# Sampling theorems with derivatives in shift-invariant spaces generated by
periodic exponential B-splines

Karlheinz Gröchenig *<EMAIL_ADDRESS>and  Irina
Shafkulovska*<EMAIL_ADDRESS>
###### Abstract.

We derive sufficient conditions for sampling with derivatives in shift-
invariant spaces generated by a periodic exponential B-spline. The sufficient
conditions are expressed with a new notion of measuring the gap between
consecutive samples. These conditions are near optimal, and, in particular,
they imply the existence of sampling sets with lower Beurling density
arbitrarily close to the necessary density.

###### Key words and phrases:

Shift-invariant spaces, splines, Chebyshev B-splines, collocation matrix,
Schoenberg-Whitney condition, nonuniform sampling, Gabor frames

###### 2010 Mathematics Subject Classification:

42C15 , 41A15, 94A20, 42C40

K. Gröchenig was supported by the Austrian Science Fund (FWF) project
P31887-N32. I. Shafkulovska was funded by the Austrian Science Fund (FWF)
project P33217 and by the Austrian Academy of Sciences through an internship
at the Acoustics Research Institute.

##  1\. Introduction

By a sampling problem, we understand the question of whether and how a
function in a given space can be recovered from discrete information, which is
usually given by point evaluations (samples). The given space is an a priori
signal model, and for a century this model has been the Paley-Wiener space of
bandlimited functions. The significance of the Paley-Wiener space and the
associated Shannon-Whittaker-Kotelnikov sampling theorem cannot be
overestimated; it is at the origin of the information theory by Shannon, it is
the technical basis for analog/digital conversion in signal processing, and it
informs a whole branch of complex analysis. From the point of view of
mathematics, the decisive results are the necessary density conditions of
Landau [26] and the sufficient conditions for sampling by Beurling [10]. Their
work remains the point of reference for the exploration of sampling problems
in analysis and signal processing. See [4, 8, 35] for some representative
expositions of modern sampling theory.

In this paper, we study sampling in a shift-invariant space generated by a
periodic exponential $B$-spline (PEB-spline). Given a PEB-spline, or more
generally a generator $\varphi$ subject to some mild decay conditions, the
shift-invariant space generated by $\varphi$ in $L^{p}(\mathbb{R})$ is defined
to be

(1.1) $V^{p}(\varphi)\coloneqq\Big{\\{}f\in
L^{p}(\mathbb{R}):f(x)=\sum\limits_{\ell\in\mathbb{Z}}c_{\ell}\varphi(x-\ell),\
(c_{\ell})_{\ell\in\mathbb{Z}}\in\ell^{p}(\mathbb{Z})\Big{\\}}\subseteq
L^{p}(\mathbb{R}).$

If $\varphi(x)=\tfrac{\sin\pi x}{\pi x}$, then $V^{2}(\varphi)=\\{f\in
L^{2}(\mathbb{R}):\operatorname{supp}\hat{f}\subseteq[-1/2,1/2]\\}$ is the
Paley-Wiener space $PW^{2}(\mathbb{R})$ of band-limited functions.

Shift-invariant spaces form a natural class of _signal models_ in sampling
theory, as they generalize the Paley-Wiener space and are useful for practical
and numerical reasons. The generator $\varphi$ can be considered a parameter
of the signal model that can be tuned to the specifics of the data
acquisition. For instance, if $\varphi$ is compactly supported, then every
point evaluation can be computed as a finite sum, and a sample $f(x)$ affects
only the coefficients $c_{k}$ in a neighbourhood of $x$. Additional local
properties of $f\in V^{p}(\varphi)$, such as differentiability and
analyticity, can be implemented by imposing them on the generator $\varphi$.
By contrast, the Paley-Wiener space consists of entire functions, and every
sample of $f$ has a long-range effect on the values of $f$. Shift-invariant
spaces are central in approximation theory [13, 22, 25, 33] and are
increasingly used in signal processing.

For the mathematical formulation of the sampling problem, we assume that the
signal space $V^{p}(\varphi)$ is known and that $f\in V^{p}(\varphi)$ is
sampled on a set $X$ together with the first derivatives
$f^{(s)}(x),s=0,\dots,\mu(x),$ for some multiplicity function $\mu$. Whereas
the standard sampling problem only takes samples on a discrete subset of
$\mathbb{R}$, the inclusion of derivatives has become important in several
applications. First derivatives model trends, and higher derivatives indicate
convexity properties. In higher dimensional spaces, i.e., for images or
physical fields, one speaks of “gradient-augmented measurements” [2], in
event-based sampling “information is transmitted only when a significant
change in the signal occurs, justifying the acquisition of a new sample” [29],
in other words, the value of the first derivative determines the position of
the next sample.

For the Paley-Wiener space, sampling with derivatives was studied early on
[16, 30, 31], the problem was taken up again recently with an emphasis on
shift-invariant spaces in [1, 15, 17, 36].

The main objective is to recover a function from its samples and derivatives
on a set $X=(x_{j})_{j\in\mathbb{Z}}$. We indicate the number of derivatives
at every point with a _multiplicity function_
$\mu_{X}:X\longrightarrow\\{0,\dots,S\\}$. The goal is to derive a sampling
inequality of type

(1.2) $A_{p}\left\lVert f\right\rVert_{p}^{p}\leq\sum\limits_{x\in
X}\sum\limits_{s=0}^{\mu_{X}(x)}\left\lvert f^{(s)}(x)\right\rvert^{p}\leq
B_{p}\left\lVert f\right\rVert_{p}^{p}\qquad\text{for all }f\in
V^{p}(\varphi).$

Note that the number of derivatives depends on the sampling point $x$. If
(1.2) is satisfied, we call $(X,\mu_{X})$ a sampling set (or a set of stable
sampling) for $V^{p}(\varphi)$. Once a sampling inequality is available, one
can use standard reconstruction algorithms from frame theory to reconstruction
$f$ from $(X,\mu_{X})$.

Our final choice is a class of _generators_ for the shift-invariant space. We
suggest the use of PEB-splines as suitable and practical generators. These
have compact support, they are well understood in approximation theory [12,
34] and in numerics. In sampling theory, they lead to $\mathcal{O}(N)$
algorithms, when $N$ is the number of available data [18]. Most importantly
and in contrast to other classes of generators, for PEB-spline generators one
can prove optimal results. This is precisely the objective of this paper.

Before proving anything at all, one should understand what is possible. How
much information is necessary to recover a function from its samples? This
question is answered by a necessary density condition. For shift-invariant
spaces the relevant quantity is the weighted Beurling density defined as

(1.3)
$D^{-}(X,\mu_{X}):=\liminf\limits_{R\rightarrow\infty}\inf\limits_{y\in\mathbb{R}}\frac{1}{2R}\sum\limits_{x\in
X\cap[y-R,y+R]}(1+\mu_{X}(x)).$

If $\mu\equiv 0$, then $D^{-}(X,\mu_{X})=D^{-}(X)$ is just the usual lower
Beurling density. The weighted Beurling density counts the average number of
data per unit interval. The necessary density condition, first proved by
Landau [27] for bandlimited functions in full generality (with arbitrary
spectra and in higher dimensions), turns out to be a universal information-
theoretic bound for sampling. In the context of sampling with derivatives in
shift-invariant spaces it can be expressed as follows [21, Prop. 3.7.].

######  Theorem 1.1.

Assume that $\mu_{X}$ is bounded by $M$ and
$\varphi,\varphi^{\prime},\dots,\varphi^{(M)}$ satisfies a mild decay
condition, e.g., $|\varphi^{s}(x)|\leq C(1+|x|)^{-1-\varepsilon}$ for some
$\varepsilon>0$.

If $(X,\mu)$ is a sampling set for $V^{p}(\varphi)$, then

(1.4) $D^{-}(X,\mu_{X})\geq 1\,.$

The problem of sufficient conditions that guarantee the reconstruction of
$f\in V^{p}(\varphi)$ from its samples $(X,\mu_{X})$ is more interesting and
also more difficult. Under minimal conditions on the smoothness and decay of
$\varphi$ one can always show with elementary tools that any sufficiently
dense set is sampling [3]. For some types of generators, the density condition
(1.4) is (almost) a characterization. If $\varphi$ is either the cardinal sine
$\frac{\sin\pi x}{\pi x}$, whence $V^{2}(\varphi)=PW$, or the Gaussian or some
related function, then $D^{-}(X,\mu)>1$ is a sufficient condition for sampling
[10, 21]. For other generators, (1.4) may not provide a characterization, in
particular, if $\varphi$ has compact support, then the maximal gap between
sampling points is limited by the size of $\operatorname{supp}\varphi$,
whereas for given Beurling density large gaps are permitted. Therefore one
often uses different notions to measure the density of a set.

Contributions. For PEB-splines as generators, we will derive a set of
conditions that allow the construction of sampling sets arbitrarily close to
the necessary condition.

Our first theorem is somewhat technical because the sufficient conditions are
expressed in terms of local Schoenberg-Whitney conditions that are well-known
in spline theory. These conditions come close to a characterization of
sampling sets with derivatives but are not complete. We postpone the precise
formulation to Section 4, and state the most important consequence.

######  Theorem 1.2.

Let $\varphi$ be a PEB-spline of order $m$ with support in $[0,m]$. For every
$\varepsilon>0$ and given multiplicity sequence
$\mu=(\mu_{j})_{j\in\mathbb{Z}}$ with values in $\\{0,\dots m-1\\}$, there
exists a set $X=(x_{j})_{j\in\mathbb{Z}}$, such that $(X,\mu_{X})$ is sampling
for $V^{p}(\varphi)$, $\mu_{X}(x_{j})=\mu_{j}$ for all $j\in\mathbb{Z}$, and
$D^{-}(X,\mu_{X})<1+\varepsilon$.

Our second result uses the maximum gap between consecutive samples, also
called the mesh width or the covering density. The maximum gap of a set
$X=(x_{j})_{j\in\mathbb{Z}}\subseteq\mathbb{R}$ consisting of consecutive
samples is defined as

(1.5) $\mathfrak{mg}(X)=\sup_{j\in\mathbb{Z}}(x_{j+1}-x_{j})\,,$

and is used frequently to formulate conditions for sampling, e.g., in [3]. Our
new contribution is a weighted maximum gap that takes into consideration the
number of derivatives at each point. It is defined as

(1.6)
$\mathfrak{mg}(X,\mu_{X})=\max\left\\{\sup\limits_{j\in\mathbb{Z}}\frac{x_{j+1}-x_{j}}{1+\mu_{X}(x_{j})}\,,\,\sup\limits_{j\in\mathbb{Z}}\frac{x_{j+1}-x_{j}}{1+\mu_{X}(x_{j+1})}\right\\}$

The rationale for this definition is that several derivatives at a point $x$,
say $f^{(s)}(x),s=0,\dots,\mu_{X}(x)$, amount to $\mu_{X}(x)+1$ data, so the
next data point should be at distance $<\mu_{X}(x)+1$ from $x$, in other
words, $\frac{x_{j+1}-x_{j}}{\mu_{X}(x)+1}<1$. The precise formulation is as
follows.

######  Theorem 1.3 (Maximum Gap Theorem).

Let $\varphi$ be a PEB-spline of order $m$ and let $X\subseteq\mathbb{R}$ be a
separated set with multiplicity function $\mu_{X}:X\to\\{0,\dots,m-2\\}$. If
the weighted maximum gap of $(X,\mu_{X})$ satisfies

(1.7) $\mathfrak{mg}(X,\mu_{X})<1,$

then $(X,\mu_{X})$ is a sampling set for $V^{p}(\varphi)$.

In contrast to the more technical theorem, the maximum gap condition is easy
to understand and to verify.

As a special case, we mention sampling with the same number of derivatives at
every point.

######  Corollary 1.4.

Let $\varphi$ be a PEB-spline of order $m\geq 2$ and let
$X\subseteq\mathbb{R}$ be a separated set. Assume that $\mu_{X}$ is constant,
$\mu_{X}(x)=s<m-1$ for all $x\in X$. If

(1.8) $\mathfrak{mg}(X)<s+1,$

then $(X,\mu_{X})$ is a sampling set for $V^{p}(\varphi)$, $1\leq
p\leq\infty$.

In particular, a lattice $X=\alpha\mathbb{Z}$ is a sampling set for
$V^{p}(\varphi),1\leq p\leq\infty$, if and only if $\alpha<s+1$.

When using $\mathfrak{mg}(X,\mu_{X})$ as the relevant density and PEB-splines
as generators, then the weighted gap condition (1.7) is best possible in the
following sense. One can show that for $a>1$ and given multiplicity sequence
$(\mu_{j})_{j\in\mathbb{Z}}$ there exist sets $X=(x_{j})$, such that
$\mathfrak{mg}(X,\mu)=a$, but $(X,\mu)$ is not a sampling set.

It is easy to see that $D^{-}(X,\mu_{X})\geq\mathfrak{mg}(X,\mu_{X})^{-1}$.
This explains why the maximum gap condition is often problematic. So far there
is only a handful of generators [4, 14, 20, 21] for which the condition
$\mathfrak{mg}(X)<1$ or $\mathfrak{mg}(X,\mu_{X})<1$ has been proved to be
sufficient for sampling. In a surprisingly large number of publications, a
sufficient condition $\mathfrak{mg}(X)\leq\delta_{0}$ has been derived for
unknown or small $\delta_{0}$. Since the comparison to the necessary condition
is missing, more research is required to understand the relevance of such
results.

In the last section, we exploit a connection between sampling in shift-
invariant spaces and the theory of Gabor frames and derive some new results
about Gabor frames with PEB windows.

Methods. Following [4] we partition $\mathbb{R}$ into intervals of finite
length and then study the local problem. Since a PEB-spline of order
$m\in\mathbb{N}$ has support in the interval $[0,m]$, the restriction of a
function $f$ in $V^{p}(\varphi)$ to an interval $[a,b),a,b\in\mathbb{Z}$, sees
only the finite sum

$f(x)=\sum_{\ell=a-m+1}^{b-1}c_{\ell}\varphi(x-\ell)\qquad\text{ for
}x\in[a,b)\,.$

This is then a finite-dimensional problem, and the intervening matrix is the
_collocation matrix_ associated to the PEB-spline $\varphi$. Its invertibility
is well understood and can be expressed with the help of the Schoenberg-
Whitney conditions on the sampling points in $[a,b]$. The characterization of
invertible collocation matrices is one of the cornerstones of spline theory
[34], and our proofs pay homage to this fundamental result. We develop two
approaches centred around the collocation matrix. In one approach we force
uniform bounds on the collocation matrix and then glue together the local
reconstructions. In an alternative approach, we use Beurling’s technique of
weak limits to arrive at a sampling theorem.

The paper is organized as follows. Section 2 introduces shift-invariant spaces
and sampling sets and prepares some technical tools for sampling with
derivatives. Section 3 is dedicated to the basic theory of ECC-systems and
B-splines with a special focus on collocation matrices for sampling with
derivatives. The first main result and the existence of sampling sets near the
optimal density are proven in Section 4. Section 5 is devoted to Beurling’s
technique adapted to sampling in shift-invariant spaces. With this at hand, we
prove the main sampling with the maximum gap condition in Section 6. We
conclude with Section 7 with a discussion of the two methods and Section 8,
where we derive the implications of the two sampling theorems for Gabor
systems. A technical result is postponed to the appendix.

##  2\. Shift-invariant spaces and sampling

We begin with properties of shift-invariant spaces and various formulations of
the sampling problem.

###  2.1. (Vector-valued) shift-invariant spaces

Let us consider a shift-invariant space $V^{p}(\varphi)$. Since each element
$f\in V^{p}(\varphi)$ is represented as
$f=\sum_{\ell\in\mathbb{Z}}c_{\ell}T_{\ell}\varphi$ for some sequence
$\left(c_{\ell}\right)_{\ell\in\mathbb{Z}}\in\ell^{p}(\mathbb{Z})$, the
desirable generators are those which come with the norm equivalence

(2.1) $A_{p}\left\lVert
c\right\rVert_{\ell^{p}}\leq\Big{\lVert}\,\sum_{\ell\in\mathbb{Z}}c_{\ell}T_{\ell}\varphi\,\Big{\rVert}_{L^{p}}\leq
B_{p}\left\lVert c\right\rVert_{\ell^{p}},\qquad c\in\ell^{p}(\mathbb{Z}),$

where the constants $A_{p},B_{p}>0$ depend only on $p$. If this holds, we say
that $\varphi$ has $p$-stable integer translates (shifts) and denote (2.1) as
$\left\lVert f\right\rVert_{L^{p}}\asymp\left\lVert c\right\rVert_{\ell^{p}}$.
If norm equivalence holds for all $p\in[1,\infty]$, we omit the reference to
$p$ and speak of stable integer translates and a stable generator.

The following theorem summarizes some well-known conditions for stability.

######  Theorem 2.1.

Let $\varphi:\mathbb{R}\to\mathbb{C}$ be a bounded, compactly supported
function. Then the following statements are equivalent:

  1. (i)

The generator $\varphi$ has stable translates.

  2. (ii)

The generator $\varphi$ has stable $\infty$-translates.

  3. (iii)

The translates $\\{T_{\ell}\,\varphi:\ell\in\mathbb{Z}\\}$ are
$\ell^{\infty}$-independent, i.e.,

(2.2) $\sum\limits_{\ell\in\mathbb{Z}}c_{\ell}\,T_{\ell}\,\varphi\not\equiv
0\qquad\text{ for all }c\in\ell^{\infty}(\mathbb{Z})\setminus\\{0\\}.$

  4. (iv)

The Fourier transform $\widehat{\varphi}$ does not have a real $1$-periodic
zero, i.e.,

(2.3)
$0<\sum\limits_{\ell\in\mathbb{Z}}\left\lvert\hat{\varphi}(\omega+\ell)\right\rvert^{2}\qquad\text{
for all }\omega\in\mathbb{R}.$

We refer to Ron’s survey article [33]. The equivalences follow from the
results in [23, Thm. 3.5] and [33, Thm. 29]. For further references, the
reader can consult [28, Eq. (9)], [23, Thm. 3.3]. For sampling in shift-
invariant spaces the stability condition is explained and used in [4, 5] and
[20, Sec. 2].

For the special case of a bounded, compactly supported generator $\varphi$
above with $\mathrm{supp}(\varphi)\subseteq[-L,L]$, the important lower
inequality is almost trivial:

(2.4) $\left\lVert
f\right\rVert_{L^{p}}^{p}\leq\int_{\mathbb{R}}\sum\limits_{\ell\in\mathbb{Z}}\left\lvert
c_{\ell}T_{\ell}\varphi(t)\right\rvert^{p}\,dt\leq\sum\limits_{k\in\mathbb{Z}}\int\limits_{K}^{K+1}\sum\limits_{\ell=K-2L+1}^{L}\left\lvert
c_{\ell}\right\rvert^{p}\left\lVert\varphi\right\rVert_{\infty}\,dt=4L\left\lVert\varphi\right\rVert_{\infty}\left\lVert
c\right\rVert_{p}^{p},$

if $p<\infty$, with the analogous estimate for $p=\infty$.

Following [21, Sec. 2], we further consider vector-valued shift-invariant
spaces

(2.5)
$V^{p}(\Phi):=\left\\{\sum\limits_{\ell\in\mathbb{Z}}c_{\ell}\,T_{\ell}\,\Phi:c\in\ell^{p}(\mathbb{Z})\right\\}\subseteq(L^{p}(\mathbb{R}))^{S+1}$

generated by a vector
$\Phi=\left(\Phi^{0},\dots,\Phi^{S}\right)^{t}\in\left(L^{1}(\mathbb{R})\right)^{S+1}$
with norm

(2.6) $\left\lVert F\right\rVert_{p}=\left(\sum\limits_{s=0}^{S}\left\lVert
F^{s}\right\rVert_{L^{p}}^{p}\right)^{1/p},$

with the usual modification for $p=\infty$. We will assume that $\Phi$ has
stable integer shifts, i.e.,

(2.7)
$\Big{\lVert}\,\sum\limits_{\ell\in\mathbb{Z}}c_{\ell}\,T_{\ell}\,\Phi\,\Big{\rVert}_{p}\asymp\left\lVert
c\right\rVert_{p}\qquad\text{ for all }c\in\ell^{p}(\mathbb{Z}).$

###  2.2. Sampling sets

To accommodate derivatives and multiplicities in sampling, we consider tuples
of sets $\overrightarrow{X}=\left(X^{0},\dots,X^{S}\right)$ with
$X^{s}\subseteq\mathbb{R}$, $0\leq s\leq S$. Assume that the generator has
stable integer shifts and that the shift-invariant space $V^{p}(\Phi)$
contains tuples of functions $F=(F_{0},\dots,F^{S})$ which are pointwise well-
defined. In the following, $V^{p}(\Phi^{s})$ contains only continuous or
piecewise continuous functions with well-defined one-sided limits at the
finitely many discontinuities. We say that $\overrightarrow{X}$ is a _sampling
set_ for $V^{p}(\Phi)$, $p\in[1,\infty]$, if

(2.8) $\sum\limits_{s=0}^{S}\sum\limits_{x\in X^{s}}\left\lvert
F^{s}(x)\right\rvert^{p}\asymp\left\lVert F\right\rVert_{p}^{p}\qquad\text{for
all }F\in V^{p}(\Phi),$

where the inequality constants depend only on $p$ and $\Phi$ (with the usual
adaptation for $p=\infty$). A weaker notion is that of a uniqueness set. We
call $\overrightarrow{X}$ a _uniqueness set_ if the linear operator
$F\mapsto(F^{0}|_{X^{0}},\dots,F^{s}|_{X^{s}})$ is injective on $V^{p}(\Phi)$,
that is,

(2.9) ${F^{s}}\big{|}_{{X^{s}}}\equiv 0\ \ \text{ for all
}s\in\\{0,\dots,S\\}\qquad\Leftrightarrow\qquad F\equiv 0.$

The case $S=0$ is the standard case of sampling sets. As usual, the upper
bound for sampling is easy to obtain in a general setting, whereas, for the
lower bound, the specific features of the generator have to be exploited.

######  Theorem 2.2.

Let $\varphi$ be a compactly supported, piecewise continuous function with
finitely many jump discontinuities and assume that $\varphi$ has stable
integer translates. Then

  1. (i)

All functions in $V^{p}(\varphi)$ are piecewise continuous. Furthermore, the
set of discontinuities is separated, consisting only of jump discontinuities.

  2. (ii)

The sampling operator $f\to f|_{X}$ is bounded on $V^{p}(\varphi)$. Here the
samples at discontinuities are to be understood as samples of the right-sided
limits.

The analogous statement holds with left-sided limits.

###### Proof.

(i) A series $\sum_{\ell\in\mathbb{Z}}c_{\ell}T_{\ell}\varphi$ is locally
finite, so it inherits the properties of the generator.

Concerning (ii), we refer the reader to [5, Thm. 3.1(iii)] for the proof
111Note that the proof does not require the continuity of $\varphi$.. ∎

To relate sampling in vector-valued spaces to sampling with derivatives with a
multiplicity function $\mu_{X}:X\to\\{0,\dots,S\\}$, we set
$\Phi\coloneqq(\varphi,\varphi^{(1)},\dots,\varphi^{(S)})$ and

(2.10) $X^{s}\coloneqq\\{x\in X:\mu_{X}(x)\geq s\\},\qquad 1\leq s\leq S.$

In this case, the inequality (2.8) for vector-valued functions is the same as
the sampling inequality

(2.11) $\sum\limits_{x\in X}\sum\limits_{s=0}^{\mu_{X}(x)}\left\lvert
f^{(s)}(x)\right\rvert^{p}\asymp\left\lVert
f\right\rVert_{p}^{p}\qquad\text{for all }f\in V^{p}(\varphi).$

##  3\. Some Spline Theory

In this section, we introduce ECC-systems and Chebyshev B-splines and cite and
prove statements about collocation matrices arising in the sampling problem.
We follow the terminology in [34].

###  3.1. ECC-systems and Chebyshev B-splines

We call a system of functions $u_{\ell}:[a,b]\longrightarrow\mathbb{C}$,
$1\leq\ell\leq m$, a Chebyshev system, if the collocation matrix with entries

(3.1) $\left(\,u_{\ell}\,(t_{i})\,\right)_{1\leq i,\ell\leq m}$

has a strictly positive determinant for all $a\leq t_{1}<\dots<t_{m}\leq b$.
It follows from the definition that a Chebyshev system is linearly
independent. The span of $\left(u_{1},\dots,u_{m}\right)$ is called a
Chebyshev space.

The definition of a Chebyshev system implies immediately the following
property of the zero set. One of the main properties of these spaces is the
number of solutions of $f(x)=0$.

######  Lemma 3.1.

If $\left(u_{1},\dots,u_{m}\right)$ is a Chebyshev system, then either
$(c_{\ell})_{1\leq\ell\leq m}=0$ or

(3.2)
$\\#\bigg{\\{}t\in[a,b]:\sum\limits_{\ell=1}^{m}c_{\ell}u_{\ell}(t)=0\bigg{\\}}\leq
m-1.$

Moving further towards splines, we choose positive weight functions $w_{s}\in
C^{m-s}[a,b]$ on an interval $[a,b]\subseteq\mathbb{R}$ and define

(3.3) $\displaystyle u_{1}(x)$ $\displaystyle=w_{1}(x),$ (3.4) $\displaystyle
u_{2}(x)$ $\displaystyle=w_{1}(x)\int\limits_{a}^{x}w_{2}(x_{2})\,dx_{2},$
(3.5) $\displaystyle\vdots$ (3.6) $\displaystyle u_{m}(x)$
$\displaystyle=w_{1}(x)\int\limits_{a}^{x}w_{2}(x_{2})\int\limits^{x_{2}}_{a}w_{3}(x_{3})\dots\int\limits_{a}^{x_{m-1}}w_{m}(x_{m})\,dx_{m}\dots\,dx_{2}.$

These are functions in $\mathbb{C}^{m-1}[a,b]$ and form a so-called _extended
complete Chebyshev (ECC) system_ on $[a,b]$. This means that for all $1\leq
D\leq m$ and all $t_{1}\leq\dots\leq t_{D}\in[a,b]$ the _collocation matrix of
Hermite interpolation_ , i.e., the matrix with entries

(3.7) $\big{(}u_{\ell}^{(d_{i})}\,(t_{i})\big{)}_{1\leq i,\ell\leq D},$

where

(3.8) $d_{i}:=\max\left\\{\ell:t_{i}=\dots=t_{i-\ell}\right\\},\qquad 1\leq
i\leq D,$

has a positive determinant. This is the matrix that arises in the Hermite
interpolation problem

(3.9) $f=\sum\limits_{\ell=1}^{D}c_{\ell}u_{\ell},\quad
f^{(d_{i})}(t_{i})=\xi_{i},\qquad 1\leq i\leq D.$

The associated ECC-space is given by
$\mathcal{U}_{m}:=\operatorname{span}_{\mathbb{C}}{(u_{1},\dots,u_{m})}.$ As
the simplest example (and the motivation), we consider $w_{1}(x)=1$ and
$w_{s}=s-1$, $2\leq s\leq m$. In this case, we obtain the polynomials
$u_{s}=(x-a)^{s-1}$, $1\leq s\leq m$, i.e., the ECC-space
$\mathcal{U}_{m}=\mathcal{P}_{m}$ is the space of polynomials of degree at
most $m-1$. One of the particularly useful properties of polynomial spaces is
the self-containment with respect to the derivatives, i.e.,

(3.10) $p^{(s)}\in\mathcal{P}_{m-s},\qquad 0\leq s\leq m-1,\quad
p\in\mathcal{P}_{m}.$

To preserve this property for ECC-systems, one replaces the usual derivatives
with related differential operators. We define $D_{0}\,f=f$ and

(3.11) $D_{s}\,f:=\tfrac{d}{dx}\,\tfrac{f}{w_{s}},\quad
L_{s}:=D_{s}D_{s-1}\dots D_{0},\quad 0\leq s\leq m.$

By Leibniz’s rule, $L_{s}$ is a differential operator of order $s$ with
variable coefficients.

Just as with the polynomials and the well-known B-splines, we can now consider
Chebyshev splines. For an ECC-system $(u_{1},\dots,u_{m})$ on $[a,b]$ and
knots $a=y_{1}<\dots<y_{D+m}=b$, a _spline_ with these knots is a function
$B\in C^{m-2}(\mathbb{R})$ satisfying

(3.12) $B|_{(y_{\ell},y_{\ell+1})}\in\mathcal{U}_{m},\qquad 1\leq\ell\leq
D+m-1.$

This condition implies that $B$ is piecewise $C^{m-1}$ and at the knots
$B^{(m-1)}$ has jump discontinuities and only one-sided derivatives exist. The
integer $m$ is the order of the spline. In this case we define $L_{m-1}B$ at
the knots $y_{i}$ to be the right-sided limit

(3.13) $L_{m-1}B(y_{i})\coloneqq\lim\limits_{x\searrow y_{i}}L_{m-1}B(x).$

A major result in spline theory is the existence of Chebyshev B-splines (CB-
splines) associated to knots $a\leq y_{1}<\dots<y_{m+D}\leq b$. Precisely, the
CB-splines are the splines $\big{(}B_{m}^{\ell}\big{)}_{1\leq\ell\leq D}$ with
controlled support

(3.14) $\operatorname{supp}(B_{m}^{\ell})=[y_{\ell},y_{\ell+m}],\qquad
1\leq\ell\leq D.$

The CB-splines $B_{m}^{\ell}$ are unique up to normalization. The traditional
construction is based on divided differences and can be found in full detail
in Schumaker’s book [34, Sec. 9.4].

We conclude by considering matrices associated to CB-splines. The definition
of ECC-systems calls for the collocation matrix with the standard derivatives.
We adapt this to the new differential operators $L_{s}$. Given $t_{1}\leq
t_{2}\leq\dots\leq t_{D}$, let

(3.15) $d_{i}:=\max\left\\{\ell:t_{i}=\dots=t_{i-\ell}\right\\}\leq m-1,\qquad
1\leq i\leq D.$

Let $(B_{m}^{1},\dots,B_{m}^{D})$ be CB-splines associated with the knots
$y_{1}<y_{2}<\dots<y_{m+D}$. We consider the collocation matrix

(3.16) $\big{(}L_{d_{i}}B_{m}^{\ell}(t_{i})\big{)}_{1\leq i,\ell\leq D}.$

This is the matrix tied to the Hermite interpolation problem

(3.17) $f=\sum\limits_{\ell=1}^{D}c_{\ell}B^{\ell}_{m},\quad
L_{d_{i}}f(t_{i})=\zeta_{i},\qquad 1\leq i\leq D.$

We can determine exactly when this matrix is invertible [34, Thm. 9.33], cf.
[34, Thm. 4.67].

######  Theorem 3.2 (Interlacing property, Schoenberg-Whitney conditions).

The determinant of the collocation matrix in (3.16) is nonnegative. It is
positive if and only if

(3.18) $t_{i}\in\begin{cases}(y_{i},y_{i+m}),&\text{ if }d_{i}<m-1\\\
[y_{i},y_{i+m}),&\text{ if }d_{i}=m-1,\end{cases}\qquad 1\leq i\leq D.$

The theorem says that it suffices for $t_{i}$ to be in the support of
$B_{m}^{i}$. The invertibility of the collocation matrix is independent of the
normalization of the CB-splines, as the normalization corresponds to
multiplying the collocation matrix with an invertible diagonal matrix.

######  Example 3.3.

As a small example, we consider CB-splines of order 5. Let the knots be
$0,1,\dots,10$. Then the splines $B^{0}_{5},\dots,B^{5}_{5}$ are supported on

(3.19) $(0,5),\quad(1,6),\quad(2,7),\quad(3,8),\quad(4,9),\quad\text{ and
}\quad(5,10),$

respectively. If we choose $t_{1}=t_{2}=0.5$, $t_{3}=4$,
$t_{4}=t_{5}=t_{6}=7.7$, then the corresponding multiplicities are

(3.20) $d=\begin{pmatrix}0&1&0&0&1&2\end{pmatrix}.$

The matrix is not invertible because $t_{2}=0.5$ is not in $(1,6)$. Visually,
the issue is clear:

(3.21)
$\displaystyle\begin{pmatrix}B_{5}^{0}(0.5)&B_{5}^{1}(0.5)&B_{5}^{2}(0.5)&B_{5}^{3}(0.5)&B_{5}^{4}(0.5)&B_{5}^{5}(0.5)\\\\[4.30554pt]
L_{1}B_{5}^{0}(0.5)&L_{1}B_{5}^{1}(0.5)&L_{1}B_{5}^{2}(0.5)&L_{1}B_{5}^{3}(0.5)&L_{1}B_{5}^{4}(0.5)&L_{1}B_{5}^{5}(0.5)\\\\[4.30554pt]
B_{5}^{0}(4)&B_{5}^{1}(4)&B_{5}^{2}(4)&B_{5}^{3}(4)&B_{5}^{4}(4)&B_{5}^{5}(4)\\\\[4.30554pt]
B_{5}^{0}(7.7)&B_{5}^{1}(7.7)&B_{5}^{2}(7.7)&B_{5}^{3}(7.7)&B_{5}^{4}(7.7)&B_{5}^{5}(7.7)\\\\[4.30554pt]
L_{1}B_{5}^{0}(7.7)&L_{1}B_{5}^{1}(7.7)&L_{1}B_{5}^{2}(7.7)&L_{1}B_{5}^{3}(7.7)&L_{1}B_{5}^{4}(7.7)&L_{1}B_{5}^{5}(7.7)\\\\[4.30554pt]
L_{2}B_{5}^{0}(7.7)&L_{1}B_{5}^{1}(7.7)&L_{2}B_{5}^{2}(7.7)&L_{2}B_{5}^{3}(7.7)&L_{2}B_{5}^{4}(7.7)&L_{2}B_{5}^{5}(7.7)\end{pmatrix}$
(3.22) $\displaystyle=$
$\displaystyle\begin{pmatrix}B_{5}^{0}(0.5)&0&0&0&0&0\\\\[4.30554pt]
L_{1}B_{5}^{0}(0.5)&0&0&0&0&0\\\\[4.30554pt]
B_{5}^{0}(4)&B_{5}^{1}(4)&B_{5}^{2}(4)&B_{5}^{3}(4)&0&0\\\\[4.30554pt]
0&0&0&B_{5}^{3}(7.7)&B_{5}^{4}(7.7)&B_{5}^{5}(7.7)\\\\[4.30554pt]
0&0&0&L_{1}B_{5}^{3}(7.7)&L_{1}B_{5}^{4}(7.7)&L_{1}B_{5}^{5}(7.7)\\\\[4.30554pt]
0&0&0&L_{2}B_{5}^{3}(7.7)&L_{2}B_{5}^{4}(7.7)&L_{2}B_{5}^{5}(7.7)\end{pmatrix}$
(3.23) $\displaystyle=$ $\displaystyle\begin{pmatrix}\ *&0&0&0&0&0\
\\\\[4.30554pt] \ *&0&0&0&0&0\ \\\\[4.30554pt] \ *&*&*&*&0&0\ \\\\[4.30554pt]
\ 0&0&0&*&*&*\ \\\\[4.30554pt] \ 0&0&0&*&*&*\ \\\\[4.30554pt] \ 0&0&0&*&*&*\
\end{pmatrix}$

If instead we sample at $t_{1}=t_{2}=1.5\in(1,5)$, $t_{3}=4$,
$t_{4}=t_{5}=t_{6}=7.7$, all points are positioned correctly: $4$ is in
$(2,7)$ and $7.7$ lies in $(3,8)$, $(4,9)$ and $(5,10)$. In this case, the
collocation matrix is invertible.

###  3.2. Periodic exponential B-splines

We have now gathered all the necessary background on CB-splines and continue
with the special case of CB-splines which are shifts of a single CB-spline.
For this, we require the following structure:

  1. (S1)

the knots are the integers $y_{i}=i-1$, $i\in\mathbb{Z},$222The choice
$y_{i}=i-1$ instead of $y_{i}=i$ is for the support condition (3.25) and
consistency of Theorem 3.2 with later applications.

  2. (S2)

the weights $w_{s}$, $1\leq s\leq m$ are given by

(3.24) $w_{s}(x)=e^{\gamma_{s}x}r_{s}(x),\qquad\gamma_{s}\in\mathbb{R},\qquad
T_{\ell}r_{s}=r_{s}>0\quad\text{for all }\ell\in\mathbb{Z}.$

CB-splines in this form are called periodic exponential B-splines (PEB-
splines) introduced in [7, 32]. If $r_{s}\equiv 1$ for all $1\leq s\leq m$,
then we refer to them as exponential B-splines (EB-splines). The benefit of
(S1) and (S2) is the fact that the CB-spline associated with the knots
$y_{\ell},\dots,y_{\ell+m}$ is (up to scaling) uniquely determined by any
other of the B-splines, e.g., by $\varphi\coloneqq B_{m}^{1}$. It is clear
that any shift $T_{\ell}\varphi$ is a Chebyshev spline with

(3.25)
$\operatorname{supp}(T_{\ell}\varphi)=[0+\ell,m+\ell],\qquad\ell\in\mathbb{Z}.$

By the uniqueness, $(T_{\ell}\varphi)_{\ell\in\mathbb{Z}}$ differs up to
positive factors $\left(\theta_{\ell}\right)_{\ell\in\mathbb{Z}}$ from the
classically constructed CB-splines with divided differences
$\left(B_{m}^{\ell}\right)_{\ell\in\mathbb{Z}}$ [34, Sec. 9.4]. The
invertibility of the collocation matrix in Theorem 3.2 is now tied to the
integers. We take a closer look at the associated differential operators. For
ease of reading, we define for all $1\leq s\leq m$ the functions

(3.26) $q_{s}(x)\coloneqq\frac{1}{r_{s}(x)},\quad\text{so that}\quad
w_{s}^{-1}(x)=e^{-\gamma_{s}x}q_{s}(x).$

Since both $q_{s}$ and $w_{s}^{-1}$ are strictly positive functions, they are
in the same differentiability class. Moreover, $q_{s}$ is periodic. A simple
calculation shows

(3.27)
$\begin{split}D_{s}f(x)&=\tfrac{d}{dx}\left[e^{-\gamma_{s}x}q_{s}(x)\,\cdot
f(x)\right]\\\ &=-\gamma_{s}e^{-\gamma_{s}x}q_{s}(x)\,\cdot
f(x)+e^{-\gamma_{s}x}q_{s}^{\prime}(x)\,\cdot
f(x)+e^{-\gamma_{s}x}q_{s}(x)\,\cdot f^{\prime}(x)\\\
&=\tfrac{1}{w_{s}(x)}\,\Big{(}\Big{(}-\gamma_{s}+\tfrac{q_{s}^{\prime}(x)}{q_{s}(x)}\Big{)}\cdot\mathrm{id}+\tfrac{d}{dx}\Big{)}f(x)\end{split}$

and

(3.28) $\displaystyle D_{s}\,T_{\ell}\,f(x)$
$\displaystyle=\tfrac{d}{dx}\left[\ e^{-\gamma_{s}x}q_{s}(x)f(x-\ell)\right]$
(3.29) $\displaystyle=\tfrac{d}{dx}\left[e^{-\gamma_{s}\ell}\cdot
e^{-\gamma_{s}(x-\ell)}q_{s}(x-\ell)f(x-\ell)\right]$ (3.30)
$\displaystyle=e^{-\gamma_{s}\ell}\
T_{\ell}\,D_{s}\,f(x),\qquad\ell\in\mathbb{Z}.$

We obtain the commutator rule

(3.31)
$L_{s}\,T_{\ell}f=e^{-\ell\,\sum_{k=1}^{s}\gamma_{k}}\,T_{\ell}\,L_{s}\,f.$

The differential operators $L_{s}$ are natural in the context of ECC-systems,
but in practice, information usually comes as samples of standard derivatives.
Therefore, in the following statements we derive the relation between the
standard derivatives and the differential operators associated with an ECC-
system.

######  Lemma 3.4.

Let $r_{s}\in C^{m-s}(\mathbb{R})$, $1\leq s\leq m$, be $1$-periodic, strictly
positive functions and $\gamma_{1},\dots,\gamma_{m}\in\mathbb{R}$. Let
$L_{1},\dots,L_{m}$ be the differential operators associated with the ECC-
system induced by

(3.32) $w_{s}(x)=e^{\gamma_{s}x}r_{s}(x),\qquad x\in\mathbb{R}.$

Set for $0\leq s\leq m$

(3.33) $v_{s}(x)=\prod\limits_{k=1}^{s-1}\frac{1}{w_{k}(x)},\qquad
x\in\mathbb{R}.$

For each $0\leq s\leq m$ there exist periodic functions $p_{s,k}\in
C^{m-1-s}(\mathbb{R})$ satisfying

(3.34)
$L_{s}f(x)=v_{s}(x)\cdot\bigg{(}f^{(s)}(x)+\sum\limits_{k=0}^{s-1}p_{s,k}(x)\cdot
f^{(k)}(x)\bigg{)}.$

In a matrix notation, this can be written as

(3.35) $\big{(}L_{0},\ L_{1},\ L_{2}\ ,\dots,\
L_{s}\big{)}^{t}=\mathcal{V}_{s}\cdot\mathcal{P}_{s}\cdot\Big{(}\operatorname{id},\
\tfrac{d}{dx},\ \tfrac{d^{2}}{dx^{2}},\ \dots,\
\tfrac{d^{s}}{dx^{s}}\Big{)}^{t},$

where $\mathcal{V}_{s}:\mathbb{R}\to\mathbb{R}^{(s+1)\times(s+1)}$ is the
diagonal matrix

(3.36) $\mathcal{V}_{s}(x)=\mathrm{diag}(v_{0}(x),\dots,v_{s}(x))$

and $\mathcal{P}_{s}:\mathbb{R}\to\mathbb{R}^{(s+1)\times(s+1)}$ is the
periodic lower triangular matrix

(3.37) $\mathcal{P}_{s}(x)=\begin{pmatrix}1&0&0&\cdots&0\\\\[4.30554pt]
p_{1,0}(x)&1&0&\cdots&0\\\\[4.30554pt]
p_{2,0}(x)&p_{2,1}(x)&1&\cdots&0\\\\[4.30554pt]
\vdots&\vdots&\vdots&\ddots&\vdots\\\\[4.30554pt]
p_{s,0}(x)&p_{s,1}(x)&\cdots&p_{s,s-1}(x)&1\end{pmatrix},\qquad
x\in\mathbb{R}.$

In particular, $\mathcal{V}_{s}(x)$ and $\mathcal{P}_{s}(x)$ are invertible
for all $x\in\mathbb{R}$ and are continuous on $\mathbb{R}$.

###### Proof.

We prove this formula by induction. Clearly, $f^{(0)}=L_{0}f$. For the
induction step we use (3.27) and $w_{k}^{-1}(x)=e^{-\gamma_{k}x}q_{k}(x)$.

(3.38) $\displaystyle L_{s+1}f$
$\displaystyle=D_{s+1}L_{s}f\overset{\scriptscriptstyle\eqref{eq:D_s_exact}}{=}\frac{1}{w_{s+1}}\,\bigg{(}\bigg{(}-\gamma_{s+1}+\frac{q_{s+1}^{\prime}}{q_{s+1}}\bigg{)}\cdot
L_{s}f+(L_{s}f)^{\prime}\bigg{)}$ (3.39)
$\displaystyle\underset{\scriptscriptstyle\text{step}}{\overset{\scriptscriptstyle\text{Induction}}{{\color[rgb]{1,1,1}=}}}$
$\displaystyle=\frac{1}{w_{s+1}}\,\bigg{\\{}\bigg{(}-\gamma_{s+1}+\frac{q_{s+1}^{\prime}}{q_{s+1}}\bigg{)}\cdot
v_{s}\cdot\sum\limits_{k=0}^{s}p_{s,k}\cdot f^{(k)}$ (3.40)
$\displaystyle\qquad\quad+v_{s}\cdot\bigg{(}\,\sum\limits_{k=0}^{s}p_{s,k}\cdot
f^{(k)}\bigg{)}^{\prime}+v_{s}^{\prime}\cdot\sum\limits_{k=0}^{s}p_{s,k}\cdot
f^{(k)}\bigg{)}$ (3.41)
$\displaystyle=\frac{1}{w_{s+1}}\,\bigg{(}-\gamma_{s+1}+\frac{q_{s+1}^{\prime}}{q_{s+1}}\bigg{)}\cdot
v_{s}\cdot\sum\limits_{k=0}^{s}p_{s,k}\cdot f^{(k)}\bigg{\\}}$ (3.42)
$\displaystyle\qquad\quad+\frac{v_{s}}{w_{s+1}}\cdot\sum\limits_{k=0}^{s}p_{s,k}^{\prime}\cdot
f^{(k)}+\frac{v_{s}}{w_{s+1}}\cdot\sum\limits_{k=0}^{s}p_{s,k}\cdot f^{(k+1)}$
(3.43)
$\displaystyle\quad\quad\quad+\frac{v_{s}}{w_{s+1}}\cdot\sum\limits_{n=1}^{s}\bigg{(}-\gamma_{n}+\frac{q_{n}^{\prime}}{q_{n}}\bigg{)}\cdot\sum\limits_{k=0}^{s}p_{s,k}\cdot
f^{(k)},$

where we used

(3.44) $v_{s}^{\prime}=\sum\limits_{n=1}^{s}{w_{n}}\cdot
v_{s}\cdot\,\left(w_{n}^{-1}\right)^{\prime}=\sum\limits_{n=1}^{s}{w_{n}}\cdot
v_{s}\cdot\,w_{n}^{-1}\bigg{(}-\gamma_{n}+\frac{q^{\prime}_{n}}{q_{n}}\bigg{)}=v_{s}\cdot\sum\limits_{n=1}^{s}\bigg{(}-\gamma_{n}+\frac{q^{\prime}_{n}}{q_{n}}\bigg{)}.$

for the last equality. We recall that $v_{s+1}=\frac{v_{s}}{w_{s+1}}$ and
$p_{s,s}=1$ by the induction hypothesis. We set $p_{s,-1}\coloneqq 0$ to
obtain

(3.45) $\displaystyle L_{s+1}f$
$\displaystyle=v_{s+1}\cdot\bigg{(}f^{(s+1)}+\sum\limits_{k=0}^{s}\left(\left(-\gamma_{s+1}+\frac{q_{s+1}^{\prime}}{q_{s+1}}\right)\cdot
p_{s,k}+p_{s,k}^{\prime}+p_{s,k-1}\right)\cdot f^{(k)}\bigg{)}$ (3.46)
$\displaystyle\quad\quad\quad+v_{s+1}\sum\limits_{n=1}^{s}\left(-\gamma_{n}+\frac{q_{n}^{\prime}}{q_{n}}\right)\cdot\sum\limits_{k=0}^{s}p_{s,k}\cdot
f^{(k)}$ (3.47) $\displaystyle=v_{s+1}\cdot
f^{(s+1)}+v_{s+1}\cdot\sum\limits_{k=0}^{s}\underbrace{\left(p_{s,k}^{\prime}+p_{s,k-1}+p_{s,k}\sum\limits_{n=1}^{s+1}\left(-\gamma_{n}+\frac{q_{n}^{\prime}}{q_{n}}\right)\,\right)}_{=:p_{s+1,k}}\cdot
f^{(k)},$

concluding the proof. ∎

######  Lemma 3.5.

Let the knots $y_{i}=i-1,i=1,\dots,D$ and the weights $w_{s}$, $1\leq s\leq m$
be given and let $B_{m}^{1},\dots,B_{m}^{D}=T_{D-1}B_{m}^{1}$ be the
associated PEB-splines. Then the determinant of the collocation matrix

(3.48) $M\begin{pmatrix}t_{1},\dots,t_{D}\\\
B_{m}^{1},\dots,B_{m}^{D}\end{pmatrix}\coloneqq\Big{(}(B_{m}^{\ell})^{(d_{i})}(t_{i})\Big{)}_{1\leq
i,\ell\leq D}=\Big{(}T_{\ell-1}(B_{m}^{1})^{(d_{i})}(t_{i})\Big{)}_{1\leq
i,\ell\leq D}$

is nonnegative. It is positive if and only if

(3.49) $t_{i}\in\begin{cases}(i-1,i+m-1),&d_{i}<m-1\\\
[i-1,i+m-1),&d_{i}=m-1,\end{cases}\qquad 1\leq i\leq D.$

###### Proof.

We first address the second equality in the definition of the matrix $M$. As
already noted in (3.25), the PEB-splines for equispaced knots are shifts of a
single PEB-spline, namely, $B_{m}^{\ell}=T_{\ell-1}B_{m}^{1}$. Since
differentiation and translation operators commute, we have
$(B_{m}^{\ell})^{(d_{i})}(t_{i})=T_{\ell-1}(B_{m}^{1})^{(d_{i})}(t_{i})$.

Let as before $d_{i}$ count the previous occurrences of the point $t_{i}$,
i.e.,

(3.50) $d_{i}=\max\left\\{\ell:t_{i}=\dots=t_{i-\ell}\right\\}\leq m-1,\qquad
1\leq i\leq D.$

Then $t=(t_{1},t_{2},t_{3},\dots,t_{D})$ can be rewritten as

$t=\Big{(}\underbrace{t_{1},\dots,t_{1}}_{{d_{1}+1}\text{-times}},\underbrace{t_{d_{1}+2},\dots,t_{d_{1}+2}}_{d_{2}+1\text{-times}},\underbrace{t_{d_{1}+d_{2}+3},\dots,t_{d_{1}+d_{2}+3}}_{d_{3}+1\text{-times}},\dots,\underbrace{t_{D},\dots,t_{D}}_{{d_{N}+1}\text{-times}}\Big{)}.$

We define the block-diagonal matrices

(3.51)
$\mathbf{P}(t):=\begin{pmatrix}\mathcal{P}_{d_{1}}(t_{1})&0&0&\cdots&0\\\\[4.30554pt]
0&\mathcal{P}_{d_{2}}(t_{d_{1}+2})&0&\cdots&0\\\\[4.30554pt]
0&0&\mathcal{P}_{d_{3}}(t_{d_{1}+d_{2}+3})&\cdots&0\\\\[4.30554pt]
\vdots&\vdots&\vdots&\ddots&\vdots\\\\[4.30554pt]
0&0&\cdots&0&\mathcal{P}_{d_{N}}(t_{D})\end{pmatrix}\in\mathbb{R}^{D\times
D},$ (3.52)
$\mathbf{V}(t):=\begin{pmatrix}\mathcal{V}_{d_{1}}(t_{1})&0&0&\cdots&0\\\\[4.30554pt]
0&\mathcal{V}_{d_{2}}(t_{d_{1}+2})&0&\cdots&0\\\\[4.30554pt]
0&0&\mathcal{V}_{d_{3}}(t_{d_{1}+d_{2}+3})&\cdots&0\\\\[4.30554pt]
\vdots&\vdots&\vdots&\ddots&\vdots\\\\[4.30554pt]
0&0&\cdots&0&\mathcal{V}_{d_{N}}(t_{D})\end{pmatrix}\in\mathbb{R}^{D\times
D}.$

Since each $\mathcal{P}_{d_{i}}$ has determinant $1$, the block matrix
$\mathbf{P}(t)$ also has determinant $1$. By construction, the matrix
$\mathbf{V}(t)$ is a diagonal matrix with strictly positive diagonal entries.
By Lemma 3.4, the collocation matrix with the associated differential
operators and the collocation matrix with the standard derivatives are related
by

(3.53) $\left(L_{d_{i}}B_{m}^{\ell}(t_{i})\right)_{1\leq i,\ell\leq
D}=\mathbf{V}(t)\cdot\mathbf{P}(t)\cdot M\begin{pmatrix}t_{1},\dots,t_{D}\\\
B_{m}^{1},\dots,B_{m}^{D}\end{pmatrix}.$

The claim follows from the multiplicativity of the determinant and Theorem 3.2
with $y_{i}=i-1$. ∎

We finally relate the collocation matrix to sampling with multiplicities.

######  Corollary 3.6.

Let $\varphi$ be a PEB-spline of order $m$ supported on the interval $[0,m]$.
Further let $x=\left(x_{1},x_{2},\dots,x_{N}\right)\in\mathbb{R}^{N}$ be a
vector of pairwise distinct sampling points, $x_{1}<\dots<x_{N}$, with
multiplicities
$\mu=\left(\mu_{1},\dots,\mu_{N}\right)\in\\{0,\dots,m-1\\}^{N}$. Set
$D=\sum_{j=1}^{N}(1+\mu_{j})$. The Hermite interpolation problem

(3.54)
$f=\sum\limits_{\ell=\ell_{0}}^{\ell_{0}+D-1}c_{\ell}T_{\ell}\varphi,\quad
f^{(s)}(x_{j})=\xi_{j,s},\quad\ 0\leq s\leq\mu_{j},\ 1\leq j\leq N$

has a unique solution if and only if

(3.55)
$\begin{split}x_{j}&\in(\mu_{j}+\ell_{0},m+\ell_{0})+\sum\limits_{n=1}^{j-1}\left(1+\mu_{n}\right),\qquad\text{if}\
\ \mu_{j}<m-1,\\\\[2.15277pt]
x_{j}&\in[\mu_{j}+\ell_{0},m+\ell_{0})+\sum\limits_{n=1}^{j-1}\left(1+\mu_{n}\right),\qquad\text{if}\
\ \mu_{j}=m-1.\end{split}$

In this case, the solution depends linearly on the interpolation data.

If the points satisfy (3.55), we say that they _satisfy the Schoenberg-Whitney
conditions_.

###### Proof.

To apply the interlacing property to the interpolation problem (3.54), we set

(3.56)
$t=\Big{(}\underbrace{x_{1},\dots,x_{1}}_{{\mu_{1}+1}\text{-times}},\underbrace{x_{2},\dots,x_{2}}_{{\mu_{2}+1}\text{-times}},\dots,\underbrace{x_{N},\dots,x_{N}}_{{\mu_{N}+1}\text{-times}}\Big{)}\in\mathbb{R}^{D},$

to be precise,

(3.57) $t_{i}=x_{j}\qquad\text{ if
}\qquad\sum\limits_{n=1}^{j-1}(1+\mu_{n})<i\leq\sum\limits_{n=1}^{j}(1+\mu_{n}).$

Then the vector of multiplicities is given by

(3.58)
$d=\Big{(}0,1,\dots,\mu_{1},0,1,\dots,\mu_{2},\dots,0,1,\dots,\mu_{N}\Big{)}\in\big{\\{}0,\dots,m-1\big{\\}}^{D}.$

Accordingly, we set $\xi\in\mathbb{C}^{D}$ to be the list of the samples
$\xi=(\xi_{i})_{i=1,\dots,D}$,

(3.59)
$\xi=\Big{(}\xi_{1,0},\,\xi_{1,1},\,\xi_{1,2},\,\dots,\,\xi_{1,\mu_{1}},\,\xi_{2,0},\,\xi_{2,1},\,\xi_{2,2},\,\dots,\,\xi_{2,\mu_{2}},\,\dots,\,\xi_{N,0},\,\xi_{N,1},\,\xi_{N,2}\dots,\,\xi_{N,\mu_{n}}\Big{)}.$

With this notation, the interpolation problem (3.54) can be recast as

(3.60)
$f=\sum\limits_{\ell=\ell_{0}}^{\ell_{0}+D-1}c_{\ell}\,T_{\ell}\varphi,\quad
f^{(d_{i})}(t_{i})=\xi_{i},\qquad 1\leq i\leq D.$

The corresponding collocation matrix is

(3.61) $M\begin{pmatrix}t_{1},\dots,t_{D}\\\
T_{\ell_{0}}\varphi,\dots,T_{\ell_{0}+D-1}\varphi\end{pmatrix}=\Big{(}T_{\ell-1}\varphi^{(d_{i})}(t_{i}-\ell_{0})\Big{)}_{1\leq
i,\ell\leq D}.$

For further reference, we denote it by

(3.62) $\mathbf{M}(x,\mu,\ell_{0})\coloneqq
M\begin{pmatrix}t_{1},\dots,t_{D}\\\
T_{\ell_{0}}\varphi,\dots,T_{\ell_{0}+D-1}\varphi\end{pmatrix}.$

By Lemma 3.5, the collocation matrix is invertible, and thus a unique solution
of (3.54) exists, if and only if $t_{i}-\ell_{0}\in(i-1,i+m-1)$ for all $1\leq
i\leq D$ (with the modification $t_{i}-\ell_{0}\in[i-1,i+m-1)$ if
$d_{i}=m-1$). Expressed with the sampling points $x_{j}$, the collocation
matrix is invertible if and only if

(3.63) $\displaystyle x_{j}\ $ $\displaystyle\in\ \bigcap_{s=0}^{\mu_{j}}\
\Big{(}\left(s,s+m\right)\
+\ell_{0}+\sum\limits_{n=1}^{j-1}\left(1+\mu_{n}\right)\Big{)}=(\mu_{j}+\ell_{0},m+\ell_{0})+\sum\limits_{n=1}^{j-1}\left(1+\mu_{n}\right).$

for all $1\leq j\leq N$. The second condition in (3.55) is the adaptation for
$\mu_{j}=m-1$. ∎

###  3.3. Stability of PEB-Splines

To use PEB-splines as generators of shift-invariant spaces, we need to verify
the stability of PEB-splines and their derivatives.

######  Lemma 3.7.

Let $\varphi$ be a PEB-spline of order $m$. Then

  1. (i)

The generator $\varphi$ has stable integer shifts.

  2. (ii)

For all $0\leq s\leq m-1$, the tuple
$\left(\varphi,\dots,\varphi^{(s)}\right)$ has stable integer shifts.

###### Proof.

(i) Without loss of generality, we can assume that $\varphi$ is supported on
$[0,m]$. By Theorem 2.1, it suffices to show the $\ell^{\infty}$-independence
of $(T_{\ell}\varphi)_{\ell\in\mathbb{Z}}$. To that end, let
$c\in\ell^{\infty}(\mathbb{Z}),c\neq 0$, and assume that $c_{\ell_{0}}\neq 0$.
Since $\operatorname{supp}(T_{\ell}\varphi)=[\ell,\ell+m]$, the restriction of
$f$ to the interval $(\ell_{0}+m-1,\ell_{0}+m)$ is given by

(3.64) $f=\sum\limits_{\ell=\ell_{0}}^{\ell_{0}+m-1}c_{\ell}T_{\ell}\varphi.$

The restriction of $f$ to $(\ell_{0}+m-1,\ell_{0}+m)$ clearly belongs to the
ECC-system. By Lemma 3.1, it has at most $m-1$ zeros. In particular, it does
not vanish on this interval.

(ii) follows from (i), because the inequality

(3.65) $A_{p}\left\lVert
c\right\rVert_{\ell^{p}}^{p}\leq\Big{\lVert}\,\sum_{\ell\in\mathbb{Z}}c_{\ell}T_{\ell}\varphi\,\Big{\rVert}_{L^{p}}^{p}\leq\sum\limits_{s=0}^{S}\Big{\lVert}\sum_{\ell\in\mathbb{Z}}c_{\ell}T_{\ell}\varphi^{(s)}\,\Big{\rVert}_{L^{p}}^{p}.$

implies that the generator $\Phi=(\varphi,\varphi^{\prime},\dots\varphi^{S})$
has stable integer shifts. The upper inequality in (2.7) are obvious. ∎

###  3.4. EB-splines

Our initial motivation to consider PEB-splines was the particular subclass of
exponential B-splines. These have already been used in time-frequency analysis
in [6, 24]. Whereas general PEB-splines are usually defined recursively via
divided differences, EB-splines also have a direct definition by their Fourier
transform as follows from [11]. Let $m\in\mathbb{N}$ and let
$\alpha=\left(\alpha_{1},\dots,\alpha_{m}\right)\in\mathbb{R}^{m}$. An EB-
spline $\mathcal{E}_{m,\alpha}:\mathbb{R}\longrightarrow\mathbb{R}$ of order
$m$ for the $m$-tuple $\alpha$ is a function of the form

(3.66)
$\mathcal{E}_{m,\alpha}(x):={\prod\limits_{s=1}^{m}}^{\mbox{\raisebox{-7.0pt}{\large$*$}}}\,e^{\alpha_{s}x}\raisebox{2.0pt}{$\chi$}_{[0,1)}(x),$

where $\prod^{\mbox{{\large$*$}}}$ denotes the convolution product. Its
Fourier transform is given by

(3.67)
$\widehat{\mathcal{E}}_{m,\alpha}(\omega)=\prod\limits_{s=1}^{m}\frac{e^{\alpha_{s}-2\pi
i\omega}-1}{\alpha_{s}-2\pi i\omega}.$

If $\alpha_{1}=\dots=\alpha_{m}=0$, then this is a classical B-spline. To
establish the connection with the original definition of PEB-splines, we set
$\gamma_{1}:=\alpha_{1}$ and $\gamma_{s}=\alpha_{s}-\alpha_{s-1}$ for $2\leq
s\leq m$. Then the weight functions $w_{s}(x)=e^{\gamma_{s}x}$ induce the EB-
spline $B_{m}=\mathcal{E}_{m,\alpha}$. For an EB-spline, the stability follows
directly from the factorization (3.67) and Theorem 2.1(iv).

##  4\. A Sampling Theorem via Uniform Bounds on the Collocation Matrices

In this section, we prove a first sampling theorem with derivatives in a
shift-invariant space generated by a PEB spline. The proof technique is
inspired by [4], and its main point is to establish uniform bounds on a family
of collocation matrices. The use of derivatives requires more spline theory
and leads to more technicalities.

Our standing assumption is that the generator $\varphi$ is a periodic
exponential B-spline of order $m$, as described in Subsection 3.2. Without
loss of generality, we assume that $\varphi$ is supported on $[0,m]$. We
always assume that $X$ is separated, i.e., $\inf_{x,y\in X,x\neq
y}|x-y|=\delta>0$, and $\delta$ is called the separation constant of $X$.

As in [4], we partition $\mathbb{R}$ into suitable intervals and analyze the
sampling problem locally. Given integers $M\in\mathbb{Z}$ and
$L\in\mathbb{N}$, we partition $\mathbb{R}$ in intervals

(4.1)
$\mathbb{R}=\bigcup\limits_{k\in\mathbb{Z}}[M+kL,M+kL+L)=\bigcup\limits_{k\in\mathbb{Z}}I_{M,L}(k).$

######  Remark.

Let $m\in\mathbb{N}$ and let $\ell_{0}\leq x_{1}<\dots<x_{N}\leq\ell_{0}+L$ be
sampling locations with multiplicities
$\mu_{1},\dots,\mu_{N}\in\\{0,1\dots,m-1\\}$ such that

$\sum\limits_{j=1}^{N}(1+\mu_{j})=L+m-1$

and $\mu_{N}\leq m-2$ if $x_{N}=\ell_{0}+L$. Then for all
$(c_{\ell})_{\ell\in\mathbb{Z}}\in\ell^{2}(\mathbb{Z})$ we have

(4.2)
$\Big{(}\sum\limits_{\ell\in\mathbb{Z}}c_{\ell}T_{\ell}\varphi\Big{)}^{(s_{j})}(x_{j})=\Big{(}\sum\limits_{\ell=\ell_{0}-m+1}^{\ell_{0}+L-1}c_{\ell}T_{\ell}\varphi\Big{)}^{(s_{j})}(x_{j}),\qquad
0\leq s_{j}\leq\mu_{j},\ 1\leq j\leq N.$

Since $\varphi$ vanishes outside of $[0,m]$, (4.2) is clear for all
$x_{1},\dots,x_{N-1}$. The additional assumption that $\mu_{N}\leq m-2$ if
$x_{N}=\ell_{0}+L$ implies that the measurements at the last sampling point do
not include a point of discontinuity of $T_{\ell_{0}+L}\varphi^{(m-1)}$. Thus
for all $0\leq s\leq\mu_{N}$ and all $\ell\geq\ell_{0}+L$,
$T_{\ell}\varphi^{(s)}(\ell_{0}+L)=0$. These observations imply (4.2).

To avoid unnecessary banalities, we first settle the easy case of PEB-splines
of order $1$.

######  Lemma 4.1.

Let $\varphi$ be a PEB-spline of order 1 supported on $[0,1)$. Assume that
$X\subseteq\mathbb{R}$ is separated. Then the following are equivalent:

  1. (i)

For every $k\in\mathbb{Z}$, there exists a point $x^{k}\in X\cap[k,k+1)$.

  2. (ii)

$(X,\mu)$ is a sampling set for $V^{p}(\varphi)$ for all $p\in[1,\infty]$.

  3. (iii)

$(X,\mu)$ is a uniqueness set for $V^{p}(\varphi)$ for all $p\in[1,\infty]$.

###### Proof.

The implication $(ii)\Rightarrow(iii)$ is trivial. A PEB-spline of the first
order is the cut-off of the continuous weight function, namely
$\varphi=w_{1}\cdot\raisebox{2.0pt}{$\chi$}_{[0,1)}$. Thus, if there exists a
$k\in\mathbb{Z}$ with $X\cap[k,k+1)=\emptyset$, then $T_{k}\varphi|_{X}=0$,
although this function is clearly not the zero function. This shows
$(iii)\Rightarrow(i)$.

Now assume that $(i)$ holds. The upper sampling bound is satisfied by Theorem
2.2. Since $w$ does not vanish, $\varphi$ has no zeros on $[0,1)$ and vanishes
outside this interval. Therefore, $\varphi(x^{k}-\ell)\neq 0$ holds if and
only if $k=\ell$. Since $w$ is continuous and is strictly positive on
$\mathbb{R}$, it attains a minimum $a>0$ on the compact interval $[0,1]$. For
all $p\in[1,\infty)$ we obtain

(4.3) $\displaystyle\sum\limits_{x\in X}\left\lvert f(x)\right\rvert^{p}$
$\displaystyle\geq\sum\limits_{k\in\mathbb{Z}}\left\lvert
f(x^{k})\right\rvert^{p}=\sum\limits_{k\in\mathbb{Z}}\left\lvert
c_{k}\varphi(x^{k}-k)\right\rvert^{p}\geq
a^{p}\sum\limits_{k\in\mathbb{Z}}\left\lvert c_{k}\right\rvert^{p},$

with an analogous estimate for $p=\infty$. This shows $(i)\Rightarrow(ii)$ and
completes the proof. ∎

From now on, we assume without loss of generality that $m\geq 2$. Our first
theorem looks a bit technical, but is definitely useful.

######  Theorem 4.2.

Let $\varphi$ be a PEB-spline of order $m\in\mathbb{N}$ supported on $[0,m]$.
Assume $X\subseteq\mathbb{R}$ with multiplicity function $\mu_{X}$ satisfies
the following properties:

  1. (i)

$X$ is separated.

  2. (ii)

$S:=\max\limits_{x\in X}\mu_{X}(x)\leq m-1$.

  3. (iii)

There exist integers $M,L>0$ and $\varepsilon>0$, such that for every
$k\in\mathbb{Z}$, there exist points $x_{1}^{k}<x_{2}^{k}<\dots<x^{k}_{N(k)}$
in $X\cap I_{M,L}(k)$ and nonnegative integers
$\mu_{1}^{k},\,\mu_{2}^{k},\,\dots,\,\mu^{k}_{N(k)}$ with the following three
properties:

(4.4) $\mu^{k}_{j}\leq\mu_{X}(x^{k}_{j})\qquad\text{ for all
}k\in\mathbb{Z},\,1\leq j\leq N(k),$ (4.5)
$\sum\limits_{j=1}^{N(k)}\left(1+\mu_{j}^{k}\right)=L+m-1\qquad\text{ for all
}k\in\mathbb{Z},$ (4.6) $x_{j}^{k}\in
X\cap\Big{(}\left[\mu^{k}_{j}-m+\varepsilon,-\varepsilon\right]+M+kL+1+\sum\limits_{n=1}^{j-1}\left(1+\mu^{k}_{n}\right)\Big{)}.$

Then $(X,\mu_{X})$ is a sampling set for $V^{p}(\varphi)$ for all
$p\in[1,\infty]$.

######  Remark.

The conditions (4.4) - (4.6) are compatible with (4.2) and in some sense are
necessary. Since the restriction of $f$ to an interval $I_{M,L}(k)$ belongs to
an $(L+m-1)$-dimensional space, we need at least $L+m-1$ samples, whence
(4.5). If more information is available, we either remove points or the
highest derivatives, which is (ii) and (4.5). As we intend to use the
invertibility of the collocation matrix (Corollary 3.6), the retained
information must be chosen diligently and involves a Schoenberg-Whitney
condition of type (4.6), which says that the support of each shift must
contain a sampling point (see Figure 1).

Figure 1. Nonvanishing shifts of $\varphi(x)={\prod_{j=1}^{\mbox{\large$*$}\
4\ }}\,e^{x}\raisebox{2.0pt}{$\chi$}_{[0,1)}(x)$. The sampling points are
$x_{1}=0.5$, $x_{2}=3$, $x_{3}=3.9$, with multiplicities
$\mu_{X}(x_{1})=\mu_{X}(x_{2})=2$, $\mu_{X}(x_{3})=0$. The first sampling
point lies in the support of the first three shifts of $\varphi$ (dot-dashed),
the second point is in the support of the next three shifts of $\varphi$
(dashed), and the last point - in the support of the last shift of $\varphi$
(solid).

###### Proof.

Fix $p\in[1,\infty)$ and let

(4.7) $f=\sum_{\ell\in\mathbb{Z}}c_{\ell}T_{\ell}\varphi\in V^{p}(\varphi).$

Since $\left\lVert f\right\rVert_{p}\asymp\left\lVert c\right\rVert_{p}$ for
$f\in V^{p}(\varphi)$ by Lemma 3.7, it suffices to show the inequality

(4.8) $\sum\limits_{x\in X}\sum\limits_{s=0}^{\mu_{X}(x)}\left\lvert
f^{(s)}(x)\right\rvert^{p}\asymp\left\lVert c\right\rVert_{\ell^{p}}^{p}.$

For given $M,\,L\in\mathbb{N}$ every interval $I_{M,L}(k)=[M+kL,M+(k+1)L)$
will be investigated separately. The aim is to use Corollary 3.6.

Step 1: The upper bound. Since $\varphi^{(s)}$, $0\leq s\leq m-1$, are
piecewise continuous, compactly supported functions with finitely many jump
discontinuities, Theorem 2.2 (ii) applies and yields the upper bound for the
sampling inequality.

Step 2: A class of collocation matrices with uniform bounds. Let $\mathcal{M}$
be the set

(4.9) $\mathcal{M}\coloneqq\Big{\\{}\mu\in\\{0,\dots,S\\}^{N}:N\in\mathbb{N},\
\sum\limits_{j=1}^{N}\left(1+\mu_{j}\right)=L+m-1\Big{\\}}.$

If $\mu\in\mathcal{M}\cap\mathbb{Z}^{N}$, then $N\leq L+m-1$. Therefore,

(4.10)
$\mathcal{M}\subseteq\bigcup\limits_{N=1}^{L+m-1}\left\\{0,\dots,S\right\\}^{N}\,,$

in particular, the set $\mathcal{M}$ is finite. Let
$\Omega_{\mu}\subseteq\mathbb{R}^{N}$ for
$\mu\in\mathcal{M}\cap\mathbb{Z}^{N}$ be the set of vectors
$z=(z_{1},z_{2},\dots,z_{N})^{t}\in\mathbb{R}^{N}$ with

(4.11) $\min\limits_{i\neq j}\left\lvert z_{i}-z_{j}\right\rvert\geq\delta$

and

(4.12) $\begin{split}z_{j}&\in[0,L]\cap\
\Big{(}\left[\mu_{j}-m+\varepsilon,-\varepsilon\right]\
+1+\sum\limits_{n=1}^{j-1}\left(1+\mu_{n}\right)\Big{)}\\\
&=\Big{[}\max\Big{\\{}0,\mu_{j}-m+1+\varepsilon+\sum\limits_{n=1}^{j-1}\left(1+\mu_{n}\right)\Big{\\}}\,,\,\min\Big{\\{}L,1-\varepsilon+\sum\limits_{n=1}^{j-1}\left(1+\mu_{n}\right)\Big{\\}}\Big{]}.\end{split}$

In other words, $\Omega_{\mu}$ contains all point configurations in $[0,L]$
that satisfy Condition (4.6). The set is bounded and closed, hence compact in
$\mathbb{R}^{N}$. For each $z\in\Omega_{\mu}$ we consider the collocation
matrix

(4.13) $\mathbf{M}(z,\mu,-m+1)\in\mathbb{R}^{(L+m-1)\times(L+m-1)}$

as defined in the proof of Corollary 3.6 in (3.62) with parameters $D=L+m-1$
and $\ell_{0}=-m+1$. By Corollary 3.6 and the positions of the points (cf.
(4.12)), the matrix is invertible. In addition,
$z\mapsto\mathbf{M}(z,\mu,-m+1)$ is continuous on the compact set
$\Omega_{\mu}$, so the matrices are bounded from above and below uniformly in
$\omega\in\Omega_{\mu}$. To see this, a trivial upper bound is a multiple of
the maximum of the entries of $\mathbf{M}(z,\mu,-m+1)$. For the lower bound,
we note that $z\mapsto\mathbf{M}(z,\mu,-m+1)^{-1}$ is continuous by Cramer’s
rule, thus we can again obtain a uniform upper bound $0<a_{p}(\mu)^{-1}$ for
$\mathbf{M}(z,\mu,-m+1)^{-1}$. Then $a_{p}(\mu)$ is a strictly positive
uniform lower bound of $\mathbf{M}(z,\mu,-m+1)$, i.e., for all
$c\in\mathbb{R}^{L+m-1}$

(4.14) $\left\lVert\mathbf{M}(z,\mu,-m+1)\cdot c\right\rVert_{p}\geq
a_{p}(\mu)\left\lVert c\right\rVert_{p}.$

Since $\mathcal{M}$ is finite, we can set

(4.15) $A_{p}\coloneqq\min\\{a_{p}(\mu):\mu\in\mathcal{M}\\}>0.$

Step 3: The lower estimate. Let $x^{k}=(x_{1}^{k},\dots,x_{N(k)}^{k})$ be the
points in $X\cap I_{M,L}(k)$ that satisfy Condition (iii), i.e., they satisfy
(4.4), (4.5), and (4.6). We set
$\mu^{k}=\left(\mu_{1}^{k},\dots,\mu^{k}_{N(k)}\right)\in\mathcal{M}$ and
$z_{j}^{k}=x_{j}^{k}-(M+kL)$. Then $z^{k}\in\Omega_{\mu^{k}}$. Let
$\textbf{F}(x^{k})$ be the complete list of evaluations

(4.16) $\displaystyle\textbf{F}(x^{k})$
$\displaystyle:=\left(f(x^{k}_{1}),\dots,f^{(\mu^{k}_{1})}(x^{k}_{1}),\dots,f(x_{N(k)}^{k}),\dots,f^{(\mu^{k}_{N(k)})}(x_{N(k)}^{k})\right)\in\mathbb{C}^{L+m-1}.$

For $p\in[1,\infty)$, we estimate from below the sum

(4.17) $\sum\limits_{x\in X}\sum\limits_{s=0}^{\mu_{X}(x)}\left\lvert
f^{(s)}(x)\right\rvert^{p}\geq\sum\limits_{k\in\mathbb{Z}}\sum\limits_{j=1}^{N(k)}\sum\limits_{s=0}^{\mu_{j}^{k}}\left\lvert
f^{(s)}(x_{j}^{k})\right\rvert^{p}=\sum\limits_{k\in\mathbb{Z}}\left\lVert\textbf{F}(x^{k})\right\rVert_{p}^{p}.$

We now consider the restriction of $f$ to the interval $I_{M,L}(k)$ and shift
it to $[0,L)$. The resulting function is

(4.18)
$T_{-(M+kL)}\big{(}f\big{|}_{I_{M,L}(k)}\big{)}=T_{-(M+kL)}\,\sum\limits_{\ell=M+kL-m+1}^{M+kL+L-1}c_{\ell}T_{\ell}\varphi=\sum\limits_{\ell=-m+1}^{L-1}c_{\ell+M+kL}T_{\ell}\varphi.$

We denote by
$\mathfrak{c}_{k}=\left(c_{M+kL-m+1},\dots,c_{M+kL+L-1}\right)^{t}$ the
extracted coefficients which are active on $I_{M,L}(k)$. By Corollary 3.6,
specifically (3.62), the measurements $\textbf{F}(x^{k})$ are given by

(4.19)
$\displaystyle\textbf{F}(x^{k})=\mathbf{M}(z^{k},\mu^{k},-m+1)\cdot\mathfrak{c}_{k}.$

From this follows

(4.20) $\begin{split}\sum\limits_{x\in
X}\sum\limits_{s=0}^{\mu_{X}(x)}\left\lvert
f^{(s)}(x)\right\rvert^{p}&\geq\sum\limits_{k\in\mathbb{Z}}\left\lVert\mathbf{M}(z^{k},\mu^{k},-m+1)\cdot\mathfrak{c}_{k}\right\rVert_{p}^{p}\geq
A_{p}^{p}\,\sum\limits_{k\in\mathbb{Z}}\left\lVert\mathfrak{c}_{k}\right\rVert_{p}^{p}\geq
A_{p}^{p}\,\left\lVert c\right\rVert_{p}^{p}\,.\end{split}$

In the last inequality, we have used the fact that every $\ell\in\mathbb{Z}$
is contained in some interval $M+kL-m+1\leq\ell\leq M+kL+L-1.$ The estimate
for $p=\infty$ is analogous, and we are done. ∎

######  Remark.

The assumptions in Theorem 4.2 are not easy to visualize. We will therefore
investigate a different set of conditions in Section 6. But Theorem 4.2 is a
powerful result, and the imposed conditions are necessary and optimal in
several regards.

(i) The restriction of $f\in V^{p}(\varphi)$ to the interval $I_{M,L}(k)$ is
given by (4.2) and is a linear combination of $L+m-1$ linearly independent
shifts $T_{\ell}\varphi$, this is Condition (4.5)).

(ii) To recover the restriction of $f$ on the interval $I_{M,L}(k)$ from
$L+m-1$ samples, the associated collocation matrix must be invertible. This is
implied by Condition (4.6). The margin $\varepsilon>0$ leads to uniform
estimates in $k$.

(iii) Condition (4.5) implies that $(X,\mu_{X})$ contains a subset
$X^{\prime}\subseteq X$ with multiplicity function $\mu_{X^{\prime}}$
satisfying $\mu_{X}^{\prime}(x)\leq\mu_{X}(x),$ $x\in X^{\prime}$, such that
$D^{-}(X^{\prime},\mu_{X^{\prime}})=1+\frac{m-1}{L}$, cf. [21, Prop. 3.7.].
Thus every sampling set subject to the assumptions of Theorem 4.2 contains a
sampling set with a weighted density close to the necessary density condition.

To illustrate the strength of Theorem 4.2, we draw two consequences. First, we
check standard sampling sets without multiplicities. It will be convenient to
label $X$ as a bi-infinite strictly increasing sequence
$X=(x_{j})_{j\in\mathbb{Z}}$, $x_{j}<x_{j+1}$. In this notation the set $X$ is
$\delta$-separated if $\inf_{j\in\mathbb{Z}}(x_{j+1}-x_{j})\geq\delta>0$, and
its maximum gap is defined as

(4.21) $\mathfrak{mg}(X)\coloneqq\sup_{j\in\mathbb{Z}}\,(x_{j+1}-x_{j})\,.$

The following result extends [4, Thm. 4] from B-splines to general PEB-
splines.

######  Corollary 4.3.

Let $\varphi$ be a PEB-spline of order $m\in\mathbb{N}$. If the sampling set
$X\subseteq\mathbb{R}$ is separated and the maximum gap satisfies
$\mathfrak{mg}(X)<1$, then $X$ is a sampling set for $V^{p}(\varphi)$.

If $\mu_{X}\equiv 0$, then the assumptions in Theorem 4.2 are identical with
those in [4, Thm. 3]. Further, the proof of [4, Thm. 4] shows that a set with
$\mathfrak{mg}(X)<1$ satisfies the conditions in [4, Thm. 4], hence, the
conditions in Theorem 4.2. We will give an alternative proof in Section 6.

To construct sampling sets with multiplicities, one can start with a sampling
set without derivatives and remove certain sampling points while adding
multiplicities at neighbouring sampling points (see Figure 2).

Finally, we show that there exist sets with a given multiplicity function that
satisfies the sufficient conditions of Theorem 4.2 with a density arbitrarily
close to the necessary condition.

(a) Starting arrangement. (b) Modification with multiplicities.

Figure 2. Local modification of a sampling set that preserves the conditions
of Thm. 4.2.

######  Theorem 4.4.

For every $\nu>0$ and given multiplicity sequence
$\mu=(\mu_{j})_{j\in\mathbb{Z}}$ with values in $\\{0,\dots m-1\\}$, there
exists a sequence $X=(x_{j})_{j\in\mathbb{Z}}$, such that $(X,\mu_{X})$ is
sampling for $V^{p}(\varphi)$, $\mu_{X}(x_{j})=\mu_{j}$ for all
$j\in\mathbb{Z}$, and $D^{-}(X,\mu_{X})<1+\nu$.

###### Proof.

The sampling set $X$ will be a suitable subset of a sufficiently dense lattice
$\alpha\mathbb{Z}$. In line with Theorem 4.2 we use a suitable partition and
argue locally. Since $X$ will be a subset of $\alpha\mathbb{Z}$, it is
separated, and Condition (ii) in Theorem 4.2 is satisfied by assumption. The
main part is to construct points $x$ and submultiplicities $\tilde{\mu}$ to
satisfy the third condition.

Step 1. Global parameters. Given $\nu>0$, choose $L\in\mathbb{N}$ such that
$\frac{2m-2}{L}<\nu$. We set $\varepsilon=\alpha=\frac{1}{(m+1)}$. We will
work with the interval partitioning $I_{L,0}(k)=[kL,(k+1)L)$. The key
observation here is the fact that
$\\#([\ell+\varepsilon,\ell+1-\varepsilon]\cap\alpha\mathbb{Z})=m$, and for a
fixed $\ell\in\mathbb{Z}$

(4.22)
$\begin{split}[\ell+\varepsilon,\ell+1-\varepsilon]\cap\Big{(}\left[\mu^{k}_{j}-m+\varepsilon,-\varepsilon\right]+kL+1+\sum\limits_{n=1}^{j-1}\left(1+\mu^{k}_{n}\right)\Big{)}\end{split}$

is non-empty for at most $m$ consecutive indices $j,j+1,\dots,j+m-1$. Thus, in
$[\ell+\varepsilon,\ell+1-\varepsilon]$ we will need at most $m$ sampling
points in $\alpha\mathbb{Z}$, which are available by choice of $\alpha$.

Step 2. Local construction. Let $N\in\mathbb{N}$ be the minimal natural number
with $\sum_{j=1}^{N}(1+\mu_{j})\geq L+m-1.$ We set

$\tilde{\mu}_{j}\coloneqq\mu_{j},\quad 1\leq j\leq
N-1,\qquad\tilde{\mu}_{N}\coloneqq(L+m-1)-\sum\limits_{j=1}^{N}(1+\mu_{j})-1.$

The minimality of $N$ implies that $0\leq\tilde{\mu}_{N}\leq\mu_{N}$, and the
definition of $\tilde{\mu}_{N}$ implies that

$\sum\limits_{j=1}^{N}(1+\tilde{\mu}_{j})=L+m-1.$

Figure 3. Choice of points (black circles) for $m=3$,
$\mu=(1+(-1)^{j})_{j\in\mathbb{Z}}$ and $\nu=\frac{1}{2}$ from the lattice
$\frac{1}{4}\mathbb{Z}$ (gray squares). The interval length is $L=13$, and the
number of points is $N=9$. The sampling locations are $\alpha
k,\,k\in\\{1,2,5,13,17,25,29,37,41\\}$. The plots represent the choice of the
first $j$ points, $j\in\\{2,5,9\\}$.

We now choose points $x_{j}\in[0,L)$ inductively (see Figure 3) and start with
$x_{1}=\alpha=\varepsilon$, which clearly satisfies (4.6). Assume now we have
picked $x_{1},\dots,x_{j-1}$. We choose $x_{j}$ to be _the minimal element in
$\alpha\mathbb{Z}$ that satisfies the Schoenberg-Whitney conditions (4.6)_.
Precisely, let

$\ell=\max\Big{\\{}0,\tilde{\mu}^{k}_{j}-m+1+\sum\limits_{n=1}^{j-1}\left(1+\tilde{\mu}^{k}_{n}\right)\Big{\\}}\,,$

and define

(4.23) $x_{j}=\min\\{\alpha k:k\in\mathbb{Z},\alpha
k\in[\ell+\varepsilon,\ell+1-\varepsilon]\\}\,.$

As observed in Step 1, we can always choose such $x_{j}$. The condition
$x_{j}\in[\ell+\varepsilon,\ell+1-\varepsilon]$ is clearly stronger than the
Schoenberg-Whitney conditions (4.6). The chosen points are in $[0,L)$ because

$0\leq\ell\leq\tilde{\mu}_{N}-m+1+\sum\limits_{n=1}^{N-1}\left(1+\tilde{\mu}_{n}\right)=L-1<L.$

Step 3. Partitioning $\mu$ and $\mathbb{R}$. We now partition the sequence
$\mu$ into blocks $M_{k}:=\\{\mu_{N_{k-1}+1},\dots,\mu_{N_{k}}\\}$. We set
$N_{-1}=0$ and construct $N_{k}$ inductively. Given $N_{k-1}$, we choose
$N_{k}\in\mathbb{Z}$ minimal with

$\sum_{n=N_{k-1}+1}^{N_{k}}(1+\mu_{j})\geq L+m-1.$

We carry out the local step for each block and obtain points
$x_{N_{k-1}+j}\in[kL,(k+1)L)$ with submultiplicities $\tilde{\mu}_{N_{k-1}+j}$
satisfy all conditions of Theorem 4.2. Consequently, the set
$\\{x_{m}:m\in\mathbb{Z}\\}=\bigcup_{k\in\mathbb{Z}}\\{x_{N_{k-1}+j}:j=1,\dots,N_{k}\\}$
with multiplicities $(\mu_{m})_{m\in\mathbb{Z}}$ form a sampling set in
$V^{p}(\varphi)$. 333Note that only at the interval ends $N_{k}$ we had to
reduce the multiplicities to apply Theorem 4.2.

Step 4. Density. Since we have

$\sum_{n=N_{k-1}+1}^{N_{k}}(1+\mu_{n})=L+m-1+(\mu_{N_{k}}-\tilde{\mu}_{N_{k}})\leq
L+2m-2$

samples in every interval $[kL,(k+1)L)$, the weighted Beurling density is
bounded by $D^{-}(X,\mu)\leq\frac{L+2m-2}{L}=1+\frac{2m-2}{L}<1+\nu$. ∎

######  Remark.

Note that a controlled perturbation of $X$ by setting
$\tilde{x_{j}}=x_{j}+\gamma_{j},j\in\mathbb{Z}$ for
$|\gamma_{j}|\leq\tau\varepsilon$ for a fixed $\tau\in(0,1)$ still satisfies
the conditions of Theorem 4.2. Additionally, there exist sampling sets with
multiplicities $(X,\mu_{X})$ which are sampling for $V^{p}(\varphi)$,
$D^{-}(X,\mu_{X})<1+\nu$, but $D^{-}(X)<1$, in particular, $X$ is not sampling
for $V^{p}(\varphi)$. A simple example is a sequence that alternates between
$s_{1}$ and $s_{2}$, $s_{1}+s_{2}>0$. If $L\in\mathbb{N}$ satisfies
$L+m-1=T(s_{1}+s_{2}+2)$ for sufficiently large $T\in\mathbb{N}$, then
$\tilde{\mu}_{j}=\mu_{j}$ for all $j\in\mathbb{Z}$, there are $2T$ sampling
locations in each interval $[kL,(k+1)L)$ and the resulting set $X$ is
$L$-periodic. Its lower Beurling density is given by

(4.24) $\displaystyle D^{-}(X)$
$\displaystyle=\lim\limits_{K\to\infty}\frac{\\#(X\cap[-kL,kL])}{2kL}=\frac{2T}{L}=\frac{2T}{T(s_{1}+s_{2}+2)-(m-1)}$
(4.25)
$\displaystyle=\Big{(}\frac{s_{1}+s_{2}}{2}+1-\frac{m-1}{2T}\Big{)}^{-1}<1,$

whenever $T>\frac{m-1}{s_{1}+s_{2}}$. This argument can be generalized to more
complicated sequences $\mu$.

##  5\. Weak limits

A particular case to consider is sampling with a constant multiplicity $S$ for
some integer $0\leq S\leq m-1$. Since every point comes with $S+1$ data, one
expects that a suitably modified gap condition might be sufficient for
sampling with derivatives. Based on Corollary 4.3, it is natural to conjecture
that

(5.1) $\frac{\mathfrak{mg}(X)}{1+S}<1$

is sufficient for sampling. This is indeed correct, but it does not follow
from Theorem 4.2. One can show that there are sets that satisfy (5.1) but
violate the conditions (4.4), (4.5), and (4.6). These examples will be
addressed in Section 7.

To study maximal gap conditions and achieve an optimal result, we will use the
technique of weak limits. These go back to Beurling [10] and have been
introduced for the investigation of shift-invariant spaces in [20, 21].

We recall Beurling’s notion of a weak limit of a sequence of sets. A sequence
$\left(X_{n}\right)_{n\in\mathbb{N}}$ of subsets of $\mathbb{R}$ converges
weakly to a set $Y\subseteq\mathbb{R}$, denoted by
$X_{n}\overset{w}{\longrightarrow}Y$, if for every open bounded interval
$(a,b)$ and every $\varepsilon>0$, there exists an $n_{0}\in\mathbb{N}$ such
that for all $n\geq n_{0}$,

(5.2) $X_{n}\cap(a,b)\subseteq
Y+(-\varepsilon,\varepsilon)\qquad\text{and}\qquad Y\cap(a,b)\subseteq
X_{n}+(-\varepsilon,\varepsilon).$

Given a tuple of sets $\overrightarrow{X}=\left(X^{0},\dots,X^{S}\right)$, a
tuple $\overrightarrow{Y}=\left(Y^{0},\dots,Y^{S}\right)$ is called a _weak
limit of integer translates_ of $\overrightarrow{X}$ if there exists a
sequence $(k_{n})_{n\in\mathbb{N}}\subseteq\mathbb{Z}$ such that
$X^{s}-k_{n}\overset{w}{\longrightarrow}Y^{s}$ for all $0\leq s\leq S$. An
important part of the definition is the fact that the sequence of translates
is the same for all $Y^{s}$. The set of weak limits of integer translates of
$X$ is denoted by $W_{\mathbb{Z}}(\overrightarrow{X})$. Weak limits enter
sampling theory through the following characterization of sampling sets [21,
Thm. 2.1].

######  Theorem 5.1.

Assume that $\Phi=\left(\Phi^{0},\dots,\Phi^{S}\right)$ consists of continuous
functions with a minimal decay property $|\Phi^{s}(x)|\leq
C(1+|x|)^{-1-\varepsilon}$ and $\Phi$ has stable integer shifts. Let
$\overrightarrow{X}=\left(X^{0},\dots,X^{S}\right)$ be a tuple of separated
sets. Then the following are equivalent.

  1. (i)

$\overrightarrow{X}$ is a sampling set for $V^{p}(\Phi)$ for some
$p\in[1,\infty]$.

  2. (ii)

$\overrightarrow{X}$ is a sampling set for $V^{p}(\Phi)$ for all
$p\in[1,\infty]$.

  3. (iii)

Every weak limit $\overrightarrow{Y}\in W_{\mathbb{Z}}(\overrightarrow{X})$ is
a sampling set for $V^{\infty}(\Phi)$.

  4. (iv)

Every weak limit $\overrightarrow{Y}\in W_{\mathbb{Z}}(\overrightarrow{X})$ is
a uniqueness set for $V^{\infty}(\Phi)$.

When sampling on $(X,\mu_{X})$, this theorem is applied to the generator
$\Phi=(\varphi,\varphi^{\prime},\dots,\varphi^{(S)})$ and the sets

(5.3) $X^{s}:=\\{x\in X:\mu_{X}(x)\geq s\\},\qquad 0\leq
s\leq\max\left\\{\mu_{X}(x):x\in X\right\\}.$

In this case, $\overrightarrow{X}-k_{n}$ can be identified with
$(X,\mu_{X})-k_{n}\coloneqq(X-k_{n},\mu_{X-k_{n}})$, where

(5.4) $\mu_{X-k_{n}}(x-k_{n})=\mu_{X}(x),\qquad x\in X.$

For separated sets, we can use an alternative characterization of weak
convergence (cf. [19, Lem. 4.4.], [21, Prop. 3.2.]).

######  Proposition 5.2.

Let $(X,\mu_{X})$ be a separated set with multiplicity function
$\mu_{X}:X\to\\{0,\dots,S\\}$, $(Y,\mu_{Y})$ be a set with multiplicity
function $\mu_{Y}$, and
$\left(k_{n}\right)_{n\in\mathbb{N}}\subseteq\mathbb{Z}$ a sequence of
integers. Then $X^{s}-k_{n}\overset{w}{\longrightarrow}Y^{s}$ as $n\to\infty$
for all $0\leq s\leq S$, if and only if

(5.5) $\sum\limits_{x\in
X}\left(1+\mu_{X}(x)\right)\delta_{x-k_{n}}\longrightarrow\sum\limits_{y\in
Y}\left(1+\mu_{Y}(y)\right)\delta_{y},\qquad\text{ as
}n\longrightarrow\infty,$

in the $(C_{c}^{*},C_{c})$-topology (where $C_{c}$ denotes the class of
continuous functions with compact support).

For splines, an additional difficulty arises that is not formally covered by
Theorem 5.1. The last derivative $\varphi^{(m-1)}$ is only piecewise
continuous with finitely many jump discontinuities. In this case, we have to
impose an additional condition on the $m$-th sampling set. The modified
version of Theorem 5.1 goes as follows:

######  Corollary 5.3.

Let $\varphi\in C^{S-1}(\mathbb{R})$ be a compactly supported, piecewise
$C^{S}$-function with finitely many jump discontinuities
$\mathcal{J}\subseteq\mathbb{R}$. Further, assume that $\varphi$ has stable
integer translates. Let $X$ be a separated set with multiplicity function
$\mu_{X}:X\to\\{0,\dots S\\}$ satisfying

(5.6) $\mathrm{dist}\big{(}\\{x\in
X:\mu_{X}(x)=S\\}\,,\mathcal{J}+\mathbb{Z}\big{)}>0.$

Then the following statements are equivalent:

  1. (i)

$(X,\mu_{X})$ is a sampling set for $V^{p}(\varphi)$ for some
$p\in[1,\infty]$.

  2. (ii)

$(X,\mu_{X})$ is a sampling set for $V^{p}(\varphi)$ for all $p\in[1,\infty]$.

  3. (iii)

Every weak limit $(Y,\mu_{Y})\in W_{\mathbb{Z}}(X,\mu_{X})$ is a sampling set
for $V^{\infty}(\varphi)$.

  4. (iv)

Every weak limit $(Y,\mu_{Y})\in W_{\mathbb{Z}}(X,\mu_{X})$ is a uniqueness
set for $V^{\infty}(\varphi)$.

######  Remark.

The equivalence of (i) and (ii) holds without condition (5.6). For PEB-splines
of order $m$, (5.6) is only needed if $X^{m-1}=\\{x\in X:\mu(x)=m-1\\}$ is
non-empty, i.e., when $(m-1)$-th derivatives are part of the data. If
$X^{m-1}=\emptyset$, then (5.6) is void and not required.

The proof of Corollary 5.3 is a small generalization of [20, Thm. 3.1]. For
the sake of completeness, we sketch the modifications in the Appendix.

When we label a separated set $X\subseteq\mathbb{R}$ as a strictly increasing
sequence, we can define the weighted maximum gap as

$\displaystyle\mathfrak{mg}(X,\mu_{X})$
$\displaystyle\coloneqq\max\bigg{\\{}\sup\limits_{j\in\mathbb{Z}}\frac{x_{j+1}-x_{j}}{1+\mu_{X}(x_{j})},\,\sup\limits_{j\in\mathbb{Z}}\frac{x_{j+1}-x_{j}}{1+\mu_{X}(x_{j+1})}\bigg{\\}}$
(5.7)
$\displaystyle=\sup\limits_{j\in\mathbb{Z}}\,\frac{x_{j+1}-x_{j}}{1+\min\\{\mu_{X}(x_{j}),\mu_{X}(x_{j+1})\\}}\,.$

We will show that $\mathfrak{mg}(X,\mu_{X})<1$ is sufficient for a set to be a
sampling set.

To apply Corollary 5.3, we first study the weighted maximum gap of a weak
limit $(Y,\mu_{Y})\in W_{\mathbb{Z}}(X,\mu_{X})$.

######  Lemma 5.4.

Let $(X,\mu_{X})$ be a separated set with a multiplicity function $\mu_{X}$
and $(Y,\mu_{Y})$ be a weak limit of $(X,\mu_{X})$. Then

(5.8) $\mathfrak{mg}(Y,\mu_{Y})\leq\mathfrak{mg}(X,\mu_{X}).$

###### Proof.

Let $(Y,\mu_{Y})$ by a weak limit of $(X,\mu_{X})$. Since $X$ is separated, we
can use Proposition 5.2 to describe the weak convergence by means of measures

(5.9) $\begin{split}\sum\limits_{x\in
X}\left(1+\mu_{X}(x)\right)\delta_{x-k_{n}}\longrightarrow\sum\limits_{y\in
Y}\left(1+\mu_{Y}(y)\right)\delta_{y},\qquad\text{
as}n\longrightarrow\infty.\end{split}$

For arbitrary $\varepsilon>0$, we choose a pair $y_{j}<y_{j+1}$ of consecutive
points in $Y$ such that

(5.10)
$\frac{y_{j+1}-y_{j}}{1+\min\\{\mu_{Y}(y_{j}),\mu_{Y}(y_{j+1})\\}}>\mathfrak{mg}(Y,\mu_{Y})-\varepsilon.$

Let $\delta>0$ be the separation constant of $X$ and $f_{0},\,f_{1}\in
C_{c}(\mathbb{R})$ be two auxiliary functions such that $0\leq f_{0},f_{1}\leq
1$,

(5.11)
$\begin{split}\operatorname{supp}(f_{0})\subseteq\left(-\tfrac{\delta}{2},\tfrac{\delta}{2}\right),&\qquad
f_{0}(x)=1\ \text{for all
}x\in\left(-\tfrac{\delta}{4},\tfrac{\delta}{4}\right),\\\
\operatorname{supp}(f_{1})\subseteq\left(y_{j},y_{j+1}\right),&\qquad
f_{1}(x)=1\ \text{for all
}x\in\left(y_{j}+\tfrac{\delta}{2},y_{j+1}-\tfrac{\delta}{2}\right).\end{split}$

Then by (5.9)

$\displaystyle\sum\limits_{x\in
X}\left(1+\mu_{X}(x)\right)f_{0}(x-k_{n}-y_{j})=$
$\displaystyle=\sum\limits_{x\in
X}\left(1+\mu_{X}(x)\right)\delta_{x-k_{n}}(T_{y_{j}}f_{0})\to\sum\limits_{y\in
Y}\left(1+\mu_{Y}(y)\right)\delta_{y}(T_{y_{j}}f_{0})=1+\mu_{Y}(y_{j}).$

Since $X$ is $\delta$-separated, $f_{0}\geq 0$ and
$\operatorname{supp}(f_{0})\subseteq\left(-\frac{\delta}{2},\frac{\delta}{2}\right)$,
for every $n\in\mathbb{N}$ there is at most one $x\in X$, say $x_{n_{j}}$,
such that

(5.12) $|x_{n_{j}}-k_{n}-y_{j}|<\tfrac{\delta}{2}.$

So $x_{n_{j}}-k_{n}\to y_{j}$ as $n$ tends to infinity and consequently
$\mu_{X}(x_{n_{j}})=\mu_{Y}(y_{j})$ for all $n\in\mathbb{N}$. Likewise, there
is a sequence $\widetilde{x}_{n_{j}}-k_{n}-y_{j+1}$ such that

(5.13) $|\widetilde{x}_{n_{j}}-k_{n}-y_{j+1}|<\tfrac{\delta}{2}$

and $\mu_{X}(\widetilde{x}_{n_{j}})=\mu(y_{j+1})$ for all $j\in\mathbb{N}$
sufficiently large.

Using the weak*-convergence for $f_{1}$, we obtain

(5.14) $\sum\limits_{x\in
X}\left(1+\mu_{X}(x)\right)f_{1}(x-k_{n})\longrightarrow\sum\limits_{y\in
Y}\left(1+\mu_{Y}(y)\right)f_{1}(y)=0,$

because $\operatorname{supp}(f_{1})\cap Y=\emptyset$. As $f_{1}\geq 0$, this
implies that for all sufficiently large $n\in\mathbb{N}$,
$\operatorname{supp}(f_{1})\cap(X-k_{n})\subseteq[y_{j},y_{j}+\frac{\delta}{2})\cap[y_{j+1}-\frac{\delta}{2},y_{j+1})$.

By (5.13) and the $\delta$-separability of the sets, these must be the points
$x_{n_{j}}-k_{n}$ and $\widetilde{x}_{n_{j}}-k_{n}$, so that necessarily
$\widetilde{x}_{n_{j}}=x_{n_{j}+1}$. Consequently, since
$\mu_{X-k_{n}}=\mu_{X}(\cdot+k_{n})$,

(5.15)
$\begin{split}\mathfrak{mg}(X,\mu_{X})\geq&\,\frac{x_{n_{j}+1}-x_{n_{j}}}{1+\min\\{\mu_{X}(x_{n_{j}}),\mu_{X}(x_{n_{j+1}})\\}}\\\
\geq&\,\frac{(x_{n_{j}+1}-k_{n})-(x_{n_{j}}-k_{n})}{1+\min\\{\mu_{X-k_{n}}(x_{n_{j}}-k_{n}),\mu_{X-k_{n}}(x_{n_{j}+1}-k_{n})\\}}\\\\[2.15277pt]
\geq&\,\frac{y_{j+1}-y_{j}}{1+\min\\{\mu_{Y}(y_{j}),\mu_{Y}(y_{j+1})\\}}\\\\[2.15277pt]
>&\,\mathfrak{mg}(Y,\mu_{Y})-\varepsilon.\end{split}$

Since $\varepsilon>0$ was arbitrary, the claim follows. ∎

##  6\. Sampling via maximal gap and weak limits

In this section, we will prove two sampling theorems that use gap conditions
for the sampling set. These conditions are easier to check than the technical
conditions of Theorem 4.2.

We first relate the weighted maximum gap to the Schoenberg-Whitney conditions
(3.55). The following statement does not make reference to splines and only
involves arrangements of points on the line.

######  Lemma 6.1 (Combinatorial lemma).

Let $m\geq 2$ and let $(X,\mu_{X})$ be a separated set with multiplicity
function $\mu_{X}$. Assume that $\mathfrak{mg}(X,\mu_{X})<1$. Then for all
$x\in X$ there exist $N,L\in\mathbb{N}$, $N,L\geq 2$, depending on $x$, and
points

$x^{*}_{1}<x^{*}_{2}<\dots<x^{*}_{N}\text{ in }X\cap\big{[}\left\lfloor
x\right\rfloor,\left\lfloor x\right\rfloor+L\big{]}$

with submultiplicities $\mu_{j}\in\mathbb{N}$,

(6.1) $\mu_{j}\leq\mu_{X}(x^{*}_{j}),\qquad\qquad 1\leq j\leq N\,,$

satisfying the following properties:

  1. (i)

dimension count:

(6.2) $\sum\limits_{j=1}^{N}(1+\mu_{j})=L+m-1,$

  2. (ii)

Schoenberg-Whitney conditions:

(6.3) $x_{j}^{*}\in(\mu_{j}-m+1,1)+\left\lfloor
x\right\rfloor+\sum\limits_{n=1}^{j-1}(1+\mu_{n}),\qquad 1\leq j\leq N$

with the usual adaptation if $\mu_{j}=m-1$,

  3. (iii)

$X\cap\big{[}\left\lfloor x\right\rfloor+L-1,\left\lfloor
x\right\rfloor+L\big{]}\neq\emptyset\,.$

$(^{*})$ If $x^{*}_{N}=\left\lfloor x\right\rfloor+L\in\mathbb{Z}$, then, in
addition, we have $\mu_{N}\leq m-2$.

Although the number of points $N$ and the length of the interval $L$ depend on
$x$, they may be bounded uniformly by

(6.4) $N\leq
1+\left\lceil(m+1)(1-\mathfrak{mg}(X,\mu_{X}))^{-1}\right\rceil=N_{0},\qquad\text{
and }L\leq m(N_{0}+m+2).$

###### Proof.

The proof is split into several steps. We make a first selection of points
$x_{1}^{*},\dots,x_{K}^{*}$ that satisfy all conditions. In the generic case,
we are done, but in some exceptional cases, we need to adapt the intermediate
parameters and add more points.

Reduction. Since the problem is invariant under integer shifts and reindexing,
we may assume without loss of generality that $\left\lfloor x\right\rfloor=0$
and $x_{1}=\min\\{x\in X:x\geq 0\\}$. We need to find $N$, $L\in\mathbb{N}$,
points $x^{*}\in[0,L]^{N}$ and submultiplicities $\mu\in\\{0,\dots,m-1\\}^{N}$
with the stated properties.

Step 1: (Preliminary) choice of $N$ and $L$. The condition
$\mathfrak{mg}(X,\mu_{X})<1$ leads to a bound for consecutive samples
$x_{j+1}-x_{j}\leq\mathfrak{mg}(X,\mu_{X})(1+\mu_{X}(x_{j}))$ for all
$j\in\mathbb{Z}$. By telescoping we obtain 444Here we use
$\frac{x_{j+1}-x_{j}}{1+\mu_{X}(x_{j})}\leq\mathfrak{mg}(X,\mu_{X})$.

(6.5)
$x_{n}-x_{1}=\sum_{j=1}^{n-1}(x_{j+1}-x_{j})\leq\mathfrak{mg}(X,\mu_{X})\sum_{j=1}^{n-1}(1+\mu_{X}(x_{j}))<\sum_{j=1}^{n-1}(1+\mu_{X}(x_{j}))\,.$

Thus the number of data $\sum_{j=1}^{n-1}(1+\mu_{X}(x_{j}))$ always exceeds
the length of the sampling interval $x_{n}-x_{1}$ by fixed factor
$\mathfrak{mg}(X,\mu_{X})^{-1}>1$. This leaves some room for manipulation.

In particular, for $n$ large enough, $(n-1)(1-\mathfrak{mg}(X,\mu_{X}))>m+1$,
we obtain

$\displaystyle\left\lceil x_{n}\right\rceil+m-1$ $\displaystyle\leq
x_{n}-x_{1}+2+m-1$
$\displaystyle\leq\mathfrak{mg}(X,\mu_{X})\sum_{j=1}^{n-1}(1+\mu_{X}(x_{j}))+m+1$
$\displaystyle\leq\mathfrak{mg}(X,\mu_{X})\sum_{j=1}^{n-1}(1+\mu_{X}(x_{j}))+(1-\mathfrak{mg}(X,\mu_{X}))(n-1)$
(6.6) $\displaystyle\leq\sum_{j=1}^{n-1}(1+\mu_{X}(x_{j}))\,.$

Now we define $K$ as the smallest integer with

(6.7) $\left\lceil
x_{K}\right\rceil+m-1\leq\sum\limits_{j=1}^{K}(1+\mu_{X}(x_{j})).$

Clearly $K\geq 1$ and by the preceding argument
$K\leq(m+1)/(1-\mathfrak{mg}(X,\mu_{X}))$. This choice of $K$ implies the
reverse inequality

(6.8) $\left\lceil
x_{j}\right\rceil+m-1>\sum\limits_{k=1}^{j}(1+\mu_{X}(x_{k}))\qquad\text{ for
}j=1,\dots,K-1\,.$

The inequalities (6.7) and (6.8) contain the key idea of the proof.

As the preliminary choice for the interval length we set

$L_{0}=\left\lceil x_{K}\right\rceil\,.$

The final choice of the number of points $N$ will be from $\\{2,3,K,K+1\\}$
and $L\leq\left\lceil x_{K}\right\rceil+m\leq(K+1)m+m$, whence we obtain their
boundedness independent of the initial choice of $x\in X$.

Step 2: Choice of the initial $x^{*}$ and their submultiplicities $\mu$ when
$K>1$. Assume first that $K>1$. The exceptional case $K=1$ will be treated
later. We choose

$x_{j}^{*}\coloneqq x_{j},\quad\ \mu_{j}\coloneqq\mu_{X}(x_{j}),\qquad\ 1\leq
j\leq K-1\,,$

and first show the Schoenberg-Whitney conditions (6.3) for these points. Since
$\left\lfloor x_{1}\right\rfloor=0$, for $j=1$ these are the inequalities
$x_{1}<1$ and $x_{1}\geq 0\geq\mu_{1}-m+1$ with equality if and only if
$\mu_{1}=m-1$.

For all $j=2,\dots,K-1$, (6.5) yields the upper inequality

(6.9) $\begin{split}x_{j}^{*}&=x_{j}=x_{j}-x_{1}+x_{1}\\\
&\leq\mathfrak{mg}(X,\mu_{X})\sum\limits_{n=1}^{j-1}(1+\mu_{X}(x_{n}))+1<1+\sum\limits_{n=1}^{j-1}(1+\mu_{n})\,.\end{split}$

The lower bound is contained in (6.8) via

$x_{j}>\left\lceil
x_{j}\right\rceil-1>-m+\sum\limits_{n=1}^{j}(1+\mu_{n})=1+\mu_{j}-m+\sum\limits_{n=1}^{j-1}(1+\mu_{n}).$

Step 3. For $j=K$ we set $x_{K}^{*}=x_{K}$ and

$\mu_{K}\coloneqq\left\lceil
x_{K}\right\rceil+m-2-\sum\limits_{j=1}^{K-1}(1+\mu_{j}).$

This choice ensures condition (6.2);

(6.10) $\sum\limits_{j=1}^{K}(1+\mu_{j})=\left\lceil
x_{K}\right\rceil+m-1=L_{0}+m-1\,.$

To check the range of $\mu_{K}$, we use (6.7) and obtain

(6.11) $\displaystyle 1+\mu_{K}$ $\displaystyle=\left\lceil
x_{K}\right\rceil+m-1-\sum\limits_{j=1}^{K-1}(1+\mu_{j})$ (6.12)
$\displaystyle\leq\sum\limits_{j=1}^{K}(1+\mu_{j})-\sum\limits_{j=1}^{K-1}(1+\mu_{j})=1+\mu_{X}(x_{K})\,,$

and by (6.8) for $j=K-1$

$\displaystyle 1+\mu_{K}$ $\displaystyle=\lceil
x_{K}\rceil+m-1-\sum\limits_{j=1}^{K-1}(1+\mu_{j})$ $\displaystyle\geq\lceil
x_{K-1}\rceil+m-1-\sum\limits_{j=1}^{K-1}(1+\mu_{j})>0\,.$

Since $\mu_{K}\in\mathbb{Z}$, we obtain $0\leq\mu_{K}\leq\mu_{X}(x_{K}^{*})$.
To show that $x_{K}^{*}$ satisfies the Schoenberg-Whitney condition (6.3), we
recall that for $j\leq K-1$, $\mu_{j}=\mu_{X}(x_{j})$, thus the estimate in
(6.9) holds for $x_{K}^{*}$ as well. The lower bound holds by choice of
$\mu_{K}$:

$\displaystyle x_{K}^{*}=x_{K}>\left\lceil
x_{K}\right\rceil-1=\mu_{K}-m+1+\sum\limits_{n=1}^{K-1}\left(1+\mu_{n}\right).$

Step 4: Generic Case $K>1,x_{K}>1$.  If $x_{K}>1$, and additionally
$x_{K-1}\notin\mathbb{Z}$ or $\mu_{K-1}\leq m-2$, then we set $N\coloneqq K$
and $L\coloneqq L_{0}=\lceil x_{K}\rceil$ so that $L-1\leq x_{N}^{*}=x_{K}\leq
L$. By construction the points $x^{*}_{1}<\dots<x^{*}_{N}$ with
submultiplicities $\mu_{1},\dots,\mu_{N}$ satisfy (i) - (iii) and we are done.

It remains to consider the exceptional cases: (i) $x_{K}\geq 1$ with
$x_{K}\in\mathbb{Z}$ and $\mu_{K}=m-1$, (ii) $x_{K}<1$, and (iii) $K=1$.

Step 5. Case $x_{K}^{*}=x_{K}\geq 1$, $x_{K}\in\mathbb{Z}$, $\mu_{K}=m-1$. In
this case, we add the next sampling point $x_{K+1}$ and set

$x_{K+1}^{*}\coloneqq x_{K+1}\quad\text{ and
}\quad\mu_{K+1}\coloneqq\left\lceil x_{K+1}\right\rceil-x_{K}-1\,.$

Since $\left\lceil x_{K+1}\right\rceil-x_{k}>0$ is an integer, we have
$\mu_{K+1}\geq 0$. We set $N=K+1$ and $L=\left\lceil x_{K+1}\right\rceil$ and
verify items (i) — (iii) after adding the new point $x^{*}_{K+1}$. We
substitute (6.10) and the definition of $\mu_{K}$ and obtain

(6.13) $\displaystyle\sum\limits_{j=1}^{K+1}(1+\mu_{j})=\lceil
x_{K}\rceil+m-1+1+\mu_{K+1}=\left\lceil x_{K+1}\right\rceil+m-1=L+m-1.$

Next,

(6.14) $\displaystyle\mu_{K+1}$ $\displaystyle=\left\lceil
x_{K+1}\right\rceil-\left\lceil
x_{K}\right\rceil-1<x_{K+1}-x_{K}<\mu_{X}(x_{K+1})+1.~{}$

This inequality of integers implies $\mu_{K+1}\leq\mu_{X}(x_{K+1})$. If also
$x_{K+1}\in\mathbb{Z}$, then

$\mu_{K+1}=x_{K+1}-x_{K}-1<\mu_{X}(x_{K+1})$

and thus $\mu_{K+1}\leq m-2$, which takes care of the extra condition
$(^{*})$.

To verify the Schoenberg-Whitney conditions (6.3), we use
$\sum\limits_{n=1}^{K}(1+\mu_{n})=\lceil x_{K}\rceil+m-1$ from (6.10) and find
the upper estimate from

$\displaystyle x_{K+1}^{*}$ $\displaystyle=x_{K+1}-x_{K}+\lceil x_{K}\rceil$
$\displaystyle<1+\mu_{X}(x_{K})+\sum\limits_{n=1}^{K}(1+\mu_{n})-m+1\leq
1+\sum\limits_{n=1}^{K}(1+\mu_{n})\,,$

whereas the lower estimate follows from

$\displaystyle\mu_{K+1}-m+1+\sum\limits_{n=1}^{K}(1+\mu_{n})$
$\displaystyle=\left\lceil x_{K+1}\right\rceil-\left\lceil
x_{K}\right\rceil-1-m+1+\lceil x_{K}\rceil+m-1$ $\displaystyle=\lceil
x_{K+1}\rceil-1<x_{K+1}\,.$

The last condition (iii) is satisfied by definition because
$x_{K+1}^{*}\in[L,L+1)$.

Step 6: Case $x_{K}<1,K>1$. Let $\widetilde{K}$ be the minimal index with
$x_{\widetilde{K}}>1$. Due to its minimality and the maximum gap condition,
the points $x_{\widetilde{K}-1},x_{\widetilde{K}}$ satisfy

(6.15) $x_{K}\leq x_{\widetilde{K}-1}\leq 1\quad\text{ and }\quad
1<x_{\widetilde{K}}<x_{\widetilde{K}-1}+m\leq 1+m.$

We set $N\coloneqq K+1$,
$L\coloneqq\big{\lceil}x_{\widetilde{K}}\big{\rceil}$, and

$x_{K+1}^{*}\coloneqq x_{\widetilde{K}},\qquad\mu_{K+1}\coloneqq\left\lceil
x_{\widetilde{K}}\right\rceil-2<x_{\tilde{K}}-1\,.$

This choice guarantees that $L=\lceil x_{\widetilde{K}}\rceil\geq 2$,
$\mu_{K+1}\geq 0$, and $x_{K+1}^{*}\in[L-1,L]\cap X$. We have

(6.16) $\mu_{K+1}<x_{\widetilde{K}}-1\leq
x_{\widetilde{K}}-x_{\widetilde{K}-1}<1+\mu_{X}(x_{\widetilde{K}})~{},$

and again $\mu_{K+1}\leq\mu_{X}(x_{K+1}^{*})$.

For the Schoenberg-Whitney condition (6.3) for $x_{K+1}$, we observe that
(6.10) says that $\sum\limits_{j=1}^{K}(1+\mu_{j})=\left\lceil
x_{K}\right\rceil+m-1=m$, so that (6.15) says that

$x_{K+1}^{*}=x_{\tilde{K}}<1+m=1+\sum\limits_{j=1}^{K}(1+\mu_{j})\,,$

and

$\mu_{K+1}-m+1+\sum\limits_{j=1}^{K}(1+\mu_{j})=\lceil
x_{\tilde{K}}\rceil-2-m+1+m=\lceil x_{K+1}^{*}\rceil-1<x_{K+1}^{*}\,.$

Step 7: Case $K=1$. In this case, (6.8) is void and cannot be used for the
Schoenberg-Whitney conditions for $j>1$. We need to modify the selection of
points. We distinguish three subcases.

If $x_{1}>0$, then the inequalities $\mu_{X}(x_{1})-m+1\leq 0<x_{1}<1$ express
the Schoenberg-Whitney condition for $x_{1}$. We proceed as in Step 6 and
choose $x_{2}^{*}=x_{\tilde{K}}$ with $x_{\tilde{K}-1}\leq 1<x_{\tilde{K}}$
and $\mu_{2}=\lceil x_{\tilde{K}}\rceil-2$. We set $N=2$ and $L=\lceil
x_{\tilde{K}}\rceil\geq 2$. Conditions (i) — (iii) are verified readily as
before.

If $x_{1}=0$, then we must have $\mu_{X}(x_{1})=m-1$. If $x_{2}\leq 1$, then
we set

$\displaystyle x_{2}^{*}$ $\displaystyle\coloneqq x_{2},\qquad\mu_{2}\coloneqq
0\,,$ $\displaystyle x_{3}^{*}$
$\displaystyle=x_{\tilde{K}}\qquad\mu_{3}=\lceil x_{\tilde{K}}\rceil-2\,.$

With $N=3$ points and interval length $L=\lceil x_{\tilde{K}}\rceil\geq 2$,
all conditions (i) — (iii) are satisfied.

If $x_{1}=0$ and $x_{2}>1$, set $N=2$ $N\coloneqq 2$, $L\coloneqq\left\lceil
x_{2}\right\rceil\geq 2$, and

$x_{2}^{*}\coloneqq x_{2},\qquad\mu_{2}\coloneqq\left\lceil
x_{2}\right\rceil-2.$

Properties (i) — (iii) are verified as in Step 6.

As we have covered all possible cases, this concludes the proof. ∎

######  Theorem 6.2.

Let $\varphi$ be a PEB-spline of order $m\in\mathbb{N}$ and
$X\subseteq\mathbb{R}$ be a separated set with a multiplicity function
$\mu_{X}:X\to\\{0,\dots,m-1\\}$. If

(6.17)
$\mathfrak{mg}(X,\mu_{X})=\max\left\\{\sup\limits_{j\in\mathbb{Z}}\frac{x_{j+1}-x_{j}}{1+\mu_{X}(x_{j})}\,,\,\sup\limits_{j\in\mathbb{Z}}\frac{x_{j+1}-x_{j}}{1+\mu_{X}(x_{j+1})}\right\\}<1,$

then $(X,\mu)$ is a uniqueness set for $V^{p}(\varphi)$ for all $1\leq
p\leq\infty$.

###### Proof.

Assume that $f^{(s)}(x)=0$ for all $0\leq s\leq\mu_{X}(x)$ for all $x\in X$.
We need to prove that $f\equiv 0$.

Step 1. Let $x\in X$ be arbitrary and set $a=\left\lfloor x\right\rfloor$. By
Lemma 6.1, there exists an integer length $L\geq 2$ and points $a\leq
x_{1}^{*}<\dots<x_{N}^{*}\leq a+L$ with multiplicities
$\mu_{1},\dots,\mu_{N}\in\\{0,\dots,m-1\\}$ satisfying properties (i) — (iii)
(correct dimension count, Schoenberg-Whitney conditions and
$x_{N}^{*}\in[L-1,L]$). By assumption on $f$, we know that

(6.18)
$0=f^{(s_{j})}(x^{*}_{j})=\Big{(}\sum\limits_{\ell\in\mathbb{Z}}c_{\ell}T_{\ell}\varphi\Big{)}^{(s_{j})}(x^{*}_{j})=\Big{(}\sum\limits_{\ell=a-m+1}^{a+L-1}c_{\ell}T_{\ell}\varphi\Big{)}^{(s_{j})}(x^{*}_{j}),\quad
0\leq s_{j}\leq\mu_{j},\ 1\leq j\leq N.$

Since the Schoenberg-Whitney conditions are satisfied for the points
$x_{j}^{*}$ with multiplicities $\mu_{j},j=1,\dots,N$, Corollary 3.6 asserts
that $f=0$ on $[a,a+L]$.

Step 2. By induction we construct a sequence of intervals
$[a_{k},a_{k}+L_{k}]$ such that

$\displaystyle\bigcup\limits_{k=1}^{\infty}\,[a_{k},a_{k}+L_{k}]$
$\displaystyle=[\left\lfloor x\right\rfloor,\infty),$
$\displaystyle[a_{k}+L_{k}-1,a_{k}+L_{k}]\cap X$ $\displaystyle\neq\emptyset$
$\displaystyle f$ $\displaystyle\equiv 0\text{ on }[a_{k},a_{k}+L_{k}]\text{
for all }k\in\mathbb{N}\,.$

We start with $a_{1}=\lfloor x\rfloor$ and $L_{1}=L$ from Step 1. Assume that
we have already constructed $K$ intervals with this property by induction. By
the induction hypothesis, there exists a point

(6.19) $\tilde{x}\in[a_{K}+L_{K}-1,a_{K}+L_{K}]\cap X\,.$

Set $a_{K+1}\coloneqq\left\lfloor\tilde{x}\right\rfloor$. By Step 1, there
exists an $L_{K+1}\geq 2$ such that $f=0$ on the interval
$[a_{K+1},a_{K+1}+L_{K+1}]$ and $[a_{K+1}+L_{K+1}-1,a_{K+1}+L_{K+1}]\cap
X\neq\emptyset.$ The sequence $a_{K}+L_{K}$ is unbounded, because
$a_{K+1}+L_{K+1}\geq L_{K}+a_{K}-1+2$. Consequently,

(6.20) $\bigcup\limits_{k=1}^{K}[a_{k},a_{k}+L_{k}]=[\left\lfloor
x\right\rfloor,a_{K+1}+L_{K+1}].$

Since $X$ is unbounded from below and $a_{K}+L_{K}$ is unbounded, $f$ vanishes
on the union $\bigcup_{x\in X,K\to\infty}[\lceil
x\rceil,a_{K}+L_{K}]=\mathbb{R}$. This shows $f\equiv 0$. ∎

We conclude the section with the proof of the sampling theorem.

######  Theorem 6.3 (Maximum Gap Theorem).

Let $\varphi$ be a PEB-spline of order $m\geq 2$ and let
$X\subseteq\mathbb{R}$ be a separated set with multiplicity function
$\mu_{X}:X\to\\{0,\dots,m-1\\}$. If the multiplicity function satisfies

(6.21) $\mathrm{dist}\left(\left\\{x\in X:\mu_{X}(x)=m-1\right\\},\
\mathbb{Z}\,\right)>0$

and the weighted maximum gap of $(X,\mu_{X})$ satisfies

(6.22) $\mathfrak{mg}(X,\mu_{X})<1,$

then $(X,\mu_{X})$ is a sampling set for $V^{p}(\varphi)$, $1\leq
p\leq\infty$.

###### Proof.

By Corollary 5.3, it suffices to prove that every weak limit of integer
translates is a uniqueness set for $V^{\infty}(\varphi)$. Let $(Y,\mu_{Y})\in
W_{\mathbb{Z}}(X,\mu_{X})$ be a weak limit of integer translates of
$(X,\mu_{X})$. Lemma 5.4 asserts that $\mathfrak{mg}(Y,\mu_{Y})<1$. By Theorem
6.2, $(Y,\mu_{Y})$ is a uniqueness set for $V^{\infty}(\varphi)$, and we are
done. ∎

Constant multiplicity. Assume that $\mu_{X}\equiv s<m-1$, i.e., we sample the
same number of derivatives at every point $x\in X$. In this case,

$\mathfrak{mg}(X,\mu_{X})=\frac{\mathfrak{mg}(X)}{s+1}\,.$

We then obtain the following consequence.

######  Corollary 6.4.

Let $\varphi$ be a PEB-spline of order $m\geq 2$ and let
$X\subseteq\mathbb{R}$ be a separated set. Assume that the multiplicity
function is constant $\mu_{X}(x)=s<m-1$ for all $x\in X$. If the maximum gap
of $X$ satisfies

(6.23) $\mathfrak{mg}(X)<s+1,$

then $(X,\mu_{X})$ is a sampling set for $V^{p}(\varphi)$, $1\leq
p\leq\infty$.

In particular, if $\mu_{X}\equiv 0$ and $\mathfrak{mg}(X)<1$, then $X$ is a
sampling set for $V^{p}(\varphi),1\leq p\leq\infty$.

For $B$-splines and multiplicity $\mu_{X}\equiv 0$, this sampling theorem was
already proved in [4]. The method with weak limits yields a different proof.

For uniform sampling on $X=\alpha\mathbb{Z}$, the maximum gap is
$\mathfrak{mg}(\alpha\mathbb{Z})=\alpha$, and $\alpha<s+1$ guarantees the
sampling inequality

$\sum_{k\in\mathbb{Z}}\sum_{j=0}^{s}|f^{(j)}(\alpha
k)|^{p}\asymp\|f\|_{p}^{p}\qquad\text{ for all }f\in V^{p}(\varphi)\,.$

Note that the weighted density in this case is
$D^{-}(\alpha\mathbb{Z},\mu)=\frac{s+1}{\alpha}>1$. As $\alpha$ approaches
$s+1$, we obtain sampling sets of density arbitrarily close to the critical
density. The condition $\alpha<s+1$ cannot be improved. The set
$\alpha\mathbb{Z}$ for $\alpha>s+1$ fails to have the necessary density and is
thus not a sampling set 777 One can show that there is always some
$x_{0}\in\mathbb{R}$, such that $x_{0}+(s+1)\mathbb{Z}$ fails to be sampling..
In this case, Theorem 6.3 is optimal and cannot be improved.

##  7\. Discussion of the methods

Comparison. Both theorems (Theorems 4.2 and 6.3) are based on similar ideas.
We choose a suitable covering of $\mathbb{R}$ and study sampling locally on
each interval of the partition by means of the active collocation matrix.

In Theorem 4.2 we control the distance of a sampling point to the boundary of
the support of $\varphi$ as a security margin. This parameter guarantees
uniform bounds for the inversion of the local collocation matrices (Step 2 in
the proof).

In Theorem 6.3, we omit that condition, and the Schoenberg-Whitney conditions
imply only the invertibility of the local collocation matrices. As a
consequence, we obtain only uniqueness sets. In this case, the theory of weak
limits and subtle criteria for sampling sets help to derive stability
estimates.

Both main theorems are optimal in some sense, but their assumptions are not
comparable. To see this, let us consider a few simple examples. In all
examples, the generator $\varphi$ is a PEB-spline of order $m$ with support in
$[0,m]$.

######  Example 7.1.

Let $X=\alpha\mathbb{Z}$ and consider $\mu_{X}(\alpha
j)=\tfrac{1}{2}(1+(-1)^{j})$ so that $\mu_{X}(\alpha j)=1$ for even $j$ and
$\mu_{X}(\alpha j)=0$ for odd $j$. The weighted maximum gap is then
$\mathfrak{mg}(X,\mu_{X})=\alpha\,.$ Consequently, by Theorem 6.3 $\alpha<1$
implies the sampling inequality

$\sum_{j\in\mathbb{Z}}|f(\alpha
j)|^{p}+\sum_{j\in\mathbb{Z}}|f^{\prime}(2\alpha
j)|^{p}\asymp\|f\|_{p}^{p}\qquad\text{ for all }f\in V^{p}(\varphi)\,.$

However, in an interval of length $K$ there are $K/(2\alpha)$ samples with
multiplicity $0$ and $K/(2\alpha)$ samples with multiplicity $1$ (up to a
bounded error term), so the weighted Beurling density of $(X,\mu_{X})$ is

$D^{-}(X,\mu_{X})=\frac{3}{2\alpha}>\frac{3}{2}\,,$

i.e., the maximum gap does not deliver an optimal estimate.

By contrast, by Theorem 4.4, for every $\varepsilon>0$, using the conditions
of Theorem 4.2, we can produce a set $(X,\mu_{X})$ with weighted Beurling
density $D^{-}(X,\mu_{X})<1+\varepsilon$ that is sampling. Hereby, $\mu_{X}$
can be chosen to be an alternating function as above, and the explicit
construction in the proof of Theorem 4.4 provides a subset of a lattice.

######  Example 7.2.

We provide an example of a set uniformly bounded away from $\mathbb{Z}$ with a
constant multiplicity function $\mu\equiv S-1>0$ and maximum gap
$\mathfrak{mg}(X)=S\,\mathfrak{mg}(X,\mu_{X})<S$ that does not satisfy the
assumptions of Theorem 4.2. While the combinatorial theorem relied heavily on
_overlaps_ , Theorem 4.2 asks for a _partition_ in intervals
$I(k)=I_{M,L}(k)$. The last assumption (4.6) implies that there exists a
sampling point, namely $x_{1}^{k}\in X\cap I_{M,L}(k)$, in the first
subinterval of $I_{M,L}(k)$ of length $1$;

(7.1)
$\begin{split}&[M+kL,M+kL+L)\cap[0-m+M+kL+1+\varepsilon,M+kL+1-\varepsilon)\\\
=&[M+kL,M+kL+1-\varepsilon)\subseteq[M+kL,M+kL+1)\eqqcolon
J_{M,L}(k).\end{split}$

Note that $M,L$ are supposed to be chosen globally. Furthermore, for any fixed
$L$, one can always assume that $0\leq M<L$, as $I_{M+L,L}(k)=I_{M,L}(k+1)$.
Hence, to prove that the claimed set exists, it suffices to construct a
separated set $X\subseteq\mathbb{R}$ with maximum gap $\mathfrak{mg}(X)<2$ and
$\mathrm{dist}(X,\mathbb{Z})>0$ such that for all
$M\in\mathbb{Z},\,L\in\mathbb{N}$, there exists an integer $k\in\mathbb{Z}$
with $X\cap J_{M,L}(k)=\emptyset$. The set can be constructed inductively by
starting with a sufficiently dense lattice $X_{0}$, e.g.,
$X_{0}=\frac{1}{n}\mathbb{Z}$ for some $n\in\mathbb{N},n\geq 2$, and for each
$L\in\mathbb{N}$ and each $0\leq M\leq L$ removing the first $n$ points of
$X_{0}$ in the interval $I_{M,L}((M+L)2^{2^{L}})$ (these are the points $X\cap
J_{M,L}(k)$), resulting in a new sampling set $X$ given by

$X\coloneqq
X_{0}\setminus\bigg{(}\bigcup\limits_{L\in\mathbb{N}}\bigcup\limits_{M=0}^{L-1}X\cap
J_{M,L}((M+L)2^{2^{L}})\bigg{)}.$

One can verify that each of the distinguished intervals is sufficiently far
apart from the others so that the maximum gap is
$\mathfrak{mg}(X)=1+\frac{1}{n}<2$. By construction of the set, there are no
integers $M,L\in\mathbb{Z}$ that provide a suitable partition of the real axis
where Theorem 4.2 is applicable. However, this is also a sampling set because
the weighted maximum gap is
$\mathfrak{mg}(X,\mu_{X})=\frac{1}{S}\mathfrak{mg}(X)<\frac{2}{S}\leq 1$, so
the conditions of Theorem 6.3 are satisfied.

##  8\. Implications for Gabor systems

In the final section, we exploit the connection between sampling in shift-
invariant spaces and Gabor frames and improve a result in [24].

We denote with $\pi(x,\omega)$, $(x,\omega)\in\mathbb{R}^{2}$, the time-
frequency shift (operator) acting on functions as $\pi(x,\omega)f(t)=e^{2\pi
i\omega t}f(t-x)$. In terms of stable expansions, given a window function
$\varphi$ and a discrete set $\Lambda\subseteq\mathbb{R}^{2}$, one asks when a
Gabor system $\mathcal{G}(\varphi,\Lambda)$, defined as

(8.1) $\mathcal{G}(g,\Lambda)=\\{\pi(\lambda)\varphi:\lambda\in\Lambda\\},$

is a (Gabor) frame, i.e., it satisfies the frame inequality

(8.2) $A\left\lVert
f\right\rVert_{2}^{2}\leq\sum\limits_{\lambda\in\Lambda}\left\lvert\left\langle
f,\pi(\lambda)\varphi\right\rangle\right\rvert^{2}\leq B\left\lVert
f\right\rVert_{2}^{2},\qquad f\in L^{2}(\mathbb{R}),$

with frame bounds $0<A\leq B<\infty$ independent of $f\in L^{2}(\mathbb{R})$.
From the general frame theory, the inequality implies a reconstruction formula

(8.3) $f=\sum\limits_{\lambda\in\Lambda}\langle
f,\pi(\lambda)\varphi\rangle\,\psi_{\lambda}=\sum\limits_{\lambda\in\Lambda}\langle
f,\psi_{\lambda}\rangle\,\pi(\lambda)\varphi,\qquad f\in L^{2}(\mathbb{R})$

with $\left(\langle f,\pi(\lambda)\varphi\rangle\right)_{\lambda\in\Lambda}$,
$\left(\langle
f,\psi_{\lambda}\rangle\right)_{\lambda\in\Lambda}\in\ell^{2}(\Lambda)$ and
$\psi_{\lambda}\in L^{2}(\mathbb{R})$ for all $\lambda\in\Lambda$. Therefore,
given $\varphi$, we are interested in determining those sets
$\Lambda\subseteq\mathbb{R}^{2}$ whose Gabor system
$\mathcal{G}(\varphi,\Lambda)$ is a frame.

While there is no manageable tool available for arbitrary point
configurations, semi-regular sets of type $\Lambda=X\times\mathbb{Z}$ can be
treated by the connection between sampling in shift-invariant spaces and Gabor
frames [20, Thm. 3.1, Thm. 3.3].

######  Theorem 8.1.

Assume that $\varphi\in C(\mathbb{R})$ decays as
$|\varphi(x)|\lesssim(1+|x|)^{-1-\varepsilon}$ for an $\varepsilon>0$, and has
stable integer shifts. Let $X\subseteq\mathbb{R}$ be a separated set. Then the
following are equivalent:

  1. (i)

The family $\mathcal{G}(\varphi,(-X)\times\mathbb{Z})$ is a frame for
$L^{2}(\mathbb{R})$.

  2. (ii)

$X$ is a sampling set of $V^{p}(\varphi)$ for some $p\in[1,\infty]$.

  3. (iii)

$X$ is a sampling set of $V^{p}(\varphi)$ for all $p\in[1,\infty]$.

The following corollary is a small extension of a result in [4] from B-splines
to PEB-splines.

######  Corollary 8.2.

Let $\varphi$ be a PEB-spline of order $m\geq 2$. Assume
$X\subseteq\mathbb{R}$ is a separated set satisfying the following property:

$(^{*})$ There exist integers $M,L>0$ and $\varepsilon>0$, such that for every
$k\in\mathbb{Z}$, there exist points
$x_{1}^{k}<x_{2}^{k}<\dots<x^{k}_{L+{m-1}}$ in $X\cap I_{M,L}(k)$ with

(8.4) $x_{j}^{k}\in[M+kL+j-m+\varepsilon,M+kL+j-\varepsilon].$

Then $\mathcal{G}(\varphi,(-X)\times\mathbb{Z})$ is a Gabor frame. In
particular, if $X$ is separated and $\mathfrak{mg}(X)<1$, then
$\mathcal{G}(\varphi,X\times\mathbb{Z})$ is a Gabor frame.

###### Proof.

We proved that $\varphi$ has stable integer shifts in the proof of Theorem
4.2. The sampling property of $X$ follows from Theorem 4.2, applied to
$(X,0)$, and Corollary 4.3. The claim follows from Theorem 8.1. ∎

A direct consequence of the last corollary is the lattice case.

######  Corollary 8.3.

Let $\varphi$ be a PEB-spline of order $m\geq 2$. Then
$\mathcal{G}(\varphi,\alpha\mathbb{Z}\times\mathbb{Z})$ is a Gabor frame if
and only if $0<\alpha<1$.

###### Proof.

The sufficient part is due to Corollary 8.2. The necessary part is due to the
Balian-Low theorem, see, e.g., [9]. ∎

##  9\. Appendix

We sketch the proof of (an extended version of) Corollary 5.3. The only
changes are required for the last, right-continuous $\varphi^{(s)}$.

######  Corollary 9.1.

Let $\varphi:\mathbb{R}\to\mathbb{C}$ be a compactly supported, piecewise
continuous function with finitely many jump discontinuities
$\mathcal{J}\subseteq\mathbb{R}$. Further, assume that $\varphi$ has stable
integer translates. Let $X$ be a separated set satisfying

(9.1)
$\varepsilon\coloneqq\mathrm{dist}\big{(}X\,,\mathcal{J}+\mathbb{Z}\big{)}>0.$

Then the following statements are equivalent:

  1. (i)

$X$ is a sampling set for $V^{p}(\varphi)$ for some $p\in[1,\infty]$.

  2. (ii)

$X$ is a sampling set for $V^{p}(\varphi)$ for all $p\in[1,\infty]$.

  3. (iii)

Every weak limit $Y\in W_{\mathbb{Z}}(X)$ is a sampling set for
$V^{\infty}(\varphi)$.

  4. (iv)

Every weak limit $Y\in W_{\mathbb{Z}}(X)$ is a uniqueness set for
$V^{\infty}(\varphi)$.

###### Proof sketch.

The sampling property can be expressed in terms of the pre-Gramian matrices

(9.2) $P_{X}(\varphi)\coloneqq\left(\varphi(x-k)\right)_{x\in
X,k\in\mathbb{Z}}.$

Precisely, $X$ is sampling for $V^{p}(\varphi)$ if and only if
$P_{X}(\varphi)$ is bounded above and below on $\ell^{p}(\mathbb{Z})$. By
Schur’s test, $P_{X}:\ell^{p}(\mathbb{Z})\to\ell^{p}(X)$ is a bounded linear
operator for all $p\in[1,\infty]$. We make use of a non-commutative version of
Wiener’s Lemma [19, Prop. 8.1].

######  Proposition 9.2.

Let $X,Y\subseteq\mathbb{R}$ be relatively separated and
$A\in\mathbb{C}^{X\times Y}$ be a matrix such that 888The decay conditions can
be weakened, e.g., $\left\lvert A_{x,y}\right\rvert\lesssim\Theta(x-y)$ for
some continuous upper bound $\Theta$ in the Wiener-amalgam space
$W(\mathbb{R})$.

(9.3) $\left\lvert A_{x,y}\right\rvert\lesssim(1+|x-y|)^{-1-\varepsilon}\qquad
x\in X,y\in Y\qquad\text{for some }\varepsilon>0.$

The operator $A$ is bounded below on some $\ell^{p_{0}}(Y)$, $1\leq
p_{0}\leq\infty$, i.e., $\left\lVert Ac\right\rVert_{p_{0}}\geq
C_{0}\left\lVert c\right\rVert_{p_{0}}$ for all $c\in\ell^{p_{o}}(Y)$, if and
only if $A$ is bounded below on _all_ $\ell^{p}(Y)$, $1\leq p\leq\infty$.

Since $\varphi$ is supported on some compact set $K\subseteq\mathbb{R}$ and
bounded,

(9.4)
$\big{|}\big{(}P_{X}(\varphi)\big{)}_{x,k}\big{|}\lesssim(1+|x-k|)^{-1-\varepsilon}\qquad
x\in X,k\in\mathbb{Z}\qquad\text{for all }\varepsilon>0.$

This gives the equivalence $(i)\Leftrightarrow(ii)$.

$(ii)\Rightarrow(iii)$ Let $Y\in W_{\mathbb{Z}}(X)$, i.e., $X-k_{n}\to Y$
weakly for some $(k_{n})_{n\in\mathbb{N}}\subseteq\mathbb{Z}$. Since $X$ is a
sampling set, $P_{X}(\varphi):\ell^{\infty}(\mathbb{Z})\to\ell^{\infty}(X)$ is
bounded from below, so its pre-adjoint
$P_{X}(g)^{\prime}:\ell^{1}(X)\to\ell^{1}(\mathbb{Z})$, given by

(9.5) $P_{X}(\varphi)^{\prime}=\left(\varphi(x-k)\right)_{k\in\mathbb{Z},x\in
X},$

is onto by the closed graph theorem. Fix now $c\in\ell^{1}(\mathbb{Z})$. For
every sequence $(c_{k-k_{n}})_{k\in\mathbb{Z}}$ there exists a sequence
$a^{n}\in\ell^{1}(X)$ with $\left\lVert
a^{n}\right\rVert_{1}\lesssim\left\lVert c\right\rVert_{1}$ and

(9.6) $c_{k-k_{n}}=\sum\limits_{x\in X}a^{n}_{x}g(x-k),\qquad k\in\mathbb{Z}.$

The weak convergence can be rewritten by means of distributions. Set
$b^{n}_{x-k_{n}}=a^{n}_{x}$ and $\mu_{n}\coloneqq\sum_{z\in
X-k_{n}}b^{n}_{z}\,\delta_{z}$. Then $\mu_{n}\in C_{0}^{*}(\mathbb{R})$ has
support in $X-k_{n}$ and satisfies

(9.7) $\left\lVert\mu_{n}\right\rVert_{C^{*}_{0}(\mathbb{R})}=\left\lVert
b^{n}\right\rVert_{1}=\left\lVert a^{n}\right\rVert_{1}\lesssim\left\lVert
c\right\rVert_{1}.$

By restricting to a subsequence and using the $w^{*}$-completeness of
$C^{*}_{0}(\mathbb{R})$, $\mu_{n}$ converges to $\mu\in C^{*}_{0}(\mathbb{R})$
in $\sigma(C_{0}^{*}(\mathbb{R}),C_{0}(\mathbb{R}))$. Since $X-k_{n}\to Y$,
the weak limit $Y$ is relatively separated and
$\operatorname{supp}(\mu)\subseteq Y$ (see [19, Lem. 4.3 and 4.5]).
Furthermore, $\mu=\sum_{y\in Y}b_{y}\delta_{y}$ for suitable
$b\in\ell^{1}(Y)$,

(9.8) $\left\lVert
b\right\rVert_{1}=\left\lVert\mu\right\rVert_{C^{*}_{0}(\mathbb{R})}\lesssim\liminf\limits_{n\to\infty}\left\lVert\mu_{n}\right\rVert_{C^{*}_{0}(\mathbb{R})}.$

Here comes the decisive modification. By assumption,
$\mathrm{dist}(X-k_{n},\mathcal{J}+\mathbb{Z})=\mathrm{dist}(X,\mathcal{J}+\mathbb{Z})>0$
for all $n\in\mathbb{Z}$ and therefore
$\mathrm{dist}(Y,\mathcal{J}+\mathbb{Z})>0$. Now we replace the generator
$\varphi$ (which has discontinuities $\mathcal{J}$) by a continuous compactly
supported function $\varphi_{0}$ such that $\varphi=\varphi_{0}$ on
$\mathbb{R}\setminus\big{(}\mathcal{J}+\mathbb{Z}+(-\varepsilon,\varepsilon)\big{)}$
for an $\varepsilon\in(0,\mathrm{dist}(X,\mathcal{J}+\mathbb{Z}))$. In
particular, $\varphi=\varphi_{0}$ on $Y$ and on $X-k_{n}$ for all
$n\in\mathbb{N}$. Hence we obtain

(9.9) $\displaystyle\sum\limits_{y\in Y}b_{y}\varphi(y-k)$
$\displaystyle=\int_{\mathbb{R}}\varphi_{0}(x-k)\,d\mu(x)=\lim\limits_{n\to\infty}\int_{\mathbb{R}}\varphi_{0}(x-k)\,d\mu_{n}(x)$
(9.10) $\displaystyle=\lim\limits_{n\to\infty}\sum\limits_{z\in
X-k_{n}}b_{z}^{n}\varphi_{0}(z-k)=\lim\limits_{n\to\infty}\sum\limits_{z\in
X-k_{n}}b_{z}^{n}\varphi(z-k)=c_{k}.$

$(iii)\Rightarrow(iv)$ is trivial. To obtain $(iv)\Rightarrow(ii)$, one
proceeds precisely as in [19], with the analogous modification with
$\varphi_{0}$ when it comes to the limits. ∎

The lift to sampling with multiplicities in Corollary 5.3 is the same as in
[21].

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