Each Blaschke product ${\mathsf{B}}$ defines invariant subspaces of
${\mathsf{H}}^{p}$. The projection on this space is given by the kernel
$\displaystyle\frac{{\mathsf{B}}(z)\overline{{\mathsf{B}}(w)}}{z-w}$. This
projection is continuous for $1<p<+\infty$.

Let $F$ be a Blaschke product of degree at least 2 with a fixed point inside
the unit disk. Its iterates define a hierarchy of nested invariant subspaces
${\mathsf{E}}_{n}=F_{n}{\mathsf{H}}^{2}$.

Due to Theorem 5.1, $\displaystyle\bigcap_{n\geq 1}{\mathsf{E}}_{n}=\\{0\\}$.

The Takenaka construction provides orthonormal bases of $E_{n}\ominus
E_{n+1}$. But this is not canonical as it depends on an ordering of the zeros
of $F_{n+1}/F_{n}$.

Figure 1 shows 1st, 3rd, and 5th iterates of $F(z)=z(z-2^{-1})/(1-2^{-1}z)$.
Figure 2 displays the phase for the fourth iterate of
$F(z)=z^{2}(z-2^{-2})/(1-2^{-2}z)$. The upper pictures display the phases
modulo $2\pi$ (values in the interval $(-\pi,\pi]$) of theses Blaschke
products while the lower pictures display minus the logarithms of their
absolute value. The coordinates $(x,y)$ correspond to the point
$\mathrm{e}^{-y+\mathrm{i}x}$. On these figures it is easy to locate the
zeros, specially by looking at the phase which then has an abrupt jump.

### Remarks on Iteration of Blaschke products as a ”Deep Neural Net”

In the upper half plane let $(a_{j})_{1\leq j}$ be a finite sequence of
complex numbers with positive imaginary parts. The corresponding Blaschke
product on the line is

${\mathsf{B}}(x)=\prod_{j\geq 0}\frac{x-a_{j}}{x-\overline{a}_{j}}.$

We can write ${\mathsf{B}}(x)=\exp\bigl{(}\mathrm{i}\theta(x)\bigr{)}$, where

$\theta(x)=\sum_{j\geq 0}\sigma\bigl{(}(x-\alpha_{j})/\beta_{j}\bigr{)}$

with $a_{j}=\alpha_{j}+\mathrm{i}\beta_{j}$ and $\sigma=\arctan x+\pi/2$ is a
sigmoid.

This is precisely the form of a single layer in a Neural Net, each unit has a
weight and bias determined by $a_{j}$. We obtain the various layers of a deep
net through the composition of each layer with a preceding layer. In our
preceding examples we took a single short layer given by a simple Blaschke
term with two zeroes in the first layer that we iterated to obtain an
orthonormal Malmquist Takenaka basis ( we could have composed different
elementary products at each layer), demonstrating the versatility of the
method to generate highly complex functional representations.

As an example let $F(z)$ be mapped from G, (2) in the section on wavelet
construction.

$F(z)=G(w)=\frac{\sin(\pi(\mathrm{i}-w))}{\sin(\pi(\mathrm{i}+w))}\text{\quad
with\quad}w=\frac{\mathrm{i}(1-z)}{(1+z)}.$

We can view the phase of F as a neural layer which when composed with itself
results in a phase which is a two layer neural net represented graphically in
fig 3.

Where each end of a color droplet corresponds to one zero or unit of the two
layer net.

We refer to Daubechies et al. ReluDNN  for a description of a similar
iteration for piecewise affine functions in which simple affine functions play
the role of a Blaschke product.

##  6  Higher dimensions, $\theta$-holomorphy

Our goal is to explore methodologies to use the remarkable analytic
approximation theorems described above to enable deeper understanding of real
analysis, in higher dimensions. We know that Blaschke factorization do not
exist, nevertheless there are remarkable bases that can be lifted.

We start by observing that
$Z_{\theta}=(x\cdot\theta+\mathrm{i}y)=t+\mathrm{i}y$ is harmonic in
$(x,y)$,(in 3 dimensions) and so is $Z_{\theta}^{k}$ This is a harmonic
homogeneous polynomial of degree k in $(x_{1},x_{2},y)$ that is constant in
the direction perpendicular to $\theta$. here we identified $\theta$ with
$(\cos\theta,\sin\theta)$. It is well known CW  that

$\frac{1}{2\pi}\int_{0}^{2\pi}\mathrm{e}^{-\mathrm{i}{l}\theta}{Z_{\theta}^{k}}\,\mathrm{d}\theta=Y_{l}^{k}(x_{1},x_{2},y)\quad(\left|l\right|\leq{k})$

is the standard orthogonal basis of spherical Harmonics in 3 dimensions.

As a consequence we see that any Harmonic function $U(x,y)$ is a superposition
over $\theta$ of holomorphic functions in $Z_{\theta}$, more explicitly a
Power series in $Z_{\theta}$ with coefficients depending on $\theta$.

$U(x,y)=\frac{1}{2\pi}\int_{0}^{2\pi}\
F_{\theta}(Z_{\theta)})\,\mathrm{d}\theta=\frac{1}{2\pi}\int_{0}^{2\pi}{\sum_{k\geq
0}a_{k}({\theta}){Z_{\theta}^{k}\mathrm{d}\theta}}\ $

where $a_{k}({\theta})$ is a trigonometric polynomial of degree $k$.

Another example, taking

$F_{\theta}(Z_{\theta})=\mathrm{e}^{-2\mathrm{i}\pi
rZ_{\theta}}{\mathrm{e}^{-2\mathrm{i}\pi k{\theta}}}.$

we get the harmonic function

$J_{k}\left(2\pi r\sqrt{x_{1}^{2}+x_{2}^{2}}\right)\mathrm{e}^{-2\mathrm{i}\pi
k{\phi}}\mathrm{e}^{-2\pi yr}$

##  Radon and Fourier in the upper Half space

This relationship between holomorphic functions in planes as generating all
harmonic functions can most easily be explored through Fourier analysis. We
define the Radon transform, and relate it to the Fourier transform to lead to
the $\theta$-holomorphic version.

Let

${\mathsf{R}}_{\theta}{f}(t)=\int_{x\in\theta^{\perp}}f(t\theta+x)\,\mathrm{d}x.$
(3)

Obviously ${\mathsf{R}}_{-\theta}{f}(t)={\mathsf{R}}_{\theta}{f}(-t).$ The
formula $\widehat{{\mathsf{R}}_{\theta}{f}}(t)=\hat{f}(u\theta)$ for Fourier
transforms is well known.

For $f\in L^{1}({\mathbb{R}}^{n})$, consider its harmonic extension $u$ to
${\mathbb{R}}_{+}^{n+1}$. For $x\in{\mathbb{R}}^{n}$ and $y>0$ we have

$\displaystyle u(x,y)$ $\displaystyle=$ $\displaystyle
f\star{\mathsf{P}}_{y}(x)=\int\mathrm{e}^{2\mathrm{i}\pi
x\cdot\xi}\mathrm{e}^{-2\pi|\xi|y}\hat{f}(\xi)\,\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle\int_{S_{n-1}}\left(\int_{0}^{\infty}\mathrm{e}^{2\mathrm{i}\pi
r(x\cdot\theta+\mathrm{i}y)}\hat{f}(r\theta)r^{n-1}\mathrm{d}r\right)\,\mathrm{d}\theta$
$\displaystyle=$
$\displaystyle\int_{S_{n-1}}F_{\theta}(x\cdot\theta+\mathrm{i}y)\,\mathrm{d}\theta,$

where

$F_{\theta}(z)=\int_{0}^{\infty}\mathrm{e}^{2\mathrm{i}\pi
rz}\hat{f}(r\theta)r^{n-1}\mathrm{d}r=\int_{0}^{\infty}\mathrm{e}^{2\mathrm{i}\pi
rz}\widehat{{\mathsf{R}}_{\theta}{f}}(r)\,r^{n-1}\mathrm{d}r.$ (4)

When $n=2$, we have

$\widehat{F_{\theta}}(t)=\widehat{{\mathsf{R}}_{\theta}{f}}(t)\,t{\large\bf
1}_{(0,+\infty)}(t)=\frac{1}{2\mathrm{i}\pi}\widehat{{\mathsf{D}}{\mathsf{R}}_{\theta}{f}}(t)\,{\large\bf
1}_{(0,+\infty)}(t).$

So, for $\Im z>0$,

$F_{\theta}(z)=-\frac{1}{4\pi^{2}}\int_{-\infty}^{\infty}\frac{\mathrm{d}({\mathsf{R}}_{\theta}{f}(t))}{t-z}\\\
=-\frac{1}{4\pi^{2}}\int_{-\infty}^{+\infty}\frac{{\mathsf{R}}_{\theta}{f}(t)}{(t-z)^{2}}\,\mathrm{d}t.$

For general $n$ we get

$F_{\theta}(z)=\frac{(n-1)!}{(2\mathrm{i}\pi)^{n}}\int_{-\infty}^{+\infty}\frac{{\mathsf{R}}_{\theta}{f}(t)}{(t-z)^{n}}\,\mathrm{d}t.$

### Some isometries

We describe some computations in the case when $n=2$ and mention the case of
higher dimension at the end of this section.

In view of (4)

$\widehat{F_{\theta}(\cdot+\mathrm{i}y)}(r)=\hat{f}(r\theta)\,\mathrm{e}^{-2\pi
ry}r{\large\bf 1}_{(0,\infty)}(r).$ (5)

Hence, the Plancherel identity yields

$\displaystyle\int_{0}^{\infty}\mathrm{d}y\int_{-\infty}^{+\infty}|F_{\theta}(t+\mathrm{i}y)|^{2}\mathrm{d}t$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\int_{0}^{\infty}|\hat{f}(r\theta)|^{2}r^{2}\mathrm{e}^{-4\pi
ry}\mathrm{d}r\mathrm{d}y$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int_{0}^{\infty}|\hat{f}(r\theta)|^{2}r\mathrm{d}r$

Let
$\displaystyle\|F\|_{B}^{2}=\int_{0}^{\infty}\int_{-\infty}^{+\infty}|F_{\theta}(t+\mathrm{i}y)|^{2}\mathrm{d}y\mathrm{d}t$
(this is the norm of the Bergman space on the upper half plane). Then

$4\pi\int_{0}^{2\pi}\|F_{\theta}\|_{B}^{2}\mathrm{d}\theta=\iint_{(0,+\infty)\times(0.2\pi)}\left|\hat{f}(r\theta)\right|^{2}r\,\mathrm{d}r\mathrm{d}\theta=\|f\|_{L^{2}({\mathbb{R}}^{n})}^{2}.$
(6)

We have

$\frac{\partial u(x,y)}{\partial
y}=-2\pi\int_{{\mathbb{R}}^{2}}\mathrm{e}^{2\mathrm{i}\pi\xi\cdot
x}|\xi|\mathrm{e}^{-2\pi|\xi|y}\hat{f}(\xi)\,\mathrm{d}\xi.$

$\displaystyle\iint_{{\mathbb{R}}_{+}^{3}}\left|\frac{\partial
u(x,y)}{\partial y}\right|^{2}\mathrm{d}x\mathrm{d}y$ $\displaystyle=$
$\displaystyle(2\pi)^{2}\int_{{\mathbb{R}}^{2}}\left(\int_{0}^{\infty}\mathrm{e}^{-4\pi|\xi|y}\mathrm{d}y\right)|\hat{f}(\xi)|^{2}|\xi|^{2}\mathrm{d}\xi$
(7) $\displaystyle=$
$\displaystyle\pi\int_{{\mathbb{R}}^{n}}|\hat{f}(\xi)|^{2}|\xi|\mathrm{d}\xi$

Equation (5) yields
$\displaystyle\int_{-\infty}^{+\infty}|F_{\theta}(t)|^{2}\,\mathrm{d}t=\int_{0}^{\infty}|\hat{f}(r\theta)|^{2}r^{2}\mathrm{d}r$,
and

$\int_{0}^{2\pi}\|F_{\theta}\|_{\mathsf{H}^{2}({\mathbb{R}})}^{2}\mathrm{d}\theta=\int_{{\mathbb{R}}^{2}}|\hat{f}(\xi)|^{2}|\xi|\mathrm{d}\xi.$
(8)

Formulas (7) and (8) together give

$\int_{0}^{2\pi}\|F_{\theta}\|_{\mathsf{H}^{2}({\mathbb{R}})}^{2}\mathrm{d}\theta=\frac{1}{\pi}\iint_{{\mathbb{R}}_{+}^{3}}\left|\frac{\partial
u(x,y)}{\partial y}\right|^{2}\mathrm{d}x\mathrm{d}y,$ (9)

In higher dimension formulas (6) and (9) become

$\int_{S_{n-1}}\mathrm{d}\theta\iint_{{\mathbb{R}}_{+}^{n+1}}|F_{\theta}(t+\mathrm{i}y)|^{2}y^{n-2}\mathrm{d}t\mathrm{d}y=\frac{1}{(4\pi)^{n-1}}|f\|_{L^{2}({\mathbb{R}}^{n})}^{2}$

and

$\iint_{{\mathbb{R}}_{+}^{n+1}}\left|\frac{\partial^{k}u(x,y)}{\partial
y^{k}}\right|y^{2k-n}\mathrm{d}x\mathrm{d}y\\\
=\frac{(4\pi)^{n-1}\Gamma(2k-n+1)}{2^{2k}}\int_{S_{n-1}}\|F_{\theta}\|_{H^{2}(\mathbb{R})}^{2}\mathrm{d}\theta.$
(10)

### Remarks; ”lifted Analysis” of Harmonic functions

One of our goals is to enable the application of some of the one dimensional
analytic approximation tools to higher dimensions. We refer to Michel Nahon’s
thesis nahon  in which he decomposes a function in the plane as a sum of
functions whose Fourier transform live in thin wedges, as a tool to extract
features ( gradients of Phase) from an image of a fingerprint. This
illustrates potential variants of our current approach.

We envision a function in two variables represented as a superposition of
$F_{\theta}(t+iy)$, each of which is approximated to error $\epsilon$ leading
to a harmonic function approximation of error $\epsilon$ in the Dirichlet
space. Similar estimations with different mix of Hilbert spaces can be easily
derived as in coifman , leading to faster rates of convergence (when more
regularity is present).

Another obvious application is the representation of a Calderón Zygmund
operator given by a Fourier multiplier homogeneous of degree $0$,
$\Omega(\theta)$, simply by averaging $\Omega(\theta)F_{\theta}(t+iy)$.

The representation of these operators is a version of the rotation method (no
parity required on $\Omega$ ). Also it provides a local representation method
for generalized conjugate functions or C-Z operators, just by using the local
spherical Harmonics version of the ${\theta}$-holomorphic representation. In
particular Riesz transforms correspond to
$\Omega(\theta)=(\cos\theta,\sin\theta)$.

### A natural ortho-basis in the Dirichlet space

We now use identity (9) to transfer an orthonormal basis of the Hardy space
$H^{2}$ to an orthonormal system in the Dirichlet space in
${\mathbb{R}}_{+}^{3}$.

Start from the basis
$\frac{\mathrm{i}}{\sqrt{\pi}}\left(\frac{z-i}{z+i}\right)^{n}\frac{1}{z+i}$
of $H^{2}$ ( this corresponds to the Fourier basis in the disc, mapped to the
upper half plane Hardy space). We consider the generating function

$F(z)=\frac{\mathrm{i}}{\sqrt{2\pi^{3}}}\sum_{k\geq
0}(z-i)^{n}t^{n}/(z+i)^{n+1}$

and compute
$G(x,y)=\displaystyle\int_{0}^{2\pi}F(z_{\theta})\,\mathrm{d}\theta$. We get

$G(x,y)=\sqrt{2/\pi}/\sqrt{(\rho^{2}+2y+1)t^{2}-2(1-\rho^{2})t+\rho^{2}-2y+1},$

where $\rho=\sqrt{x_{1}^{2}+x_{2}^{2}+(y+1)^{2}}$.

This also can be written as

$G(x,y)=\frac{\sqrt{2/\pi}}{\sqrt{\rho^{2}+2y+1}\,\sqrt{a^{2}t^{2}-2bat+1}},$

if one sets $a=\sqrt{\frac{\rho^{2}-2y+1}{\rho^{2}+2y+1}}$ and
$b=\frac{\rho^{2}-1}{\sqrt{(\rho^{2}+1)^{2}-4y^{2}}}.$

It results that the functions
$\displaystyle\frac{\sqrt{2/\pi}a^{n}P_{n}(b)}{\sqrt{\rho^{2}+2y+1}}$, where
the $P_{n}$ are the Legendre polynomials, form an orthonormal system in the
Dirichlet space.

To get an orthonormal basis of the Dirichlet space in 3 dimensions, it
suffices to take
$\displaystyle\frac{\sqrt{2/\pi}2a^{n}P_{n}(b)\,{\mathrm{e}}^{\mathrm{i}k\theta}}{\sqrt{\rho^{2}+2y+1}}$,
with $k\in\mathbb{Z}$ and $n\geq 0$.

Of course such computation can be done in higher dimension: isometry (10)
allows to transfer orthonormal bases of $H^{2}$ to orthonormal systems in a
suitable Dirichlet space.

###  Concluding remarks and potential applications

As we all know complex methods, such as interpolation of operators, or the
remarkable proofs by Calderón of the boundedness in $L^{2}$ of commutators
with the Hilbert transform, or the Cauchy integral on Lipschitz curves are
powerful tools. Over the years the goal has been to convert them into real
variable methods. In parallel the quest for higher dimensional complex tools
is continuing, see the examples CW  in which various systems generalizing
holomorphic functions to higher dimension are studied. The point here, is that
the infinite dimensional $\theta$-holomorphic functions generate all of these
systems through the choice of appropriate multipliers (as described for the
Riesz system) .

Our goal here was to describe recent nonlinear analytic tools in the classical
setting. and transfer them to the higher dimensional real setting. Together
with Guido Weiss we had observed CW  that all harmonic functions in higher
dimensions are combinations of holomorphic functions on subplanes which are
constant in normal directions. The recent developments in one dimension as
well as the isometries described here, and the corresponding efficient
approximation methods, open the door for applications in higher dimensions,
such as image denoising. See CSW  for the impact of unwinding on precision
Doppler analysis in 1 dimension, which we expect to carry over to 2 or 3
dimensions.

Observe also that, for simplicity, we restricted our attention to 2 dimensions
in cylindrical coordinates. We could have defined more generally power series
in the variable $Z_{\epsilon}=(x\cdot{\epsilon)}$ where ${\epsilon}$
satisfying; $(\epsilon\cdot\epsilon)=0$ , represents a point on the complex
quadric with $|\Re{\epsilon}|=1,|\Im{\epsilon}|=1$. , or a two dimensional
plane spanned by $\Re{\epsilon},\Im{\epsilon}$.

Clearly we can extend the preceding discussion to this setting. Where;
$Z_{\epsilon}=(x\cdot{\epsilon)}$ is the point $t+\mathrm{i}s$ in the complex
plane with coordinate $t\Re{\epsilon}+\mathrm{i}s\Im{\epsilon}$.

## References

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Figure 1:  The argument and the absolute value of $F(z)$, $F^{(3)}(z)$, and
$F^{(5}(z)$,
with $F(z)=\frac{z(z-2^{-1})}{1-2^{-1}z}$ and $z=\exp(-y+\mathrm{i}x)$. Figure
2: The multiscale view of the argument of $F^{(4)}(z)$, with
$F(z)=\frac{z^{2}(z-2^{-2})}{1-2^{-2}z}$ and $z=\exp(-y+\mathrm{i}x)$. Figure
3:  Two iterations of
$F(z)=G(w)=\frac{\sin(\pi(\mathrm{i}-w))}{\sin(\pi(\mathrm{i}+w))}\text{\quad
with\quad}w=\frac{\mathrm{i}(1-z)}{(1+z)}.$