# On numerical realizations of Shannon’s sampling theorem

Melanie Kircheis111Corresponding author<EMAIL_ADDRESS>Chemnitz University of Technology, Faculty of Mathematics, D–09107 Chemnitz,
Germany Daniel<EMAIL_ADDRESS>Chemnitz University of
Technology, Faculty of Mathematics, D–09107 Chemnitz, Germany Manfred
<EMAIL_ADDRESS>University of Rostock, Institute of
Mathematics, D–18051 Rostock, Germany

###### Abstract

In this paper, we discuss some numerical realizations of Shannon’s sampling
theorem. First we show the poor convergence of classical Shannon sampling sums
by presenting sharp upper and lower bounds of the norm of the Shannon sampling
operator. In addition, it is known that in the presence of noise in the
samples of a bandlimited function, the convergence of Shannon sampling series
may even break down completely. To overcome these drawbacks, one can use
oversampling and regularization with a convenient window function. Such a
window function can be chosen either in frequency domain or in time domain. We
especially put emphasis on the comparison of these two approaches in terms of
error decay rates. It turns out that the best numerical results are obtained
by oversampling and regularization in time domain using a $\sinh$-type window
function or a continuous Kaiser–Bessel window function, which results in an
interpolating approximation with localized sampling. Several numerical
experiments illustrate the theoretical results.

_Key words_ : Shannon sampling sums, Whittaker–Kotelnikov–Shannon sampling
theorem, bandlimited function, regularization with window function,
regularized Shannon sampling formulas, error estimates, numerical robustness.

AMS _Subject Classifications_ : 94A20, 65T50.

##  1 Introduction

The classical Whittaker–Kotelnikov–Shannon sampling theorem  (see [32, 11,
25]) plays a fundamental role in signal processing, since it describes the
close relation between a bandlimited function and its equidistant samples. A
function $f\in L^{2}(\mathbb{R})$ is called bandlimited with bandwidth
$\frac{N}{2}$, if the support of its Fourier transform

${\hat{f}}(v)\coloneqq\int_{\mathbb{R}}f(t)\,{\mathrm{e}}^{-2\pi{\mathrm{i}}tv}\,{\mathrm{d}}t\,,\quad
v\in\mathbb{R}\,,$

is contained in $\big{[}-\tfrac{N}{2},\,\tfrac{N}{2}\big{]}$. Let the space of
all bandlimited functions with bandwidth $\frac{N}{2}$ be denoted by

${\mathcal{B}}_{N/2}(\mathbb{R})\coloneqq\Big{\\{}f\in
L^{2}(\mathbb{R})\colon\,\mathrm{supp}\,\hat{f}\subseteq\big{[}-\tfrac{N}{2},\,\tfrac{N}{2}\big{]}\Big{\\}}\,,$
(1.1)

which is also known as the _Paley–Wiener space_. By definition (1.1), the
Paley–Wiener space ${\mathcal{B}}_{N/2}(\mathbb{R})$ consists of equivalence
classes of almost equal functions. However, it can be shown (see e.g. [10, p.
6]) that there is always a smooth representation, since for each
$r\in\mathbb{N}_{0}$ we have by inverse Fourier transform that

$\displaystyle
f^{(r)}(t)=\int_{-N/2}^{N/2}\hat{f}(v)\,(2\pi\mathrm{i}v)^{r}\,\mathrm{e}^{2\pi\mathrm{i}vt}\,\mathrm{d}t$

and $(2\pi\mathrm{i}\,\cdot)^{r}\hat{f}\in
L^{1}\big{(}\big{[}-\tfrac{N}{2},\,\tfrac{N}{2}\big{]}\big{)}$ and therefore
$f^{(r)}\in C_{0}(\mathbb{R})$, where $C_{0}(\mathbb{R})$ denotes the Banach
space of continuous functions $f\colon\mathbb{R}\to\mathbb{C}$ vanishing as
$|t|\to\infty$ with the norm

$\|f\|_{C_{0}(\mathbb{R})}\coloneqq\max_{t\in\mathbb{R}}|f(t)|\,.$

That is to say, we have ${\mathcal{B}}_{N/2}(\mathbb{R})\subseteq
L^{2}(\mathbb{R})\cap C_{0}(\mathbb{R})\cap C^{\infty}(\mathbb{R})$.

Then the Whittaker–Kotelnikov–Shannon sampling theorem states that any
bandlimited function $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ can be recovered
from its samples $f\big{(}\tfrac{k}{L}\big{)}$, $k\in\mathbb{Z}$, with $L\geq
N$ as

$f(t)=\sum_{k\in\mathbb{Z}}f\big{(}\tfrac{k}{L}\big{)}\,\mathrm{sinc}(L\pi
t-\pi k)\,,\quad t\in\mathbb{R}\,,$ (1.2)

where the $\mathrm{sinc}$ function is given by

$\mathrm{sinc}\,x\coloneqq\left\\{\begin{array}[]{ll}\frac{\sin
x}{x}&\quad\colon x\in\mathbb{R}\setminus\\{0\\}\,,\\\\[4.30554pt]
1&\quad\colon x=0\,.\end{array}\right.$

It is well known that the series in (1.2) converges absolutely and uniformly
on whole $\mathbb{R}$. Unfortunately, the practical use of this sampling
theorem is limited, since it requires infinitely many samples, which is
impossible in practice. Furthermore, the $\mathrm{sinc}$ function decays very
slowly such that the _Shannon sampling series_

$\sum_{k\in\mathbb{Z}}f\big{(}\tfrac{k}{L}\big{)}\,\mathrm{sinc}(L\pi
t-k\pi)\,,\quad t\in\mathbb{R}\,,$ (1.3)

with $L\geq N$ has rather poor convergence, as can be seen from the sharp
upper and lower bounds of the norm of the Shannon sampling operator (see
Theorem 2.2). In addition, it is known (see [6]) that in the presence of noise
in the samples $f\big{(}\frac{k}{L}\big{)}$, $k\in\mathbb{Z}$, of a
bandlimited function $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$, the convergence of
Shannon sampling series may even break down completely. To overcome these
drawbacks, many applications employ _oversampling_ , i.e., a function $f\in
L^{2}(\mathbb{R})$ of bandwidth $\frac{N}{2}$ is sampled on a finer grid
$\frac{1}{L}\,\mathbb{Z}$ with $L>N$, where the oversampling is measured by
the _oversampling parameter_ $\lambda\coloneqq\frac{L-N}{N}\geq 0$. In
addition, we consider several regularization techniques, where a so-called
_window function_ is used. Since this window function can be chosen in
frequency domain or in spatial domain, we study both approaches and compare
the theoretical and numerical approximation properties in terms of decay
rates.

On the one hand, we investigate the regularization with a window function in
frequency domain (called _frequency window function_), cf. e.g. [5, 15, 24,
17, 28]. Here we use a suitable function of the form

${\hat{\psi}}(v)\coloneqq\left\\{\begin{array}[]{ll}1&\colon|v|\leq\tfrac{N}{2}\,,\\\
\chi(|v|)&\colon\tfrac{N}{2}<|v|<\tfrac{L}{2}\,,\\\
0&\colon|v|\geq\tfrac{L}{2}\,,\end{array}\right.$

where $\chi\colon\big{[}\tfrac{N}{2},\,\tfrac{L}{2}\big{]}\to[0,\,1]$ is
frequently chosen as some monotonously decreasing, continuous function with
$\chi\big{(}\frac{N}{2}\big{)}=1$ and $\chi\big{(}\frac{L}{2}\big{)}=0$.
Applying inverse Fourier transform, we determine the corresponding function
$\psi$ in time domain. Since $\hat{\psi}$ is compactly supported, the
uncertainty principle (cf. [18, p. 103, Lemma 2.39]) yields
$\mathrm{supp}\,\psi=\mathbb{R}$. Then it is known that the function $f$ can
be represented in the form

$f(t)=\sum_{k\in\mathbb{Z}}f\big{(}\tfrac{k}{L}\big{)}\,\frac{1}{L}\,\psi\big{(}t-\tfrac{k}{L}\big{)}\,,\quad
t\in\mathbb{R}\,.$

Using uniform truncation, we approximate a function
$f\in\mathcal{B}_{N/2}(\mathbb{R})$ by the $T$-th partial sum

$(P_{\psi,T}f)(t)\coloneqq\sum_{k=-T}^{T}f\big{(}\tfrac{k}{L}\big{)}\,\frac{1}{L}\,\psi\big{(}t-\tfrac{k}{L}\big{)}\,,\quad
t\in[-1,\,1]\,.$

On the other hand, we examine the regularization with a window function in
time domain (called _time window function_), cf. e.g. [21, 13, 14, 10]. Here a
suitable window function $\varphi\colon\mathbb{R}\to[0,\,1]$ with compact
support $\big{[}-\frac{m}{L},\,\frac{m}{L}\big{]}$ belongs to the set
$\Phi_{m/L}$ (as defined in Section 4) with some
$m\in\mathbb{N}\setminus\\{1\\}$. Then we recover a function
$f\in\mathcal{B}_{N/2}(\mathbb{R})$ by the _regularized Shannon sampling
formula_

$\displaystyle(R_{\varphi,m}f)(t)\coloneqq\sum_{k\in\mathbb{Z}}f\big{(}\tfrac{k}{L}\big{)}\,{\mathrm{sinc}}\big{(}L\pi
t-\pi k\big{)}\,\varphi\big{(}t-\tfrac{k}{L}\big{)}\,,\quad t\in\mathbb{R}\,,$

with $L\geq N$. By defining the set $\Phi_{m/L}$ of window functions
$\varphi$, only small truncation parameters $m$ are needed to achieve high
accuracy, resulting in short sums being computable very fast. In other words,
this approach uses localized sampling. Moreover, this method is an
interpolating approximation, since for all $n,\,k\in\mathbb{Z}$ we have

$\displaystyle{\mathrm{sinc}}\big{(}L\pi t-\pi
k\big{)}\,\varphi\big{(}t-\tfrac{k}{L}\big{)}\big{|}_{t=\frac{n}{L}}=\delta_{n,k}\,.$

In this paper we propose new estimates of the uniform approximation errors

$\displaystyle\max_{t\in[-1,\,1]}\big{|}f(t)-(P_{\psi,T}f)(t)\big{|}\quad\text{
and }\quad\max_{t\in\mathbb{R}}\big{|}f(t)-(R_{\varphi,m}f)(t)\big{|}\,,$

where we apply several window functions $\hat{\psi}$ and $\varphi$.  Note that
for the frequency window functions $\hat{\psi}$ only error estimates on
certain compact interval (such as for example $[-1,\,1]$) can be given, while
results on whole $\mathbb{R}$ are not feasible. It is shown in the subsequent
sections that the uniform approximation error decays algebraically with
respect to $T$, if $\hat{\psi}$ is a frequency window function. Otherwise, if
$\varphi\in\Phi_{m/L}$ is chosen as a time window function such as the
$\sinh$-type or continuous Kaiser–Bessel window function, then the uniform
approximation error decays exponentially with respect to $m$.

To this end, this paper is organized as follows. First, in Section 2 we show
the poor convergence of classical Shannon sampling sums and improve results on
the upper and lower bounds of the norm of the Shannon sampling operator.
Consequently, we study the different regularization techniques. In Section 3
we start with the regularization using a frequency window function. After
recapitulating a general result in Theorem 3.3, we consider window functions
of different regularity and present the corresponding algebraic decay results
in Theorems 3.4 and 3.7. Subsequently, in Section 4 we proceed with the
regularization using a time window function. Here we also review the known
general result in Theorem 4.1 and afterwards demonstrate the exponential decay
of the considered $\sinh$-type and continuous Kaiser–Bessel window functions
in Theorems 4.2 and 4.3. Finally, in Section 5 we compare the previously
considered approaches from Sections 3 and 4 to illustrate our theoretical
results.

##  2 Poor convergence of Shannon sampling sums

In order to show that the Shannon sampling series (1.3) has rather poor
convergence, we truncate the series (1.3) with $T\in\mathbb{N}$, and consider
the $T$-_th_ _Shannon sampling sum_

$(S_{T}f)(t)\coloneqq\sum_{k=-T}^{T}f\big{(}\tfrac{k}{L}\big{)}\,\mathrm{sinc}(L\pi
t-k\pi)\,,\quad t\in\mathbb{R}\,.$ (2.1)

Obviously, this operator can be formed for each $f\in C_{0}(\mathbb{R})$.

######  Lemma 2.1.

The linear operator $S_{T}\colon\,C_{0}(\mathbb{R})\to C_{0}(\mathbb{R})$ has
the norm

$\|S_{T}\|=\max_{t\in\mathbb{R}}\sum_{k=-T}^{T}\big{|}\mathrm{sinc}(L\pi
t-k\pi)\big{|}\,.$ (2.2)

_Proof_. For each $f\in C_{0}(\mathbb{R})$ and $t\in\mathbb{R}$ we have

$\displaystyle\big{|}(S_{T}f)(t)|$
$\displaystyle\leq\sum_{k=-T}^{T}\big{|}f\big{(}\tfrac{k}{L}\big{)}\big{|}\,\big{|}\mathrm{sinc}(L\pi
t-k\pi)\big{|}\leq\sum_{k=-T}^{T}\big{|}\mathrm{sinc}(L\pi
t-k\pi)\big{|}\,\|f\|_{C_{0}(\mathbb{R})}\,,$

such that

$\|S_{T}f\|_{C_{0}(\mathbb{R})}\leq\max_{t\in\mathbb{R}}\sum_{k=-T}^{T}\big{|}\mathrm{sinc}(L\pi
t-k\pi)\big{|}\,\|f\|_{C_{0}(\mathbb{R})}\,.$

By defining the even nonnegative function

$s_{T}(t)\coloneqq\sum_{k=-T}^{T}\big{|}\mathrm{sinc}(L\pi
t-k\pi)\big{|}\,,\quad t\in\mathbb{R}\,,$ (2.3)

which is contained in $C_{0}(\mathbb{R})$, and assuming that $s_{T}$ has its
maximum in $t_{0}\in\mathbb{R}$, this yields

$\|S_{T}\|=\sup\big{\\{}\|S_{T}f\|_{C_{0}(\mathbb{R})}\colon\|f\|_{C_{0}(\mathbb{R})}=1\big{\\}}\leq
s_{T}(t_{0})\,.$

The other way around, we consider the linear spline $g\in C_{0}(\mathbb{R})$
with nodes in $\frac{1}{L}\,\mathbb{Z}$, where

$g\big{(}\tfrac{k}{L}\big{)}=\left\\{\begin{array}[]{ll}\mathrm{sign}\big{(}\mathrm{sinc}(L\pi
t_{0}-k\pi)\big{)}&\colon k=-T\,,\,\ldots,\,T\,,\\\\[1.29167pt] 0&\colon
k\in\mathbb{Z}\setminus\\{-T\,,\,\ldots,\,T\\}\,.\end{array}\right.$

Obviously, we have $\|g\|_{C_{0}(\mathbb{R})}=1$ and

$(S_{T}g)(t)=\sum_{k=-T}^{T}\mathrm{sign}\big{(}\mathrm{sinc}(L\pi
t_{0}-k\pi)\big{)}\,\mathrm{sinc}(L\pi t-k\pi)\big{)}\leq s_{T}(t)\leq
s_{T}(t_{0})\,.$

Then

$(S_{T}g)(t_{0})=\sum_{k=-T}^{T}\big{|}\mathrm{sinc}(L\pi
t_{0}-k\pi)\big{|}=\max_{t\in\mathbb{R}}s_{T}(t)=s_{T}(t_{0})$

implies

$\|S_{T}\|\geq\|S_{T}g\|_{C_{0}(\mathbb{R})}=\max_{t\in\mathbb{R}}\big{|}(S_{T}g)(t)\big{|}=s_{T}(t_{0})$

and hence (2.2).

Now we show that the norm $\|S_{T}\|$ is unbounded with respect to $T$.

######  Theorem 2.2.

The norm of the operator $S_{T}\colon\,C_{0}(\mathbb{R})\to C_{0}(\mathbb{R})$
can be estimated by

$\frac{2}{\pi}\,\big{[}\ln T+2\ln 2+\gamma\big{]}-\frac{1}{\pi
T\,(2T+1)}<\|S_{T}\|<\frac{2}{\pi}\,\big{[}\ln T+2\ln
2+\gamma\big{]}+\frac{T+2}{\pi T\,(T+1)}\,,$ (2.4)

where we use _Euler’s constant_

$\gamma\coloneqq\lim_{T\to\infty}\Bigg{(}\sum_{k=1}^{T}\frac{1}{k}-\ln
T\Bigg{)}=0.57721566\dots\,.$

_Proof_. As  suggested in [27,  p. 142, Problem 3.1.5], we represent  (2.3) in
the form

$s_{T}(t)=\sum_{k=1}^{T+1}a_{k}(t)\,,\quad t\in\mathbb{R}\,,$

with

$a_{k}(t)\coloneqq\left\\{\begin{array}[]{ll}\big{|}\mathrm{sinc}(L\pi
t-k\pi)\big{|}+\big{|}\mathrm{sinc}(L\pi t+(k-1)\pi)\big{|}&\colon
k=1\,,\,\ldots,\,T\,,\\\\[4.30554pt] \big{|}\mathrm{sinc}(L\pi
t+T\pi)\big{|}&\colon k=T+1\,.\end{array}\right.$

Since (2.3) is even, we estimate the maximum of $s_{T}(t)$ only for $t\geq 0$.
For
$k={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}1},\,\ldots,\,T,$
we have  $a_{k}(0)=a_{k}\big{(}\frac{1}{L}\big{)}=0$ and by trigonometric
identities we obtain for $t\in\big{(}0,\,\tfrac{1}{L}\big{)}$ that

$a_{k}(t)=\frac{\sin(L\pi
t)}{\pi}\,\bigg{(}\frac{1}{k-1+Lt}+\frac{1}{k-Lt}\bigg{)}=\frac{(2k-1)\,\sin(L\pi
t)}{\pi\,\big{[}(k-1)k+Lt\,(1-Lt)\big{]}}\,.$

We define the functions $b_{k}\colon\,(0,\,1)\to\mathbb{R}$,
$k={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}1},\,\ldots,\,T$,
via

$b_{k}(x)\coloneqq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{\pi}{(2k-1)}}\,a_{k}\big{(}\tfrac{x}{L}\big{)}=\frac{\sin(\pi
x)}{(k-1)k+x(1-x)}\,,\quad x\in(0,\,1)\,,$

such that the symmetry relation $b_{k}(x)=b_{k}(1-x)$ is fulfilled, i.e., each
$b_{k}$ is symmetric with reference to $\frac{1}{2}$. Furthermore, by
$b_{k}^{\prime}(x)\geq 0$ for
$x\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\big{(}}0,\,\frac{1}{2}\big{]}$,
the function $b_{k}$ is increasing on
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\big{(}}0,\,\frac{1}{2}\big{]}$
and therefore has its maximum at $x=\tfrac{1}{2}$. Thus, the function
$a_{k}\colon\,\big{[}0,\,\tfrac{1}{L}\big{]}\to\mathbb{R}$ has its maximum at
$t=\tfrac{1}{2L}$, i.e.,

$\max_{t\in[0,\,1/L]}a_{k}(t)=a_{k}\big{(}\tfrac{1}{2L}\big{)}=\frac{4}{(2k-1)\pi}\,.$

Since $a_{T+1}(t)$ can be written as

$a_{T+1}(t)=\frac{\sin(L\pi t)}{\pi\,(T+Lt)}$

for $t\in\big{[}0,\,\tfrac{1}{L}\big{]}$, we obtain

$0<\max_{t\in[0,\,1/L]}a_{T+1}(t)<\frac{1}{\pi T}\,.$

Hence, in summary this yields

$\frac{4}{\pi}\,\sum_{k=1}^{T}\frac{1}{2k-1}<\max_{t\in[0,\,1/L]}s_{T}(t)<\frac{4}{\pi}\,\sum_{k=1}^{T}\frac{1}{2k-1}+\frac{1}{\pi
T}\,.$

For $t\in\big{[}\tfrac{n}{L},\,\tfrac{n+1}{L}\big{]}$ with arbitrary
$n\in\mathbb{N}$, the sum $s_{T}(t)$ is less than it is for
$t\in\big{[}0,\,\tfrac{1}{L}\big{]}$, since for each $n\in\mathbb{N}$ and
$t\in\big{(}0,\,\tfrac{1}{L}\big{)}$ we have

$\sum_{k=-T}^{T}\frac{\sin(L\pi t)}{|L\pi
t-(k-n)\pi|}<\sum_{k=-T}^{T}\frac{\sin(L\pi t)}{|L\pi t-k\pi|}$

and therefore $s_{T}\big{(}\tfrac{n}{L}+t\big{)}<s_{T}(t)$. Thus, we obtain

$\frac{4}{\pi}\,\sum_{k=1}^{T}\frac{1}{2k-1}<\max_{t\in\mathbb{R}}s_{T}(t)=\|S_{T}\|<\frac{4}{\pi}\,\sum_{k=1}^{T}\frac{1}{2k-1}+\frac{1}{\pi
T}\,.$ (2.5)

Note that for $T\gg 1$ the value

$s_{T}\big{(}\tfrac{1}{2L}\big{)}=\frac{4}{\pi}\,\sum_{k=1}^{T}\frac{1}{2k-1}+\frac{2}{\pi\,(2T+1)}$
(2.6)

is a good approximation of the norm $\|S_{T}\|$.

Now we estimate $\|S_{T}\|$ by $\ln T$. For this purpose we denote the
$T$-_th_ _harmonic number_ by

$H_{T}\coloneqq\sum_{k=1}^{T}\frac{1}{k}\,,\quad T\in\mathbb{N}\,,$

such that

$\sum_{k=1}^{T}\frac{1}{2k-1}=\sum_{k=1}^{T}\bigg{(}\frac{1}{2k-1}+\frac{1}{2k}\bigg{)}-\sum_{k=1}^{T}\frac{1}{2k}=H_{2T}-\frac{1}{2}\,H_{T}\,.$
(2.7)

Using _Euler’s constant_

$\gamma=\lim_{T\to\infty}\big{(}H_{T}-\ln T\big{)}\,,$

the estimates

$\frac{1}{2T+2}<H_{T}-\ln T-\gamma<\frac{1}{2T}$ (2.8)

are known (see [33]). From (2.7) and (2.8) we conclude that

$\frac{1}{2}\,\ln T+\ln
2+\frac{1}{2}\,\gamma-\frac{1}{4T\,(2T+1)}<H_{2T}-\frac{1}{2}\,H_{T}<\frac{1}{2}\,\ln
T+\ln 2+\frac{1}{2}\,\gamma+\frac{1}{4T\,(T+1)}\,.$ (2.9)

Therefore, applying (2.5), (2.7), and (2.9) yields the assertion (2.4).

Note that Theorem 2.2 improves a former result of [27, p. 142, Problem 3.1.5],
which only contains a coarse proof sketch for the upper bound, while (2.4)
gives a very precise nesting, see also Figure 2.1. Additionally, we remark
that  Theorem 2.2 immediately implies

$\lim_{T\to\infty}\bigg{(}\|S_{T}\|-\frac{2}{\pi}\,\ln
T\bigg{)}=\frac{4}{\pi}\,\ln 2+\frac{2\gamma}{\pi}\,.$

######  Remark 2.3.

By Theorem 2.2 and the theorem of Banach–Steinhaus, an arbitrary function
$f\in C_{0}(\mathbb{R})$ cannot be represented in the form (1.3), since the
norm of the linear operator $S_{T}\colon\,C_{0}(\mathbb{R})\to
C_{0}(\mathbb{R})$ is unbounded with respect to $T$. However, as known from
the Whittaker–Kotelnikov–Shannon sampling theorem for bandlimited functions
$f\in{\mathcal{B}}_{N/2}(\mathbb{R})$, the series (1.3) converges absolutely
and uniformly on whole $\mathbb{R}$.
Nevertheless, this convergence is very slow due to the poor decay of the
$\mathrm{sinc}$ function, as can be seen from the sharp upper and lower bounds
of the norm of the Shannon sampling operator, see Lemma 2.2. More precisely,
rigorous analysis of the approximation error, cf. [8, Theorem 1], shows that
on the fixed interval $[-1,\,1]$ we have a convergence rate of only
$(T-L)^{-1/2}$, which was also mentioned in [34, 29].
Note that general results on whole $\mathbb{R}$ are not feasible for arbitrary
$f\in{\mathcal{B}}_{N/2}(\mathbb{R})$. Rather, the uniform approximation error
$\|f-S_{T}f\|_{C_{0}(\mathbb{R})}$ can only be studied under the additional
assumption that $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ satisfies certain decay
conditions, see [12, 9].

Now let $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ be a given bandlimited function
with bandwidth $\tfrac{N}{2}$ and let $T\in\mathbb{N}$ be sufficiently large.
For given samples $f\big{(}\tfrac{k}{L}\big{)}$ with $k\in\mathbb{Z}$ and
$L\geq N$ we consider finitely many erroneous samples

${\tilde{f}}_{k}\coloneqq\left\\{\begin{array}[]{ll}f\big{(}\tfrac{k}{L}\big{)}+\varepsilon_{k}&\colon
k=-T,\,\ldots,\,T\,,\\\\[4.30554pt] f\big{(}\tfrac{k}{L}\big{)}&\colon
k\in\mathbb{Z}\setminus\\{-T,\,\ldots,\,T\\}\,,\end{array}\right.$

with error terms $\varepsilon_{k}$ which are bounded by
$|\varepsilon_{k}|\leq\varepsilon$ for $k=-T,\,\ldots,\,T$. Then for the
approximation

${\tilde{f}}(t)\coloneqq\sum_{k\in\mathbb{Z}}{\tilde{f}}_{k}\,\mathrm{sinc}(L\pi
t-k\pi)=f(t)+\sum_{k=-T}^{T}\varepsilon_{k}\,\mathrm{sinc}(L\pi
t-k\pi)\,,\quad t\in\mathbb{R}\,,$

the following error bounds can be shown.

######  Theorem 2.4.

Let $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ be an arbitrary bandlimited function
with bandwidth $\tfrac{N}{2}$. Further let $L\geq N$, $T\in\mathbb{N}$, and
$\varepsilon>0$ be given.
Then we have

$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\|\tilde{f}-f\|_{C_{0}(\mathbb{R})}<\varepsilon\,\bigg{(}\frac{2}{\pi}\,\ln
T+\frac{5}{4}+\frac{1}{2T}\bigg{)}\,.}$ (2.10)

Moreover, for the special error terms

$\displaystyle\varepsilon_{k}$
$\displaystyle=\varepsilon\,\mathrm{sign}\big{(}\mathrm{sinc}(\tfrac{\pi}{2}-k\pi)\big{)}=\varepsilon\,(-1)^{k+1}\,{\mathrm{sign}}(2k-1)\,,\quad
k=-T,\ldots,T\,,$

we have

$\|{\tilde{f}}-f\|_{C_{0}(\mathbb{R})}\geq\varepsilon\,\bigg{(}\frac{2}{\pi}\,\ln
T+\frac{4}{\pi}\,\ln
2+\frac{2\gamma}{\pi}\bigg{)}>\varepsilon\,\bigg{(}\frac{2}{\pi}\,\ln
T+\frac{5}{4}\bigg{)}\,,$ (2.11)

such that the Shannon sampling series is not numerically robust in the worst
case analysis.

_Proof_.  By (2.3), (2.5) and (2.9) we are given the upper error bound

$\displaystyle\|\tilde{f}-f\|_{C_{0}(\mathbb{R})}\leq\varepsilon\,\max_{t\in\mathbb{R}}s_{T}(t)<\frac{4\varepsilon}{\pi}\,\sum_{k=1}^{T}\frac{1}{2k-1}+\frac{\varepsilon}{\pi
T}<\frac{2\varepsilon}{\pi}\,\bigg{(}\ln T+2\ln
2+\gamma+\frac{T+2}{2T(T+1)}\bigg{)}\,.$

Since

$\frac{4}{\pi}\,\ln 2+\frac{2\gamma}{\pi}=1.2500093\ldots$ (2.12)

and $\mu_{T}\coloneqq\frac{T+2}{T+1}=1+\frac{1}{T+1}$ is monotonously
decreasing with $\max_{T\in\mathbb{N}}\mu_{T}=\mu_{1}=\tfrac{3}{2}$, we have

$\displaystyle\frac{4}{\pi}\,\ln 2+\frac{2\gamma}{\pi}+\frac{1}{\pi
T}\cdot\mu_{T}\leq\frac{5}{4}+\frac{1}{2T}\,,$

which yields (2.10).

Next, we proceed with the lower bound. Due to the special choice of the error
terms $\varepsilon_{k}$ we obtain

${\tilde{f}}(t)-f(t)=\varepsilon\,\sum_{k=-T}^{T}\mathrm{sign}\big{(}\mathrm{sinc}(\tfrac{\pi}{2}-k\pi)\big{)}\;\mathrm{sinc}(L\pi
t-k\pi)\,,\quad t\in\mathbb{R}\,.$ (2.13)

By (2.6) and (2.9) we conclude that

$\displaystyle\|{\tilde{f}}-f\|_{C_{0}(\mathbb{R})}$
$\displaystyle\geq\big{|}{\tilde{f}}(\tfrac{1}{2L})-f(\tfrac{1}{2L})\big{|}=\varepsilon\,\sum_{k=-T}^{T}\big{|}\mathrm{sinc}(\tfrac{\pi}{2}-k\pi)\big{|}=\varepsilon\,s_{T}\big{(}\tfrac{1}{2L}\big{)}$
$\displaystyle=\frac{4\varepsilon}{\pi}\,\sum_{k=1}^{T}\frac{1}{2k-1}+\frac{2\varepsilon}{(2T+1)\pi}>\varepsilon\,\bigg{(}\frac{2}{\pi}\,\ln
T+\frac{4}{\pi}\,\ln 2+\frac{2\gamma}{\pi}\bigg{)}\,,$

such that (2.12) completes the proof.

Note that (2.10) specifies a corresponding remark of [6, p. 681] as it makes
the constant explicit. In addition, we remark that the norm
$\|\tilde{f}-f\|_{C_{0}(\mathbb{R})}$ does not depend on the special choice of
the function $f$ or the oversampling parameter $\lambda$, see Figure 2.1.
Furthermore, Figure 2.1 also illustrates that  for $T\to\infty$ the error
behavior shown in Theorem 2.4 is not satisfactory. Thus, in the presence of
noise in the samples $f\big{(}\tfrac{{k}}{L}\big{)}$, ${k}\in\mathbb{Z}$, the
convergence of the Shannon sampling series (1.3) may even break down
completely.

Figure 2.1: The norm $\|\tilde{f}-f\|_{C_{0}(\mathbb{R})}$ of (2.13) as well
as its lower/upper bounds (2.11) and (2.10) for several  $T\in\mathbb{N}$ and
$L=N(1+\lambda)$ with $\lambda\in\\{0,0.5,1,2\\}$, where $N=128$ and
$\varepsilon=10^{-3}$ are chosen.

######  Remark 2.5.

In the above worst case analysis we have seen that the approximation of
$f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ by the $T$-th partial sum (2.1) of its
Shannon sampling series with $L\geq N$ is not numerically robust in the
deterministic sense. Otherwise, a simple average case study (see [31]) shows
that this approximation is numerically robust in the stochastic sense.
Therefore, we compute (2.1) as an inner product of the real
$(2T+1)$-dimensional vectors

$\Big{(}f\big{(}\tfrac{k}{L}\big{)}\Big{)}_{k=-T}^{T}\quad\text{ and
}\quad\big{(}\mathrm{sinc}(L\pi t-k\pi)\big{)}_{k=-T}^{T}\,.$

Now assume that instead of the exact samples $f\big{(}\frac{k}{L}\big{)}$,
$k=-T,\ldots,T$, only perturbed samples
${\tilde{f}}\big{(}\frac{k}{L}\big{)}=f\big{(}\frac{k}{L}\big{)}+X_{k}$,
$k=-T,\ldots,T$, are given, where the real random variables $X_{k}$ are
uncorrelated, each having expectation ${\mathbb{E}}(X_{k})=0$ and constant
variance ${\mathbb{V}}(X_{k})={\mathbb{E}}(|X_{k}|^{2})=\rho^{2}$ with
$\rho>0$. Then we consider the stochastic approximation error

$\displaystyle\Delta_{T}$ $\displaystyle:=$
$\displaystyle\sum_{k=-T}^{T}{\tilde{f}}\big{(}\tfrac{k}{L}\big{)}\,\mathrm{sinc}(L\pi
t-k\pi)-\sum_{k=-T}^{T}f\big{(}\tfrac{k}{L}\big{)}\,\mathrm{sinc}(L\pi
t-k\pi)$ $\displaystyle=$
$\displaystyle\sum_{k=-T}^{T}X_{k}\,\mathrm{sinc}(L\pi t-k\pi)\,.$

Obviously, this error term $\Delta_{T}$ has the expectation

${\mathbb{E}}(\Delta_{T})=\sum_{k=-T}^{T}\mathrm{sinc}(L\pi
t-k\pi)\,{\mathbb{E}}(X_{k})=0$

and the variance

${\mathbb{V}}(\Delta_{T})=\sum_{k=-T}^{T}\big{|}\mathrm{sinc}(L\pi
t-k\pi)\big{|}^{2}\,{\mathbb{V}}(X_{k})=\rho^{2}\,\sum_{k=-T}^{T}\big{|}\mathrm{sinc}(L\pi
t-k\pi)\big{|}^{2}\,.$

From [27, p. 89, Problem 1.10.9] it follows that

$\sum_{k\in\mathbb{Z}}\big{|}\mathrm{sinc}(L\pi t-k\pi)\big{|}^{2}=1\,,$

such that ${\mathbb{V}}(\Delta_{T})\leq\rho^{2}$.

##  3 Regularization with a frequency window function

To overcome the drawbacks of poor convergence and numerical instability, one
can apply regularization with a convenient window function either in the
frequency domain or in the time domain. Often one employs _oversampling_ ,
i.e., a bandlimited function $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ of
bandwidth $\tfrac{N}{2}$ is sampled on a finer grid $\frac{1}{L}\,\mathbb{Z}$
with $L=N\,(1+\lambda)$, where $\lambda>0$ is the oversampling parameter.

First, together with oversampling, we consider the _regularization with a
frequency window function_ of the form

${\hat{\psi}}(v)\coloneqq\left\\{\begin{array}[]{ll}1&\colon|v|\leq\tfrac{N}{2}\,,\\\
\chi(|v|)&\colon\tfrac{N}{2}<|v|<\tfrac{L}{2}\,,\\\
0&\colon|v|\geq\tfrac{L}{2}\,,\end{array}\right.$ (3.1)

cf. [5, 17, 28], where
$\chi\colon\big{[}\tfrac{N}{2},\,\tfrac{L}{2}\big{]}\to[0,\,1]$ is frequently
chosen as some monotonously decreasing, continuous function with
$\chi\big{(}\frac{N}{2}\big{)}=1$ and $\chi\big{(}\frac{L}{2}\big{)}=0$.
Applying the inverse Fourier transform, we determine the corresponding
function in time domain as

$\psi(t)=\int_{\mathbb{R}}{\hat{\psi}}(v)\,{\mathrm{e}}^{2\pi{\mathrm{i}}\,vt}\,{\mathrm{d}}v=2\,\int_{0}^{L/2}{\hat{\psi}}(v)\,\cos(2\pi\,vt)\,{\mathrm{d}}v\,.$
(3.2)

######  Example 3.1.

A simple example of a frequency window function is the _linear frequency
window function_ (cf. [5, pp. 18–19] or [17, pp. 210–212])

${\hat{\psi}_{\mathrm{lin}}}(v)\coloneqq\left\\{\begin{array}[]{ll}1&\colon|v|\leq\tfrac{N}{2}\,,\\\
1-\tfrac{2|v|-N}{L-N}&\colon\tfrac{N}{2}<|v|<\tfrac{L}{2}\,,\\\
0&\colon|v|\geq\tfrac{L}{2}\,.\end{array}\right.$ (3.3)

Note that in a trigonometric setting a function of the form (3.3) is also
often referred to as trapezoidal or de La Vallée-Poussin type window function,
respectively. Obviously, ${\hat{\psi}_{\mathrm{lin}}}(v)$ is a continuous
linear spline supported on $\big{[}-\tfrac{L}{2},\,\tfrac{L}{2}\big{]}$, see
Figure 3.1 (a). By (3.2) we receive $\psi_{\mathrm{lin}}(0)=\tfrac{N+L}{2}$.
For $t\in\mathbb{R}\setminus\\{0\\}$ we obtain

$\displaystyle\psi_{\mathrm{lin}}(t)$
$\displaystyle=2\,\int_{0}^{N/2}\cos(2\pi\,vt)\,{\mathrm{d}}v+2\,\int_{N/2}^{L/2}\bigg{(}1-\frac{2v-N}{L-N}\bigg{)}\,\cos(2\pi\,vt)\,{\mathrm{d}}v$
$\displaystyle=\frac{N+L}{2}\,\mathrm{sinc}\big{(}\tfrac{N+L}{2}\,\pi
t\big{)}\,\mathrm{sinc}\big{(}\tfrac{L-N}{2}\,\pi t\big{)}\,.$ (3.4)

This function $\tfrac{1}{L}\,\psi_{\mathrm{lin}}$ is even, supported on whole
$\mathbb{R}$, has its maximum at $t=0$ such that

$\Big{\|}\frac{1}{L}\,\psi_{\mathrm{lin}}\Big{\|}_{C_{0}(\mathbb{R})}=\frac{1}{L}\,\psi_{\mathrm{lin}}(0)=\frac{2+\lambda}{2+2\lambda}<1\,.$

In addition, $\tfrac{1}{L}\,\psi_{\mathrm{lin}}(t)$ has a faster decay than
$\mathrm{sinc}(N\pi t)$ for $|t|\to\infty$, cf. Figure 3.1 (b). Note that we
have

$\lim_{L\to+N}\frac{1}{L}\,\psi_{\mathrm{lin}}(t)=\mathrm{sinc}(N\pi t)\,.$

(a) $\hat{\psi}_{\mathrm{lin}}$ in (3.3)

(b) $\frac{1}{L}\,\psi_{\mathrm{lin}}$ in (3.1)

Figure 3.1: The frequency window function (3.3) and its scaled inverse Fourier
transform (3.1).

######  Remark 3.2.

Note that
$\big{\\{}\tfrac{1}{L}\,\psi_{\mathrm{lin}}\big{(}\cdot\,-\tfrac{k}{L}\big{)}\big{\\}}_{k\in\mathbb{Z}}$
is a _Bessel sequence in $L^{2}(\mathbb{R})$ with the bound_ $\tfrac{1}{L}$,
i.e., for all $f\in L^{2}(\mathbb{R})$ we have

$\sum_{k\in\mathbb{Z}}\big{|}\langle
f,\,\tfrac{1}{L}\,\psi_{\mathrm{lin}}\big{(}\cdot\,-\tfrac{k}{L}\big{)}\rangle_{L^{2}(\mathbb{R})}\big{|}^{2}\leq\tfrac{1}{L}\,\|f\|_{L^{2}(\mathbb{R})}^{2}\,.$

However,
$\big{\\{}\tfrac{1}{L}\,\psi_{\mathrm{lin}}\big{(}\cdot\,-\tfrac{k}{L}\big{)}\big{\\}}_{k\in\mathbb{Z}}$
is not an orthonormal sequence and also not a Riesz sequence. To see this, we
consider the 1-periodic function

$\Psi(v)\coloneqq\frac{1}{L^{2}}\,\sum_{k\in\mathbb{Z}}\big{|}{\hat{\psi}_{\mathrm{lin}}}(Lv+Lk)\big{|}^{2}\,,\quad
v\in\mathbb{R}\,.$

By (3.3) we have

${\hat{\psi}_{\mathrm{lin}}}(Lv)=\left\\{\begin{array}[]{ll}1&\colon|v|\leq\tfrac{1}{2+2\lambda}\,,\\\\[4.30554pt]
1-\tfrac{(2+2\lambda)|v|-1}{\lambda}&\colon\tfrac{1}{2+2\lambda}<|v|<\tfrac{1}{2}\,,\\\\[4.30554pt]
0&\colon|v|\geq\tfrac{1}{2}\,,\end{array}\right.$

i.e., $\Psi(v)\leq\frac{1}{L^{2}}$ for all $v\in\mathbb{R}$ and $\Psi(v)=0$
for $v\in\tfrac{1}{2}+\mathbb{Z}$. Then [4, Theorem 9.2.5] yields the result.

Analogous to [5, p. 19] and [17, Theorem 7.2.5], we obtain the following
representation result, see also [28, p. 4].

######  Theorem 3.3.

Let $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ be a bandlimited function with
bandwidth $\tfrac{N}{2}$, $N\in\mathbb{N}$, and let $L=N\,(1+\lambda)$ with
$\lambda>0$ be given.  Assume that the samples $f\big{(}\frac{k}{L}\big{)}$,
$k\in\mathbb{Z}$, fulfill the condition

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sum_{k\in\mathbb{Z}}\big{|}f\big{(}\tfrac{k}{L}\big{)}\big{|}<\infty\,.}$
(3.5)

Using oversampling and regularization with a frequency window function
$\hat{\psi}$ of the form (3.1), the function $f$ can be represented as

$f(t)=\sum_{k\in\mathbb{Z}}f\big{(}\tfrac{k}{L}\big{)}\,\frac{1}{L}\,\psi\big{(}t-\tfrac{k}{L}\big{)}\,,\quad
t\in\mathbb{R}\,,$ (3.6)

where the series (3.6) converges absolutely and uniformly on $\mathbb{R}$.

_Proof_. Since by assumption $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$, we have
$\mathrm{supp}\,\hat{f}\subseteq\big{[}-\tfrac{N}{2},\,\tfrac{N}{2}\big{]}\subset\big{[}-\tfrac{L}{2},\,\tfrac{L}{2}\big{]}$
and therefore the function ${\hat{f}}$ restricted on
$\big{[}-\tfrac{L}{2},\,\tfrac{L}{2}\big{]}$ can be represented by its
$L$-periodic Fourier series

${\hat{f}}(v)=\sum_{k\in\mathbb{Z}}c_{k}(\hat{f})\,{\mathrm{e}}^{2\pi{\mathrm{i}}\,kv/L}\,,\quad
v\in\big{[}-\tfrac{L}{2},\,\tfrac{L}{2}\big{]}\,,$ (3.7)

with the Fourier coefficients

$c_{k}(\hat{f})=\frac{1}{L}\,\int_{-L/2}^{L/2}{\hat{f}}(v)\,{\mathrm{e}}^{-2\pi{\mathrm{i}}\,kv/L}\,{\mathrm{d}}v\,.$

Using the inverse Fourier transform, we see that

$c_{k}(\hat{f})=\frac{1}{L}\,\int_{\mathbb{R}}{\hat{f}}(v)\,{\mathrm{e}}^{-2\pi{\mathrm{i}}\,kv/L}\,{\mathrm{d}}v=\frac{1}{L}\,f\big{(}-\tfrac{k}{L}\big{)}\,.$

Hence, we may write $\hat{f}$ given by (3.7) as

${\hat{f}}(v)=\frac{1}{L}\,\sum_{k\in\mathbb{Z}}f\big{(}\tfrac{k}{L}\big{)}\,{\mathrm{e}}^{-2\pi{\mathrm{i}}\,kv/L}\,,\quad
v\in\big{[}-\tfrac{L}{2},\,\tfrac{L}{2}\big{]}\,.$ (3.8)

By the Weierstrass M-test and assumption (3.5), the Fourier series (3.8)
converges absolutely and uniformly on
$\big{[}-\tfrac{L}{2},\,\tfrac{L}{2}\big{]}$. Additionally, we have
${\hat{\psi}}(v)=1$ for $v\in\big{[}-\tfrac{N}{2},\,\tfrac{N}{2}\big{]}$ by
(3.1) as well as
$\mathrm{supp}\,\hat{f}\subseteq\big{[}-\tfrac{N}{2},\,\tfrac{N}{2}\big{]}$ by
assumption, such that ${\hat{f}}(v)={\hat{f}}(v)\,{\hat{\psi}}(v)$ for all
$v\in\mathbb{R}$. Therefore, we obtain

$\displaystyle f(t)$
$\displaystyle=\int_{\mathbb{R}}{\hat{f}}(v)\,{\mathrm{e}}^{2\pi{\mathrm{i}}\,tv}\,{\mathrm{d}}v=\int_{-L/2}^{L/2}{\hat{f}}(v)\,{\mathrm{e}}^{2\pi{\mathrm{i}}\,tv}\,{\mathrm{d}}v=\int_{-L/2}^{L/2}{\hat{f}}(v)\,{\hat{\psi}}(v)\,{\mathrm{e}}^{2\pi{\mathrm{i}}\,tv}\,{\mathrm{d}}v$
$\displaystyle=\sum_{k\in\mathbb{Z}}\frac{1}{L}\,f\big{(}\tfrac{k}{L}\big{)}\,\int_{-L/2}^{L/2}{\hat{\psi}}(v)\,{\mathrm{e}}^{2\pi{\mathrm{i}}\,(t-k/L)\,v}\,{\mathrm{d}}v=\sum_{k\in\mathbb{Z}}f\big{(}\tfrac{k}{L}\big{)}\,\frac{1}{L}\,\psi\big{(}t-\tfrac{k}{L}\big{)}\,,\quad
t\in\mathbb{R}\,,$

where summation and integration may be interchanged by the theorem of
Fubini–Tonelli, since we have (3.5) and $|\hat{\psi}(v)|\leq 1$ by definition
(3.1). Additionally, note that from (3.1) and (3.2) it follows that

$|\psi(t)|\leq 2\,\int_{0}^{L/2}\big{|}\cos(2\pi
vt)\big{|}\,{\mathrm{d}}v<L\,,\quad t\in\mathbb{R}\,.$

Hence, we have $\frac{1}{L}\,\big{|}\psi\big{(}t-\frac{k}{L}\big{)}\big{|}<1$
for all $t\in\mathbb{R}$ and $k\in\mathbb{Z}$, and consequently the series
(3.6) converges absolutely and uniformly on $\mathbb{R}$ by (3.5) and the
Weierstrass M-test.

Note that (3.6) is not an interpolating approximation, since in general we
have

$\displaystyle\frac{1}{L}\,\,\psi\big{(}t-\tfrac{k}{L}\big{)}\,\bigg{|}_{t=\frac{n}{L}}\not=\delta_{n,k}\,,\quad
n,k\in\mathbb{Z}\,.$

Moreover, since the frequency window function $\hat{\psi}$ in (3.1) is
compactly supported, the uncertainty principle (cf. [18, p. 103, Lemma 2.39])
yields $\mathrm{supp}\,\psi=\mathbb{R}$, such that (3.6) does not imply
localized sampling for any choice of $\hat{\psi}$. In other words, the
representation (3.6) still requires infinitely many samples
$f\big{(}\tfrac{k}{L}\big{)}$. Thus, for practical realizations we need to
consider a truncated version of (3.6) and hence for $T\in\mathbb{N}$ we
introduce the $T$-th partial sum

$(P_{\psi,T}f)(t)\coloneqq\sum_{k=-T}^{T}f\big{(}\tfrac{k}{L}\big{)}\,\frac{1}{L}\,\psi\big{(}t-\tfrac{k}{L}\big{)}\,,\quad
t\in[-1,\,1]\,.$ (3.9)

Then for the linear frequency window function (3.3) we show the following
convergence result.

######  Theorem 3.4.

Let $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ be a bandlimited function with
bandwidth $\tfrac{N}{2}$, $N\in\mathbb{N}$, and let $L=N\,(1+\lambda)$ with
$\lambda>0$ be given. Assume that the samples $f\big{(}\tfrac{k}{L}\big{)}$,
$k\in\mathbb{Z}$, fulfill the condition (3.5). Using oversampling and
regularization with the linear frequency window function (3.3), the $T$-th
partial sums $P_{{\mathrm{lin}},T}f$ converge uniformly to $f$ on $[-1,\,1]$
as $T\to\infty$. For $T>L$ the following estimate holds

$\max_{t\in[-1,\,1]}\big{|}f(t)-(P_{\mathrm{lin},T}f)(t)\big{|}\leq\sqrt{\frac{2L}{3}}\,\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}2}\,(1+\lambda)}{\pi^{2}\lambda}\,({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}T-L})^{-3/2}\,\|f\|_{L^{2}(\mathbb{R})}\,.$
(3.10)

_Proof_. By (3.6) and (3.9) we have

$f(t)-(P_{\mathrm{lin},T}f)(t)=\sum_{|k|>T}f\big{(}\tfrac{k}{L}\big{)}\,\frac{1}{L}\,\psi_{\mathrm{lin}}\big{(}t-\tfrac{k}{L}\big{)}\,,$

such that Cauchy–Schwarz inequality implies

$\big{|}f(t)-(P_{\mathrm{lin},T}f)(t)\big{|}\leq\bigg{(}\sum_{|k|>T}\big{|}f\big{(}\tfrac{k}{L}\big{)}\big{|}^{2}\bigg{)}^{1/2}\,\bigg{(}\sum_{|k|>T}\big{|}\tfrac{1}{L}\,\psi_{\mathrm{lin}}\big{(}t-\tfrac{k}{L}\big{)}\big{|}^{2}\bigg{)}^{1/2}\,.$
(3.11)

For $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ the Parseval equality implies

$\frac{1}{L}\,\sum_{k\in\mathbb{Z}}\big{|}f\big{(}\tfrac{k}{L}\big{)}\big{|}^{2}=\|f\|_{L^{2}(\mathbb{R})}^{2}\,,$

see [10, Formula (3.8)], and thereby we have

$\bigg{(}\sum_{|k|>T}\big{|}f\big{(}\tfrac{k}{L}\big{)}\big{|}^{2}\bigg{)}^{1/2}\leq\sqrt{L}\,\|f\|_{L^{2}(\mathbb{R})}\,.$
(3.12)

It can easily be seen that (3.1) satisfies the decay condition

$\big{|}\tfrac{1}{L}\,\psi_{\mathrm{lin}}(x)\big{|}\leq\frac{2}{LN\lambda\,\pi^{2}}\,x^{-2}\,,\quad
x\in\mathbb{R}\setminus\\{0\\}\,,$

and thereby

$\big{|}\tfrac{1}{L}\,\psi_{\mathrm{lin}}\big{(}t-\tfrac{k}{L}\big{)}\big{|}^{2}\leq\frac{4\,(1+\lambda)^{2}}{\lambda^{2}\pi^{4}}\,(Lt-k)^{-4}\,.$

Thus, for $T>L$ and $t\in[-1,\,1]$ we obtain

$\displaystyle\bigg{(}\sum_{|k|>T}\big{|}\tfrac{1}{L}\,\psi_{\mathrm{lin}}\big{(}t-\tfrac{k}{L}\big{)}\big{|}^{2}\bigg{)}^{1/2}$
$\displaystyle\leq\frac{2\,(1+\lambda)}{\lambda\pi^{2}}\,\bigg{(}\sum_{|k|>T}(Lt-k)^{-4}\bigg{)}^{1/2}$
$\displaystyle\leq\frac{2\sqrt{2}\,(1+\lambda)}{\lambda\pi^{2}}\,\bigg{(}\sum_{k=T+1}^{\infty}(k-L)^{-4}\bigg{)}^{1/2}\,.$

Using the integral test for convergence of series, we conclude

$\displaystyle\sum_{k=T+1}^{\infty}(k-L)^{-4}$
$\displaystyle\leq\int_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}T}}^{\infty}(t-L)^{-4}\,{\mathrm{d}}t=\tfrac{1}{3}\,(T-L)^{-3}\,,$

which yields

$\bigg{(}\sum_{k=T+1}^{\infty}(k-L)^{-4}\bigg{)}^{1/2}\leq\tfrac{1}{\sqrt{3}}\,(T-L)^{-3/2}\,.$
(3.13)

Therefore, (3.11), (3.12), and (3.13) imply the estimate (3.10).

######  Example 3.5.

Next, we visualize the error bound of Theorem 3.4, i.e., for a given function
$f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ with $L=N\,(1+\lambda)$, $\lambda>0$, we
show that the approximation error satisfies (3.10). For this purpose, the
error

$\max_{t\in[-1,\,1]}\big{|}f(t)-(P_{\mathrm{lin},T}f)(t)\big{|}$ (3.14)

is estimated by evaluating the given function $f$ as well as its approximation
$P_{\mathrm{lin},T}f$, cf. (3.9), at equidistant points $t_{s}\in[-1,\,1]$,
$s=1,\dots,S$, with $S=10^{5}$. Here we study the function
$f(t)=\sqrt{N}\,\mathrm{sinc}(N\pi t)$, $t\in\mathbb{R}$, such that
$\|f\|_{L^{2}(\mathbb{R})}=1$. We fix $N=128$ and consider the error behavior
for increasing $T\in\mathbb{N}$. More specifically, in this experiment we
choose several oversampling parameters $\lambda\in\\{0.5,1,2\\}$ and
truncation parameters  $T=2^{c}$ with $c\in\\{0,\dots,15\\}$. The
corresponding results are depicted in Figure 3.2. Note that the error bound in
(3.10) is only valid for $T>L$. Therefore, we have additionally marked the
point $T=L$ for each $\lambda$ as a vertical dash-dotted line. It can easily
be seen that also the error results are much better when $T>L$. Note, however,
that increasing the oversampling parameter $\lambda$ requires a much larger
truncation parameter $T$ to obtain errors of the same size.

Figure 3.2: Maximum approximation error (3.14) (solid) and error constant
(3.10) (dashed) using the linear frequency window $\psi_{\mathrm{lin}}$ from
(3.1) in (3.9) for the function $f(t)=\sqrt{N}\,\mathrm{sinc}(N\pi t)$ with
$N=128$,  $T=2^{c}$, $c\in\\{0,\dots,15\\}$, and $\lambda\in\\{0.5,1,2\\}$.

In order to obtain convergence rates better than the one in Theorem 3.4, one
may consider frequency window functions (3.1) of higher smoothness.

######  Example 3.6.

Next, we construct a continuously differentiable frequency window function by
polynomial interpolation. Since the frequency window function (3.1) is even,
it suffices to consider only
$\chi\colon\big{[}\tfrac{N}{2},\,\tfrac{L}{2}\big{]}\to[0,\,1]$ at the
interval boundaries $\tfrac{N}{2}$ and $\tfrac{L}{2}$. Clearly, the linear
frequency window function $\hat{\psi}_{\mathrm{lin}}$ in (3.3) fulfills

$\lim_{v\to\frac{N}{2}}\chi(v)=1\,,\quad\lim_{v\to\frac{L}{2}}\chi(v)=0\,.$

Thus, to obtain a smoother frequency window function, we need to additionally
satisfy the first order conditions

$\lim_{v\to\frac{N}{2}}\chi^{\prime}(v)=0\,,\quad\lim_{v\to\frac{L}{2}}\chi^{\prime}(v)=0\,.$

Then the corresponding interpolation polynomial yields the _cubic frequency
window function_

${\hat{\psi}_{\mathrm{cub}}}(v)\coloneqq\left\\{\begin{array}[]{ll}1&\colon|v|\leq\tfrac{N}{2}\,,\\\
\frac{16}{\left(L-N\right)^{3}}\,\big{(}|v|-\frac{L}{2}\big{)}^{2}\big{(}|v|-\frac{3N-L}{4}\big{)}&\colon\tfrac{N}{2}<|v|<\tfrac{L}{2}\,,\\\
0&\colon|v|\geq\tfrac{L}{2}\,,\end{array}\right.$ (3.15)

see Figure 3.3 (a). By (3.2) we see that the inverse Fourier transform of
(3.15) is given by

$\displaystyle{\psi}_{\mathrm{cub}}(t)$
$\displaystyle={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{L+N}{2}\,\mathrm{sinc}\big{(}\tfrac{L+N}{2}\,\pi
t\big{)}\cdot\frac{12\left(\mathrm{sinc}\big{(}\tfrac{L-N}{2}\,\pi
t\big{)}-\cos\big{(}\tfrac{L-N}{2}\,\pi
t\big{)}\right)}{\pi^{2}t^{2}(L-N)^{2}}}\,,$ (3.16)

for $t\in\mathbb{R}\setminus\\{0\\}$ and
$\psi_{\mathrm{cub}}(0)=\frac{L+N}{2}$, cf. Figure 3.3 (b).

Analogous to Theorem 3.4, the following error estimate can be derived.

######  Theorem 3.7.

Let $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ be a bandlimited function with
bandwidth $\tfrac{N}{2}$, $N\in\mathbb{N}$, and let $L=N\,(1+\lambda)$ with
$\lambda>0$ be given. Assume that the samples $f\big{(}\tfrac{k}{L}\big{)}$,
$k\in\mathbb{Z}$, fulfill the condition (3.5). Using oversampling and
regularization with the cubic frequency window function (3.15), the $T$-th
partial sums $P_{{\mathrm{cub}},T}f$ converge uniformly to $f$ on $[-1,\,1]$
as $T\to\infty$. For $T>L$ the following estimate holds

$\max_{t\in[-1,\,1]}\big{|}f(t)-(P_{\mathrm{cub},T}f)(t)\big{|}\leq\sqrt{\frac{2L}{5}}\,\frac{24\,(1+\lambda)^{2}}{\pi^{3}\lambda^{2}}\,({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}T-L})^{-5/2}\,\|f\|_{L^{2}(\mathbb{R})}\,.$
(3.17)

######  Example 3.8.

Another continuously differentiable frequency window function is given in [24]
as the _raised cosine frequency window function_

${\hat{\psi}_{\cos}}(v)\coloneqq\left\\{\begin{array}[]{ll}1&\colon|v|\leq\tfrac{N}{2}\,,\\\
\frac{1}{2}+\frac{1}{2}\,\cos\Big{(}\tfrac{2|v|-N}{L-N}\pi\Big{)}&\colon\tfrac{N}{2}<|v|<\tfrac{L}{2}\,,\\\
0&\colon|v|\geq\tfrac{L}{2}\,,\end{array}\right.$ (3.18)

see Figure 3.3 (a). By (3.2) the corresponding function in time domain can be
determined as

$\displaystyle{\psi}_{\cos}(t)$
$\displaystyle={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{L+N}{2}\,\mathrm{sinc}\big{(}\tfrac{L+N}{2}\,\pi
t\big{)}\cdot\frac{\cos\big{(}\tfrac{L-N}{2}\,\pi
t\big{)}}{1-t^{2}(L-N)^{2}},\quad
t\in\mathbb{R}\setminus\big{\\{}\\!\pm\tfrac{1}{L-N}\big{\\}}},$ (3.19)

and
$\psi_{\cos}\left(\pm\tfrac{1}{L-N}\right)=\tfrac{L-N}{4}\,\cos\big{(}\tfrac{N\pi}{L-N}\big{)}$,
see Figure 3.3 (b). Note that since ${\hat{\psi}_{\cos}}$ in (3.18) possesses
the same regularity as ${\hat{\psi}_{\mathrm{cub}}}$ in (3.15), both frequency
window functions meet the same error bound (3.17), cf. Figure 5.1.

(a) $\hat{\psi}_{\mathrm{cub}}$ and $\hat{\psi}_{\cos}$

(b) $\frac{1}{L}\,\psi_{\mathrm{cub}}$ and $\frac{1}{L}\,\psi_{\cos}$

Figure 3.3: The frequency window functions (3.15) and (3.18), and their scaled
inverse Fourier transforms.

Note that by (3.1) and the convolution property of the Fourier transform, for
$L>N$ the linear frequency window function (3.3) can be written as

$\displaystyle{\hat{\psi}}_{\mathrm{lin}}(v)=\frac{2}{L-N}\,\big{(}{\mathbf{1}}_{(N+L)/4}\ast{\mathbf{1}}_{({L-N})/4}\big{)}(v)\,.$

Therefore, instead of determining smooth frequency window functions of the
form (3.1) by means of interpolation as in Example 3.6, they can also be
constructed by convolution, cf. [15].

######  Lemma 3.9.

Let $L>N$ be given. Assume that $\rho\colon\mathbb{R}\to[0,\,\infty)$ is an
even integrable function with
$\mathrm{supp}\,\rho=\big{[}-\frac{L-N}{4},\,\frac{L-N}{4}\big{]}$ and
$\int_{\mathbb{R}}\rho(v)\,{\mathrm{d}}v=1$.
Then the convolution

$\displaystyle{\hat{\psi}_{\mathrm{conv}}}(v)=\big{(}{\mathbf{1}}_{(N+L)/4}\ast\rho\big{)}(v)\,,\quad
v\in\mathbb{R}\,,$ (3.20)

is a frequency window function of the form (3.1).

_Proof._ By assumptions we have

$\displaystyle{\hat{\psi}}_{\mathrm{conv}}(v)=\big{(}{\mathbf{1}}_{(N+L)/4}\ast\rho\big{)}(v)=\int_{-(N+L)/4}^{(N+L)/4}\rho(v-w)\,{\mathrm{d}}w=\int_{v-(N+L)/4}^{v+(N+L)/4}\rho(w)\,{\mathrm{d}}w\,.$
(3.21)

Since the convolution of two even functions is again an even function, it
suffices to consider (3.21) only for $v\geq 0$. For
$v\in\big{[}0,\,\frac{N}{2}\big{]}$ we have
$v-\frac{L+N}{4}\leq-\frac{L-N}{4}<\frac{L-N}{4}\leq v+\frac{L+N}{4}$ and
therefore

$\displaystyle{\hat{\psi}}_{\mathrm{conv}}(v)=\int_{-(L-N)/4}^{(L-N)/4}\rho(w)\,{\mathrm{d}}w=1\,.$

For $v\in\big{[}\frac{N}{2},\,\frac{L}{2}\big{]}$ we can write

$\displaystyle{\hat{\psi}}_{\mathrm{conv}}(v)=\int_{v-(L+N)/4}^{(L-N)/4}\rho(w)\,{\mathrm{d}}w\geq
0\,,$

such that ${\hat{\psi}}_{\mathrm{conv}}\big{(}\frac{N}{2}\big{)}=1$,
${\hat{\psi}}_{\mathrm{conv}}\big{(}\frac{L}{2}\big{)}=0$, and
${\hat{\psi}}_{\mathrm{conv}}\colon\big{[}\frac{N}{2},\,\frac{L}{2}\big{]}\to[0,\,1]$
is monotonously non-increasing, since $\rho(w)\geq 0$ by assumption. For
$v\in\big{[}\frac{L}{2},\,\infty\big{)}$ we have
$v-\frac{L+N}{4}\geq\frac{L-N}{4}$, which implies by assumption
$\mathrm{supp}\,\rho=\big{[}-\frac{L-N}{4},\,\frac{L-N}{4}\big{]}$ that

$\displaystyle{\hat{\psi}}_{\mathrm{conv}}(v)=\int_{v-(L+N)/4}^{v+(L+N)/4}\rho(w)\,{\mathrm{d}}w=0\,.$

This completes the proof.

Given such a frequency window function $\hat{\psi}_{\mathrm{conv}}$ as in
(3.20), its inverse Fourier transform (3.2) is known by the convolution
property of the Fourier transform as

$\displaystyle\psi_{\mathrm{conv}}(t)=\frac{N+L}{2}\,\mathrm{sinc}\big{(}\tfrac{N+L}{2}\,\pi
t\big{)}\,{\check{\rho}}(t)\,.$ (3.22)

Thus, to obtain a suitable window function (3.20), we need to assure that the
inverse Fourier transform $\check{\rho}$ of $\rho$ is explicitly known. Since
$\rho$ is even by assumption, we have $\check{\rho}=\hat{\rho}$ with

$\displaystyle\check{\rho}(t)=\int_{\mathbb{R}}\rho(v)\,{\mathrm{e}}^{2\pi{\mathrm{i}}\,tv}\,{\mathrm{d}}v=2\,\int_{0}^{\infty}\rho(v)\,\cos(2\pi\,vt)\,{\mathrm{d}}v\,.$

Note that the convolutional approach (3.20) has the substantial advantage that
the smoothness of (3.20) is determined by the smoothness of the chosen
function $\rho$.

######  Remark 3.10.

The frequency window functions ${\hat{\psi}}_{\mathrm{cub}}$ in (3.15) and
${\hat{\psi}}_{\cos}$ in (3.18) lack a convolutional representation (3.20).
Although the corresponding functions (3.16) and (3.19) in spatial domain are
of the form (3.22), for both frequency windows the Fourier transform of the
respective function $\check{\rho}$ is only known in the sense of tempered
distributions.

######  Example 3.11.

For the special choice of
$\rho(v)=\frac{2n}{L-N}\,M_{n}\big{(}\frac{2n}{L-N}\,v\big{)}$ with
$n\in\mathbb{N}$, where $M_{n}$ is the centered cardinal B-spline of order
$n$, we have

$\displaystyle{\check{\rho}}(t)=\Big{(}\mathrm{sinc}\big{(}\tfrac{L-N}{2n}\,\pi
t\big{)}\Big{)}^{n}\,.$

Using $n=1$ this again yields (3.1), whereas for $n=2$ we obtain

$\displaystyle{\psi}_{\mathrm{conv},2}(t)=\frac{N+L}{2}\,\mathrm{sinc}\big{(}\tfrac{N+L}{2}\,\pi
t\big{)}\,\Big{(}\mathrm{sinc}\big{(}\tfrac{L-N}{4}\,\pi
t\big{)}\Big{)}^{2}\,.$ (3.23)

Note that the frequency window function $\hat{\psi}_{\mathrm{conv},2}$, cf.
(3.23), possesses the same regularity as $\hat{\psi}_{\mathrm{cub}}$ in (3.15)
and $\hat{\psi}_{\mathrm{cos}}$ in (3.18), and therefore they all meet the
same error bound (3.17), cf. Figure 5.1.

######  Example 3.12.

In [15] the infinitely differentiable function

$\displaystyle\rho_{\infty}(v)=\left\\{\begin{array}[]{ll}c\,\exp\big{(}\big{[}\big{(}\frac{4v}{L-N}\big{)}^{2}-1\big{]}^{-1}\big{)}&\colon|v|<\frac{L-N}{4}\,,\\\
0&\colon\text{otherwise}\,,\end{array}\right.$

with the scaling factor

$\displaystyle
c=\frac{1}{2}\,\bigg{(}\int_{0}^{(L-N)/4}\exp\big{(}\big{[}\big{(}\tfrac{4v}{L-N}\big{)}^{2}-1\big{]}^{-1}\big{)}\,{\mathrm{d}}v\bigg{)}^{-1}\,.$

is considered. The corresponding frequency window function (3.20) is denoted
by $\hat{\psi}_{\infty}$. However, since for this function $\rho_{\infty}$ the
inverse Fourier transform $\check{\rho}_{\infty}$ cannot explicitly be stated,
the function (3.22) in time domain can only be approximated, which was done by
a piecewise rational approximation $\check{\rho}_{\mathrm{rat}}$ in [15]. We
remark that because of this additional approximation a numerical decay of the
expected rate is doubtful, since the issue of robustness of the corresponding
regularized Shannon series remains unclear. This effect can also be seen in
Figure 5.1, where the corresponding frequency window function (3.20), denoted
by $\hat{\psi}_{\mathrm{rat}}$, shows similar behavior as the classical
Shannon sampling sums (2.1).

The same comment also applies to [28], where an infinitely differentiable
window function $\hat{\psi}$ is aimed for as well. Since such $\hat{\psi}$
with explicit inverse Fourier transform (3.2) is not known, in [28] the
function $\psi$ is estimated by some Gabor approximation. Although an
efficient computation scheme via fast Fourier transform (FFT) was introduced
in [29], the numerical nonrobustness by this approximation seems to be
neglected in this work.

Finally, we remark that already in [5, p. 19] it was stated that a faster
decay than for $\hat{\psi}_{\mathrm{lin}}$ from (3.3) can be obtained by
choosing $\hat{\psi}$ in (3.1) smoother, but at the price of a very large
constant. This can also be seen in Figure 5.1, where the results for the
window functions ${\hat{\psi}_{\mathrm{cub}}}$ in (3.15),
${\hat{\psi}_{\cos}}$ in (3.18), ${\hat{\psi}_{\mathrm{conv},2}}$ in (3.23),
and $\hat{\psi}_{\mathrm{rat}}$ from Example 3.12 are plotted as well. For
this reason many authors restricted themselves to the linear frequency window
function $\hat{\psi}_{\mathrm{lin}}$ in (3.3). Furthermore, the numerical
results in Figure 5.1 encourage the suggestion that in practice only algebraic
decay rates are achievable by regularization with a frequency window function.

##  4 Regularization with a time window function

To preferably obtain better decay rates, we now consider a second
regularization technique, namely regularization with a convenient window
function in the time domain. To this end, for $L>N$ and any
$m\in{\mathbb{N}}\setminus\\{1\\}$ with $2m\ll L$, we introduce the set
$\Phi_{m/L}$ of all window functions $\varphi\colon\,\mathbb{R}\to[0,\,1]$
with the following properties:

  * $\quad\bullet$

Each $\varphi\in\Phi_{m/L}$ is supported on
$\big{[}-\frac{m}{L},\,\frac{m}{L}\big{]}$. Further, $\varphi$ is even and
continuous on $\big{[}-\frac{m}{L},\,\frac{m}{L}\big{]}$.

  * $\bullet$

Each $\varphi\in\Phi_{m/L}$ restricted on $\big{[}0,\,\frac{m}{L}\big{]}$ is
monotonously non-increasing with $\varphi(0)=1$.

As examples of such window functions we consider the $\sinh$-_type window
function_

$\displaystyle\varphi_{\sinh}(t)\coloneqq\begin{cases}\frac{1}{\sinh\beta}\,\sinh\Big{(}\beta\,\sqrt{1-\big{(}\tfrac{Lt}{m}\big{)}^{2}}\,\Big{)}&\quad\colon
t\in\big{[}-\tfrac{m}{L},\,\tfrac{m}{L}\big{]}\,,\\\ 0&\quad\colon
t\in\mathbb{R}\setminus\big{[}-\tfrac{m}{L},\,\tfrac{m}{L}\big{]}\,,\end{cases}$
(4.1)

with $\beta=\frac{\pi m\,(L-N)}{L}$, and the _continuous Kaiser–Bessel window
function_

$\displaystyle\varphi_{\mathrm{cKB}}(t)\coloneqq\begin{cases}\frac{1}{I_{0}(\beta)-1}\,\bigg{(}I_{0}\Big{(}\beta\,\sqrt{1-\big{(}\tfrac{Lt}{m}\big{)}^{2}}\,\Big{)}-1\bigg{)}&\quad\colon
t\in\big{[}-\tfrac{m}{L},\,\tfrac{m}{L}\big{]}\,,\\\ 0&\quad\colon
t\in\mathbb{R}\setminus\big{[}-\tfrac{m}{L},\,\tfrac{m}{L}\big{]}\,,\end{cases}$
(4.2)

with $\beta=\frac{\pi m\,(L-N)}{L}$, where $I_{0}(x)$ denotes the _modified
Bessel function of first kind_ given by

$I_{0}(x)\coloneqq\sum_{k=0}^{\infty}\frac{1}{\big{(}(2k)!!\big{)}^{2}}\,x^{2k}\,,\quad
x\in\mathbb{R}\,.$

Both window functions are well-studied in the context of the nonequispaced
fast Fourier transform (NFFT), see e.g. [19] and references therein.

A visualization of the continuous Kaiser–Bessel window function (4.2) as well
as the corresponding regularization
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\xi}_{\mathrm{cKB}}(t)\coloneqq\mathrm{sinc}(L\pi
t)\,\varphi_{\mathrm{cKB}}(t)$ of the $\mathrm{sinc}$ function can be found in
Figure 4.1. We remark that in contrast to Figure 3.1 here the function
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\xi}_{\mathrm{cKB}}$
in time domain is compactly supported and its Fourier transform
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\xi}}_{\mathrm{cKB}}$
is supported on whole $\mathbb{R}$, where for the frequency window function
(3.3) it is vice versa (see [18, p. 103, Lemma 2.39]). Note that a
visualization for the $\sinh$-type window function (4.1) would look the same
as Figure 4.1.

(a)
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\xi}}_{\mathrm{cKB}}$

(b) Window function (4.2)

Figure 4.1: The regularized $\mathrm{sinc}$ function
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\xi}_{\mathrm{cKB}}(t)\coloneqq\mathrm{sinc}(L\pi
t)\,\varphi_{\mathrm{cKB}}(t)$ using the continuous Kaiser–Bessel window
function (4.2) and its Fourier transform
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\xi}}_{\mathrm{cKB}}$.

Then we  approximate a bandlimited function
$f\in{\mathcal{B}}_{N/2}({\mathbb{R}})$ by the _regularized Shannon sampling
formula_

$(R_{\varphi,m}f)(t)\coloneqq\sum_{k\in{\mathbb{Z}}}f\big{(}\tfrac{k}{L}\big{)}\,\mathrm{sinc}(L\pi
t-\pi k)\,\varphi\big{(}t-\tfrac{k}{L}\big{)}\,,\quad t\in{\mathbb{R}}\,,$
(4.3)

with $L\geq N$. Since by assumption $\mathrm{sinc}(\pi(n-k))=\delta_{n,k}$ for
all $n,\,k\in\mathbb{Z}$ with the Kronecker delta $\delta_{n,k}$ and
$\varphi(0)=1$, this procedure is an _interpolating approximation_ of $f$,
because

$\mathrm{sinc}(L\pi t-\pi
k)\,\varphi\big{(}t-\tfrac{k}{L}\big{)}\,\Big{|}_{t=\frac{n}{L}}=\delta_{n,k}\,.$

Furthermore, the use of the compactly supported window function
$\varphi\in\Phi_{m/L}$ leads to _localized sampling_ of the bandlimited
function $f\in{\mathcal{B}}_{N/2}({\mathbb{R}})$, i.e., the computation of
$(R_{\varphi,m}f)(t)$ for
$t\in{\mathbb{R}}\setminus\tfrac{1}{L}\,{\mathbb{Z}}$ requires only $2m+1$
samples $f\big{(}\frac{k}{L}\big{)}$, where $k\in{\mathbb{Z}}$ fulfills the
condition $|k-Lt|\leq m$. Consequently, for given
$f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ and $L\geq N$, the reconstruction of $f$
on the interval $[-1,\,1]$ requires $2m+2L+1$ samples
$f\big{(}\frac{k}{L}\big{)}$ with $k=-m-L,\,\dots,\,m+L$. In addition, we
again employ oversampling of the bandlimited function
$f\in{\mathcal{B}}_{N/2}({\mathbb{R}})$, i.e., $f$ is sampled on a finer grid
$\tfrac{1}{L}\,{\mathbb{Z}}$ with $L>N$.

This concept of regularized Shannon sampling formulas with localized sampling
and oversampling has already been studied by various authors. A survey of
different approaches for window functions can be found in [22], while the
prominent Gaussian window function was e.g. considered in [21, 23, 26, 30,
13]. Since this Gaussian window function has also been studied in [10], where
superiority of the $\sinh$-type window function (4.1) was shown, we now focus
on time window functions $\varphi\in\Phi_{m/L}$, such as (4.1) and (4.2).

Similar as in [10], for given $f\in{\mathcal{B}}_{N/2}({\mathbb{R}})$ and
$\varphi\in\Phi_{m/L}$ the uniform approximation error
$\|f-R_{\varphi,m}f\|_{C_{0}({\mathbb{R}})}$ of the regularized Shannon
sampling formula can be estimated as follows.

######  Theorem 4.1.

Let $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ be a bandlimited function with
bandwidth $\tfrac{N}{2}$, $N\in\mathbb{N}$, and let $L>N$, and
$m\in{\mathbb{N}}\setminus\\{1\\}$ be given. Further let
$\varphi\in\Phi_{m/L}$.
Then the regularized Shannon sampling formula (4.3) satisfies

$\displaystyle\|f-R_{\varphi,m}f\|_{C_{0}(\mathbb{R})}\leq\big{(}E_{1}(m,N,L)+E_{2}(m,L)\big{)}\,\|f\|_{L^{2}(\mathbb{R})}\,,$

with the corresponding error constants

$\displaystyle E_{1}(m,N,L)$
$\displaystyle\coloneqq\sqrt{N}\,\max_{\mathbf{v}\in[-N/2,\,N/2]}\bigg{|}1-\int_{v-L/2}^{v+L/2}{\hat{\varphi}}(u)\,{\mathrm{d}}u\,\bigg{|}\,,$
(4.4) $\displaystyle E_{2}(m,L)$
$\displaystyle\coloneqq\frac{\sqrt{2L}}{\pi\,m}\,\varphi\big{(}\tfrac{m}{L}\big{)}\,.$
(4.5)

_Proof_. For a proof of Theorem 4.1 see [10, Theorem 3.2].

Note that it is especially beneficial for the estimation of the error constant
(4.4), if the Fourier transform

${\hat{\varphi}}(v)=\int_{\mathbb{R}}\varphi(t)\,{\mathrm{e}}^{-2\pi{\mathrm{i}}\,vt}\,{\mathrm{d}}t=2\,\int_{0}^{m/L}\varphi(t)\,\cos(2\pi
vt)\,{\mathrm{d}}t\,,\quad v\in\mathbb{R}\,,$ (4.6)

of $\varphi\in\Phi_{m/L}$ is explicitly known.

Now we specify the result of Theorem 4.1 for certain window functions. To this
end, assume that $f\in{\mathcal{B}}_{N/2}({\mathbb{R}})$ with $N\in\mathbb{N}$
and $L=N\,(1+\lambda)$, $\lambda>0$. Additionally, we choose
$m\in\mathbb{N}\setminus\\{1\\}$. We demonstrate that for the window functions
(4.1) and (4.2) the approximation error of the regularized Shannon sampling
formula (4.3) decreases exponentially with respect to $m$.  For the
$\sinh$-type window function (4.1) the following result is already known.

######  Theorem 4.2.

Let $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ be a bandlimited function with
bandwidth $\tfrac{N}{2}$, $N\in\mathbb{N}$, and let $L=N\,(1+\lambda)$ with
$\lambda>0$ and $m\in\mathbb{N}\setminus\\{1\\}$ be given.
Then the regularized Shannon sampling formula (4.3) with the $\sinh$-type
window function (4.1) satisfies the error estimate

$\|f-R_{\sinh,m}f\|_{C_{0}({\mathbb{R}})}\leq\sqrt{N}\,{\mathrm{e}}^{-m\pi\lambda/(1+\lambda)}\,\|f\|_{L^{2}({\mathbb{R}})}\,.$
(4.7)

_Proof_. For a proof of Theorem 4.2 see [10, Theorem 6.1].

Next, we continue with the continuous Kaiser–Bessel window function (4.2).

######  Theorem 4.3.

Let $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ be a bandlimited function with
bandwidth $\tfrac{N}{2}$, $N\in\mathbb{N}$, and let $L=N\,(1+\lambda)$ with
$\lambda\geq\frac{1}{m-1}$ and $m\in\mathbb{N}\setminus\\{1\\}$ be given.
Then the regularized Shannon formula (4.3) with the continuous Kaiser–Bessel
window function (4.2) satisfies the error estimate

$\|f-R_{\mathrm{cKB},m}f\|_{C_{0}({\mathbb{R}})}\leq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{7\sqrt{N}\,m\pi\lambda\,(1+\lambda+4m\lambda)}{4\,(1+\lambda)^{2}}\,{\mathrm{e}}^{-m\pi\lambda/(1+\lambda)}}\,\|f\|_{L^{2}({\mathbb{R}})}\,.$
(4.8)

_Proof_. By means of Theorem 4.1 we only have to compute the error constants
(4.4) and (4.5). Note that (4.2) implies
$\varphi_{\mathrm{cKB}}\big{(}\tfrac{m}{L}\big{)}=0$, such that the error
constant (4.5) vanishes. For computing the error constant (4.4) we introduce
the function
$\eta\colon\,\big{[}-\tfrac{N}{2},\,\tfrac{N}{2}\big{]}\to\mathbb{R}$ given by

$\eta(v)=1-\int_{v-L/2}^{v+L/2}{\hat{\varphi}}_{\mathrm{cKB}}(u)\,{\mathrm{d}}u\,.$
(4.9)

As known by [16, p. 3, 1.1, and p. 95, 18.31], the Fourier transform of (4.2)
has the form

$\displaystyle{\hat{\varphi}}_{\mathrm{cKB}}(v)=\frac{2m}{\big{(}I_{0}(\beta)-1\big{)}\,L}\cdot\begin{cases}\Big{(}\frac{\sinh\big{(}\beta\sqrt{1-w^{2}}\,\big{)}}{\beta\sqrt{1-w^{2}}}-\mathrm{sinc}(\beta
w)\Big{)}&\quad\colon|w|<1\,,\\\
\big{(}\mathrm{sinc}\big{(}\beta\sqrt{w^{2}-1}\,\big{)}-\mathrm{sinc}(\beta
w)\big{)}&\quad\colon|w|\geq 1\,,\end{cases}$ (4.10)

with the scaled frequency $w=\frac{2\pi m}{\beta L}\,v$. Thus, substituting
$w=\frac{2\pi m}{\beta L}\,u$ in (4.9) yields

$\eta(v)=1-\frac{\beta L}{2\pi
m}\,\int_{-a(-v)}^{a(v)}{\hat{\varphi}}_{\mathrm{cKB}}\big{(}\tfrac{\beta
L}{2m\pi}\,w\big{)}\,{\mathrm{d}}w$

with the increasing linear function $a(v)\coloneqq\tfrac{2m\pi}{\beta
L}\,\big{(}v+\tfrac{L}{2}\big{)}$. By the choice of the parameter
$\beta=\tfrac{m\pi\lambda}{1+\lambda}$ with  $\lambda\geq\frac{1}{m-1}$ we
have $a\big{(}-\tfrac{N}{2}\big{)}=1$ and $a(v)\geq 1$ for all
$v\in\big{[}-\tfrac{N}{2},\,\tfrac{N}{2}\big{]}$. Using (4.10), we decompose
$\eta(v)$ in the form

$\eta(v)=\eta_{1}(v)-\eta_{2}(v)\,,\quad
v\in\big{[}-\tfrac{N}{2},\,\tfrac{N}{2}\big{]}\,,$

with

$\displaystyle\eta_{1}(v)$
$\displaystyle=1-\frac{\beta}{\pi\,\big{(}I_{0}(\beta)-1\big{)}}\,\int_{-1}^{1}\bigg{(}\frac{\sinh\big{(}\beta\sqrt{1-w^{2}}\,\big{)}}{\beta\sqrt{1-w^{2}}}-\mathrm{sinc}(\beta
w)\bigg{)}\,{\mathrm{d}}w\,,$ $\displaystyle\eta_{2}(v)$
$\displaystyle=\frac{\beta}{\pi\,\big{(}I_{0}(\beta)-1\big{)}}\,\bigg{(}\int_{-a(-v)}^{-1}+\int_{1}^{a(v)}\bigg{)}\Big{(}\mathrm{sinc}\big{(}\beta\sqrt{w^{2}-1}\,\big{)}-\mathrm{sinc}(\beta
w)\Big{)}\,{\mathrm{d}}w\,.$

By [7, 3.997–1] we have

$\displaystyle\int_{-1}^{1}\frac{\sinh\big{(}\beta\sqrt{1-w^{2}}\,\big{)}}{\beta\sqrt{1-w^{2}}}\,{\mathrm{d}}w$
$\displaystyle=\frac{2}{\beta}\,\int_{0}^{1}\frac{\sinh\big{(}\beta\sqrt{1-w^{2}}\,\big{)}}{\sqrt{1-w^{2}}}\,{\mathrm{d}}w$
$\displaystyle=\frac{2}{\beta}\,\int_{0}^{\pi/2}\sinh(\beta\cos
s)\,{\mathrm{d}}s=\frac{\pi}{\beta}\,{\textbf{L}}_{0}(\beta)\,,$

where ${\textbf{L}}_{0}(x)$ denotes the _modified Struve function_ given by
(see [1, 12.2.1])

${\textbf{L}}_{0}(x)\coloneqq\sum_{k=0}^{\infty}\frac{(x/2)^{2k+1}}{\big{(}\Gamma\big{(}k+\tfrac{3}{2}\big{)}\big{)}^{2}}=\frac{2x}{\pi}\,\sum_{k=0}^{\infty}\frac{x^{2k}}{\big{(}(2k+1)!!\big{)}^{2}}\,.$

Note that the function $I_{0}(x)-{\textbf{L}}_{0}(x)$ is completely monotonic
on $[0,\,\infty)$ (see [3, Theorem 1]) and tends to zero as $x\to\infty$.
Applying the _sine integral function_

$\mathrm{Si}(x)\coloneqq\int_{0}^{x}\frac{\sin
w}{w}\,{\mathrm{d}}w=\int_{0}^{x}\mathrm{sinc}\,w\,{\mathrm{d}}w\,,\quad
x\in\mathbb{R}\,,$

implies

$\int_{-1}^{1}\mathrm{sinc}(\beta
w)\,{\mathrm{d}}w=2\,\int_{0}^{1}\mathrm{sinc}(\beta
w)\,{\mathrm{d}}w=\frac{2}{\beta}\,\mathrm{Si}(\beta)\,.$

Hence, we obtain

$\displaystyle\eta_{1}(v)$
$\displaystyle=1-\frac{1}{I_{0}(\beta)-1}\,\bigg{(}{\textbf{L}}_{0}(\beta)-\frac{2}{\pi}\,\mathrm{Si}(\beta)\bigg{)}=\frac{1}{I_{0}(\beta)-1}\,\bigg{(}I_{0}(\beta)-{\textbf{L}}_{0}(\beta)-1+\frac{2}{\pi}\,\mathrm{Si}(\beta)\bigg{)}\,.$

Note that for suitable $\beta=\tfrac{m\pi\lambda}{1+\lambda}$ we find
$\big{[}I_{0}(\beta)-{\textbf{L}}_{0}(\beta)-1+\frac{2}{\pi}\,\mathrm{Si}(\beta)\big{]}\in(0,\,1)$,
cf. Figure 4.2. In addition, it is known that $I_{0}(x)\geq 1$,
$x\in\mathbb{R}$, such that $\eta_{1}(v)>0$.

Figure 4.2: Visualization of
$\big{[}I_{0}(\beta)-{\textbf{L}}_{0}(\beta)-1+\frac{2}{\pi}\,\mathrm{Si}(\beta)\big{]}\in(0,\,1)$
for suitable $\beta=\tfrac{m\pi\lambda}{1+\lambda}$.

Now we estimate $\eta_{2}(v)$ for
$v\in\big{[}-\tfrac{N}{2},\,\tfrac{N}{2}\big{]}$ by the triangle inequality as

$|\eta_{2}(v)|\leq\frac{\beta}{\pi\,\big{(}I_{0}(\beta)-1\big{)}}\,\bigg{(}\int_{-a(-v)}^{-1}+\int_{1}^{a(v)}\bigg{)}\Big{|}\mathrm{sinc}\big{(}\beta\sqrt{w^{2}-1}\,\big{)}-\mathrm{sinc}(\beta
w)\Big{|}\,{\mathrm{d}}w\,.$

By [20, Lemma 4] we have for $|w|\geq 1$ that

$\Big{|}\mathrm{sinc}\big{(}\beta\sqrt{w^{2}-1}\,\big{)}-\mathrm{sinc}(\beta
w)\Big{|}\leq\frac{2}{w^{2}}\,.$

Thus, we conclude

$|\eta_{2}(v)|\leq\frac{4\beta}{\pi\,\big{(}I_{0}(\beta)-1\big{)}}\,\int_{1}^{\infty}\frac{1}{w^{2}}\,{\mathrm{d}}w=\frac{4\beta}{\pi\,\big{(}I_{0}(\beta)-1\big{)}}$

and using Figure 4.2 we therefore obtain

$\displaystyle|\eta(v)|$
$\displaystyle\leq\eta_{1}(v)+|\eta_{2}(v)|\leq\frac{1}{I_{0}(\beta)-1}\,\bigg{(}I_{0}(\beta)-{\textbf{L}}_{0}(\beta)-1+\frac{2}{\pi}\,\mathrm{Si}(\beta)+\frac{4\beta}{\pi}\bigg{)}$
$\displaystyle\leq\frac{1}{I_{0}(\beta)-1}\,\bigg{(}1+\frac{4\beta}{\pi}\bigg{)}\,.$

By [2] the function $\frac{{\mathrm{e}}^{x}}{I_{0}(x)}$ is strictly decreasing
on $[0,\infty)$ and tends to zero as $x\to\infty$. Numerical experiments have
shown that $\frac{{\mathrm{e}}^{x}}{x\,(I_{0}(x)-1)}$ is strictly decreasing
on $[\pi,\infty)$, too. By the assumption $\lambda\geq\frac{1}{m-1}$ we have
$\beta=\frac{\pi m\lambda}{1+\lambda}\geq\pi$ for all
$m\in\mathbb{N}\setminus\\{1\\}$. Hence, it follows that

$\displaystyle\frac{{\mathrm{e}}^{\beta}}{\beta\,(I_{0}(\beta)-1)}\leq\frac{{\mathrm{e}}^{\pi}}{\pi\,(I_{0}(\pi)-1)}=1.644967\ldots<\frac{7}{4}$

and thus we conclude that

$\displaystyle\frac{1}{I_{0}(\beta)-1}\,\bigg{(}1+\frac{4\beta}{\pi}\bigg{)}$
$\displaystyle<\frac{7\beta}{4}\,\bigg{(}1+\frac{4\beta}{\pi}\bigg{)}\,{\mathrm{e}}^{-\beta}=\frac{7\pi
m\lambda}{4\,(1+\lambda)}\,\bigg{(}1+\frac{4m\lambda}{1+\lambda}\bigg{)}\,{\mathrm{e}}^{-\pi
m\lambda/(1+\lambda)}$

for all $m\in\mathbb{N}\setminus\\{1\\}$. This completes the proof.

As seen in Theorem 2.4, if the samples $f\big{(}\frac{k}{L}\big{)}$,
$k\in\mathbb{Z}$, of a bandlimited function
$f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ are not known exactly, i.e., only
erroneous samples ${\tilde{f}}_{k}\coloneqq
f\big{(}\frac{k}{L}\big{)}+\varepsilon_{k}$ with
$|\varepsilon_{k}|\leq\varepsilon$, $k\in\mathbb{Z}$, with $\varepsilon>0$ are
known, the corresponding Shannon sampling series (1.3) may differ appreciably
from $f$. Here we denote the regularized Shannon sampling formula with
erroneous samples ${\tilde{f}}_{k}$ by

$\displaystyle(R_{\varphi,m}{\tilde{f}})(t)=\sum_{k\in\mathbb{Z}}\,\tilde{f}_{k}\
\mathrm{sinc}(L\pi t-\pi k)\,\varphi\big{(}t-\tfrac{k}{L}\big{)},\quad
t\in\mathbb{R}.$

Then, in contrast to the Shannon sampling series (1.3), the regularized
Shannon sampling formula (4.3) is numerically robust in the worst case
analysis, i.e., the uniform perturbation error
$\|R_{\varphi,m}{\tilde{f}}-R_{\varphi,m}f\|_{C_{0}(\mathbb{R})}$ is small.

######  Theorem 4.4.

Let $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ be a bandlimited function with
bandwidth $\tfrac{N}{2}$, $N\in\mathbb{N}$, and let $L>N$, and
$m\in{\mathbb{N}}\setminus\\{1\\}$ be given. Further let
$\varphi\in\Phi_{m/L}$ as well as
${\tilde{f}}_{k}=f(\frac{k}{L})+\varepsilon_{k}$, where
$|\varepsilon_{k}|\leq\varepsilon$ for all $k\in\mathbb{Z}$, with
$0<\varepsilon\ll 1$.
Then the regularized Shannon sampling sum (4.3) satisfies

$\displaystyle\|R_{\varphi,m}{\tilde{f}}-R_{\varphi,m}f\|_{C_{0}(\mathbb{R})}$
$\displaystyle\leq\varepsilon\,\big{(}2+L\,\hat{\varphi}(0)\big{)},$ (4.11)
$\displaystyle\|f-R_{\varphi,m}{\tilde{f}}\|_{C_{0}(\mathbb{R})}$
$\displaystyle\leq\|f-R_{\varphi,m}{f}\|_{C_{0}(\mathbb{R})}+\varepsilon\,\big{(}2+L\,\hat{\varphi}(0)\big{)}.$

_Proof_. For a proof of Theorem 4.4 see [10, Theorem 3.4].

Note that it is especially beneficial for obtaining explicit error estimates,
if the Fourier transform (4.6) of $\varphi\in\Phi_{m/L}$ is explicitly known.
In the following, we demonstrate that for the window functions (4.1) and (4.2)
the perturbation error of the regularized Shannon sampling formula (4.3) only
grows as $\mathcal{O}(\sqrt{m})$.

######  Theorem 4.5.

Let $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ be a bandlimited function with
bandwidth $\tfrac{N}{2}$, $N\in\mathbb{N}$, and let $L=N\,(1+\lambda)$ with
$\lambda>0$ and $m\in{\mathbb{N}}\setminus\\{1\\}$ be given. Further let
${\tilde{f}}_{k}=f\big{(}\frac{k}{L}\big{)}+\varepsilon_{k}$, with
$|\varepsilon_{k}|\leq\varepsilon$ for all $k\in\mathbb{Z}$ and
$0<\varepsilon\ll 1$.
Then the regularized Shannon sampling formula (4.3) with the $\sinh$-type
window function (4.1) satisfies the error estimate

$\displaystyle\|R_{\sinh,m}{\tilde{f}}-R_{\sinh,m}f\|_{C_{0}(\mathbb{R})}\leq\varepsilon\left(2+\sqrt{\frac{2+2\lambda}{\lambda}}\,\frac{1}{1-\mathrm{e}^{-2\beta}}\,\sqrt{m}\right).$

_Proof_. For a proof of Theorem 4.5 see [10, Theorem 6.3].

######  Theorem 4.6.

Let $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ be a bandlimited function with
bandwidth $\tfrac{N}{2}$, $N\in\mathbb{N}$, and let $L=N\,(1+\lambda)$ with
$\lambda>0$ and $m\in{\mathbb{N}}\setminus\\{1\\}$ be given. Further let
${\tilde{f}}_{k}=f\big{(}\frac{k}{L}\big{)}+\varepsilon_{k}$, with
$|\varepsilon_{k}|\leq\varepsilon$ for all $k\in\mathbb{Z}$ and
$0<\varepsilon\ll 1$.
Then the regularized Shannon sampling formula (4.3) with the continuous
Kaiser–Bessel window function (4.2) satisfies the error estimate

$\displaystyle\|R_{\mathrm{cKB},m}{\tilde{f}}-R_{\mathrm{cKB},m}f\|_{C_{0}(\mathbb{R})}\leq\varepsilon\left(2+\sqrt{\frac{2+2\lambda}{\lambda}}\,\sqrt{m}\right).$
(4.12)

_Proof_. By Theorem 4.4 we have to compute $\hat{\varphi}_{\mathrm{cKB}}(0)$
for the continuous Kaiser–Bessel window function (4.2), which is given by
(4.10) as

$\displaystyle\hat{\varphi}_{\mathrm{cKB}}(0)$
$\displaystyle=\frac{2m}{\big{(}I_{0}(\beta)-1\big{)}\,L}\left(\frac{\sinh(\beta)}{\beta}-1\right)=\frac{2m}{L\sqrt{\beta}}\cdot\frac{\sinh(\beta)-\beta}{\sqrt{\beta}\,\big{(}I_{0}(\beta)-1\big{)}}.$

By [1, 9.7.1] we have

$\displaystyle\lim_{x\to\infty}\sqrt{2\pi x}\,\mathrm{e}^{-x}I_{0}(x)=1$

and hence

$\displaystyle\lim_{x\to\infty}\sqrt{2\pi
x}\,\mathrm{e}^{-x}\,\big{(}I_{0}(x)-1\big{)}=1.$

Moreover, for $\beta>0$ the term
$\frac{\sinh(\beta)-\beta}{\sqrt{\beta}\,\big{(}I_{0}(\beta)-1\big{)}}$ is
monotonously increasing with

$\displaystyle\lim_{\beta\to\infty}\,\frac{\sinh(\beta)-\beta}{\sqrt{\beta}\,\big{(}I_{0}(\beta)-1\big{)}}=\lim_{\beta\to\infty}\,\frac{1-\mathrm{e}^{-2\beta}-2\beta\,\mathrm{e}^{-\beta}}{2\sqrt{\beta}\,\mathrm{e}^{-\beta}\,\big{(}I_{0}(\beta)-1\big{)}}=\sqrt{\frac{\pi}{2}}\,,$

such that (4.11) and $\beta=\frac{\pi m\lambda}{1+\lambda}$ yields the
assertion (4.12).

##  5 Comparison of the two regularization methods

Finally, we compare the behavior of the regularization methods presented in
Sections 3 and 4 to the classical Shannon sampling sums (2.1). For a given
function $f\in{\mathcal{B}}_{N/2}(\mathbb{R})$ with $L=N\,(1+\lambda)$,
$\lambda>0$, we consider the approximation errors

$\max_{t\in[-1,\,1]}\big{|}f(t)-(S_{T}f)(t)\big{|}\quad\text{ and
}\quad\max_{t\in[-1,\,1]}\big{|}f(t)-(P_{\psi,T}f)(t)\big{|}$ (5.1)

for
$\psi\in\\{\psi_{\mathrm{lin}},\,\psi_{\mathrm{cub}},\,\psi_{\cos},\,{\psi}_{\mathrm{conv},2},\,\psi_{\mathrm{rat}}\\}$,
cf. (3.1), (3.16), (3.19), (3.23), and Example 3.12, as well as the
corresponding error constants (3.10) and (3.17). In addition, we study the
approximation error

$\max_{t\in[-1,\,1]}|f(t)-(R_{\varphi,m}f)(t)|$ (5.2)

with $\varphi\in\\{\varphi_{\sinh},\,\varphi_{\mathrm{cKB}}\\}$, cf. (4.1) and
(4.2), and the corresponding error constants (4.7) and (4.8). By the
definition of the regularized Shannon sampling formula in (4.3) we have

$(R_{\varphi,m}f)(t)=\sum_{k=-L-m}^{L+m}f(\tfrac{k}{L})\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\xi}(t-\tfrac{k}{L})\,,\quad
t\in[-1,1]\,,$ (5.3)

with the regularized $\mathrm{sinc}$ function

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\xi}(t)\coloneqq\mathrm{sinc}(L\pi
t)\,\varphi(t)\,.$ (5.4)

Thus, to compare (5.3) to $S_{T}f$ in (2.1) and $P_{\psi,T}f$ in (3.9), we set
$T=L+m$, such that all approximations use the same number of samples. As in
Example 3.5 the errors (5.1) and (5.2) shall be estimated by evaluating a
given function $f$ and its approximation at equidistant points
$t_{s}\in[-1,\,1]$, $s=1,\dots,S,$ with $S=10^{5}$. Analogous to [16, Section
IV, C], we choose the function

$f(t)=\sqrt{\tfrac{4N}{5}}\left[\mathrm{sinc}(N\pi
t)+\tfrac{1}{2}\,\mathrm{sinc}(N\pi(t-1))\right]\,,\quad t\in\mathbb{R}\,,$
(5.5)

with $\|f\|_{L^{2}(\mathbb{R})}=1$. We fix $N=256$ and consider several values
of $m\in\mathbb{N}\setminus\\{1\\}$ and $\lambda>0$.

The associated results are displayed in Figure 5.1. We see that for all window
functions the theoretical error behavior perfectly coincides with the
numerical outcomes.  In this regard, see also Table 5.1 which summarizes the
theoretical results. Moreover, it can clearly be seen that for higher
oversampling parameter $\lambda$ and higher truncation parameter $m$, the
error results using (4.3) get much better than the ones using (3.6), due to
the exponential error decay rate shown for (4.3).

This is to say, our numerical results show that regularization with a time
window function performs much better than regularization with a frequency
window function, since an exponential decay can (up to now) only be realized
using a time window function. Furthermore, the great importance of an explicit
representation of the Fourier transform of the regularizing window function
can be seen, cf. Example 3.12.

window function | error decay rate | see
---|---|---
$\mathrm{sinc}(L\pi\,\boldsymbol{\cdot})$ | $(T-L)^{-1/2}$ | [8, Theorem 1]
$\hat{\psi}_{\mathrm{lin}}$ in (3.3) | $(T-L)^{-3/2}$ | Theorem 3.4
$\hat{\psi}_{\mathrm{cub}}$ in (3.15) | $(T-L)^{-5/2}$ | Theorem 3.7
$\hat{\psi}_{\mathrm{cos}}$ in (3.18) | $(T-L)^{-5/2}$ | Example 3.8
$\hat{\psi}_{\mathrm{conv},2}$, cf. (3.23) | $(T-L)^{-5/2}$ | Example 3.11
$\varphi_{\sinh}$ in (4.1) | ${\mathrm{e}}^{-m\pi\lambda/(1+\lambda)}$ | Theorem 4.2
$\varphi_{\mathrm{cKB}}$ in (4.2) | ${\mathrm{e}}^{-m\pi\lambda/(1+\lambda)}$ | Theorem 4.3
Table 5.1:  Summary of the theoretical results on decay rates for the window
functions considered in Sections 3 and 4.

(a) $\lambda=0.5$

(b) $\lambda=1$

(c) $\lambda=2$

Figure 5.1: Maximum approximation error (solid) and error constants (dashed)
using  classical Shannon sampling sums compared to regularizations (3.6) with
several frequency window functions and regularizations (4.3) with time window
functions $\varphi_{\sinh}$ and $\varphi_{\mathrm{cKB}}$, cf. (5.4), for the
function (5.5) with $N=256$, $m\in\\{2,3,\ldots,10\\}$, and
$\lambda\in\\{0.5,1,2\\}$.

Note that the code files for this and all the other experiments are available
on https://github.com/melaniekircheis/On-numerical-realizations-of-Shannons-
sampling-theorem.

In summary, we found that the regularized Shannon sampling formula with the
$\sinh$-type time window function is the best of the considered methods, since
this approach is the most accurate, easy to compute, robust in the worst case
error, and requires less data (for comparable accuracy) than the classical
Shannon sampling sums or the regularization with a frequency window function.

## Acknowledgments

Melanie Kircheis gratefully acknowledges the support from the BMBF grant
01$\mid$S20053A (project SA$\ell$E).

Moreover, the authors thank the referees and the editor for their very useful
suggestions for improvements.

## Conflict of interest statement

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

## References

  * [1] M. Abramowitz and I.A. Stegun, editors.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.  Dover, New York, 1972.
  * [2] Á. Baricz.  Bounds for modified Bessel functions of the first and second kinds.  Proc. Edinb. Math. Soc. (2), 53:575–599, 2010.
  * [3] Á. Baricz and T.K. Pogány.  Functional inequalities for modified Struve functions II.  Math. Inequal. Appl., 17:1387–1398, 2014.
  * [4] O. Christensen.  An Introduction to Frames and Riesz Bases.  Second edition, Birkhäuser/Springer, Basel, 2016.
  * [5] I. Daubechies.  Ten Lectures on Wavelets.  SIAM, Philadelphia, 1992.
  * [6] I. Daubechies and R. DeVore.  Approximating a bandlimited function using very coarsely quantized data: A family of stable sigma-delta modulators of arbitrary order.  Ann. of Math. (2), 158:679–710, 2003.
  * [7] I.S. Gradshteyn and I.M. Ryzhik.  Table of Integrals, Series, and Products.  Academic Press, New York, 1980.
  * [8] D. Jagerman.  Bounds for truncation error of the sampling expansion.  SIAM J. Appl. Math., 14(4):714–723, 1966.
  * [9] L. Jingfan and F. Gensun.  On uniform truncation error bounds and aliasing error for multidimensional sampling expansion.  Sampl. Theory Signal Image Process., 2(2):103–115, 2003.
  * [10] M. Kircheis, D. Potts, and M. Tasche.  On regularized Shannon sampling formulas with localized sampling.  Sampl. Theory Signal Process. Data Anal., 20(2): Paper No. 20, 34 pp., 2022.
  * [11] V. A. Kotelnikov.  On the transmission capacity of the “ether” and wire in electrocommunications.  In Modern Sampling Theory: Mathematics and Application, pages 27–45. Birkhäuser, Boston, 2001.  Translated from Russian.
  * [12] X. M. Li.  Uniform bounds for sampling expansions.  J. Approx. Theory, 93(1):100–113, 1998.
  * [13] R. Lin and H. Zhang.  Convergence analysis of the Gaussian regularized Shannon sampling formula.  Numer. Funct. Anal. Optim., 38(2):224–247, 2017.
  * [14] C.A. Micchelli, Y. Xu, and H. Zhang.  Optimal learning of bandlimited functions from localized sampling.  J. Complexity, 25(2):85–114, 2009.
  * [15] F. Natterer.  Efficient evaluation of oversampled functions.  J. Comput. Appl. Math., 14(3):303–309, 1986.
  * [16] F. Oberhettinger.  Tables of Fourier Transforms and Fourier Transforms of Distributions.  Springer, Berlin, 1990.
  * [17] J. R. Partington.  Interpolation, Identification, and Sampling.  Clarendon Press, New York, 1997.
  * [18] G. Plonka, D. Potts, G. Steidl, and M. Tasche.  Numerical Fourier Analysis.  Second edition, Birkhäuser/Springer, Cham, 2023.
  * [19] D. Potts and M. Tasche.  Uniform error estimates for nonequispaced fast Fourier transforms.  Sampl. Theory Signal Process. Data Anal. 19(2): Paper No. 17, 42 pp., 2021.
  * [20] D. Potts and M. Tasche.  Continuous window functions for NFFT.  Adv. Comput. Math. 47(2): Paper 53, 34 pp., 2021.
  * [21] L. Qian.  On the regularized Whittaker–Kotelnikov–Shannon sampling formula.  Proc. Amer. Math. Soc., 131(4):1169–1176, 2003.
  * [22] L. Qian.  The regularized Whittaker-Kotelnikov-Shannon sampling theorem and its application to the numerical solutions of partial differential equations.  PhD thesis, National Univ. Singapore, 2004.
  * [23] L. Qian and D.B. Creamer.  Localized sampling in the presence of noise.  Appl. Math. Letter, 19:351–355, 2006.
  * [24] T. S. Rappaport.  Wireless Communications: Principles and Practice.  Prentice Hall, New Jersey, 1996.
  * [25] C. E. Shannon.  Communication in the presence of noise.  Proc. I.R.E., 37:10–21, 1949.
  * [26] G. Schmeisser and F. Stenger.  Sinc approximation with a Gaussian multiplier.  Sampl. Theory Signal Image Process., 6(2):199–221, 2007.
  * [27] F. Stenger.  Numerical Methods Based on Sinc and Analytic Functions.  Springer, New York, 1993.
  * [28] T. Strohmer and J. Tanner.  Implementations of Shannon’s sampling theorem, a time–frequency approach.  Sampl. Theory Signal Image Process., 4(1):1–17, 2005.
  * [29] T. Strohmer and J. Tanner.  Fast reconstruction methods for bandlimited functions from periodic nonuniform sampling.  SIAM J. Numer. Anal., 44(3):1071-1094, 2006.
  * [30] K. Tanaka, M. Sugihara, and K. Murota.  Complex analytic approach to the sinc-Gauss sampling formula.  Japan J. Ind. Appl. Math., 25:209–231, 2008.
  * [31] M. Tasche and H. Zeuner.  Worst and average case roundoff error analysis for FFT.  BIT, 41(3):563–581, 2001.
  * [32] E.T. Whittaker.  On the functions which are represented by the expansions of the interpolation theory.  Proc. R. Soc. Edinb., 35:181–194, 1915.
  * [33] R.M. Young.  Euler’s constant.  Math. Gaz., 75:187–190, 1991.
  * [34] A. I. Zayed.  Advances in Shannon’s Sampling Theory.  CRC Press, Boca Raton, 1993.