Abstract
The exponential sampling formula has some limitations. By incorporating a Mellin bandlimited multiplier, we extend it to a wider class of functions with a series that converges faster. This series is a generalized exponential sampling series with some interesting properties. Moreover, under a side condition, any generalized exponential sampling series that is interpolating can be generated by a Mellin bandlimited multiplier. For an error analysis, we consider a truncated series with \(2N+1\) terms and look for a highest speed of convergence as \(N\rightarrow \infty \). We show by using a certain nonbandlimited multiplier, which introduces in addition an aliasing error, that we can achieve a higher rate of convergence to the function, namely \(\mathcal {O}(e^{\alpha N})\) with \(\alpha >0\), than with the truncated series of an exact formula. The results are illustrated by three examples.
1 Introduction
The exponential sampling formula was introduced in a formal way in [13, 24] in connection with problems related to optical physics phenomena like lightscattering, diffraction and others. Later, a rigorous treatment was first performed in [15, 16] for functions f belonging to the Mellin–Paley–Wiener spaces \(B^1_{c, \pi T}\) and \(B^2_{c, \pi T}\), with \(c \in \mathbb {R}\) and \(T>0\) (see Sect. 2 for the definition of these spaces). The latter formula is contained in the following theorem, which may serve as the starting point of this paper. We present this result in a notation that will be specified in Sect. 2; also see [16, Theorem 5.2] or [3, Theorem 1].
Theorem A
Let f belong to the Mellin–Paley–Wiener space \(B_{c,\pi T}^2\), where \(c\in \mathbb {R}\) and \(T>0\). Then
The series converges uniformly on compact subsets of the positive part of the real line.
In Mellin analysis, this is the counterpart of the familiar sampling formula of Whittaker–Kotel’nikov–Shannon (WKS for short). As the latter, Theorem A is of central importance in several ways but it also has some limitations in theoretical as well as in practical use. For example, the assumption that the function f is bandlimited in the sense of Mellin, imposes a severe restriction, as one recognizes by looking at the Mellin version of the Paley–Wiener theorem (see [4, 5]). It excludes all functions f for which \(\limsup _{r\rightarrow +\infty } \left r^cf(r)\right >0\) or \(\limsup _{r\rightarrow 0+} \left r^cf(r)\right >0\) let alone unbounded functions. Furthermore, the series may converge very slowly. We shall see that these deficiencies can be overcome by incorporating a Mellin bandlimited multiplier in the series.
This technique can be considered within a more general framework. Motivated by the development in the study of the WKS sampling series, there was introduced in [11] (see also [10]) the socalled generalized exponential sampling series, in which the function \(\textrm{lin}_{c/T}\) is replaced by a general continuous function \(\varphi \) satisfying suitable assumptions (see Definition 4 below). Thus the generalized exponential sampling series of a function f with w instead of T takes the form
The function \(\varphi \) is called the kernel of \((S^\varphi _wf)(r).\)
In general, such a series does not reproduce f as the series in Theorem A, but it approximates continuous, bounded functions f in the sense that
(see [11, Theorem 3.2]). For finite w, we have \(f(r)=(S^\varphi _wf)(r)+(E_wf)(r)\) with an approximation error \((E_wf)(r)\). A further error occurs in computations since we will have to truncate the series \((S^\varphi _wf)(r)\).
The present paper can also be seen as a study of the following questions:

1.
How can we find kernels \(\varphi \) for which the corresponding generalized exponential sampling series improves upon the series of Theorem A?

2.
What is an appropriate kernel \(\varphi \) for establishing an efficient algorithm by truncating the corresponding generalized exponential sampling series?
We shall answer both questions by adopting a multiplier technique that was effectively developed for the WKS sampling series (see [17,18,19,20, 22, 25,26,27,28,29]).
This paper is organized as follows. After fixing our notation and recalling some preliminary results in the next section, we devote Sect. 3 to Mellin bandlimited multipliers \(\psi \) such that \(\psi (1)=1\), obtaining a formula
that holds for a wider class of functions f than the corresponding formula of Theorem A (see Theorem 3 below). The generalized exponential sampling series \((S_w^{\psi {{\,\textrm{lin}\,}}}f)(r)\), occurring on the righthand side for \(c=0\), has several remarkable properties (see Theorem 5) and is even of interest in the case of functions f for which (2) is violated since this series always interpolates with respect to the sample points. Moreover, there is a converse result: Any generalized sampling series (1) which interpolates with respect to the sample points and has a Mellin bandlimited kernel \(\varphi \) satisfying a mild side condition can be obtained from the series of Theorem A by incorporating a Mellin bandlimited multiplier \(\psi \), that is, \(\varphi = \psi {{\,\textrm{lin}\,}}\) (Theorem 6). This establishes an interesting link between the classical and the generalized sampling series. We also study the approximation power of \((S_w^{\psi {{\,\textrm{lin}\,}}}f)(r)\) for bounded, continuous functions f and express it in terms of best approximation by functions from a Mellin–Bernstein space (Proposition 7).
Particular attention is paid to the truncation error which occurs when the series \((S_w^{\psi {{\,\textrm{lin}\,}}}f)(r)\) is reduced to the sum \((S_{w,N}^{\psi {{\,\textrm{lin}\,}}}f)(r)\) containing the \(2N+1\) terms whose sample points are closest to r. The obtained results include classes of unbounded functions f (Theorem 9). As one would suggest, the speed of convergence to zero of the truncation error, which may be interpreted as the speed of an algorithm that approximates f(r) from \(2N+1\) samples, depends on the decay of \(\psi (r)\) as \(r\rightarrow 0+\) and \(r\rightarrow +\infty \). This leads us to the question as to how fast a Mellin bandlimited function \(\psi \) can decay if \(\psi (1)=1\). With an answer adopted from [21], we shall see that one can achieve convergence of order \(\mathcal {O}(e^{N/(\log N)^\gamma })\) with \(\gamma >1\) as \(N\rightarrow \infty \), but it is in general not possible to have \(\mathcal {O}(e^{\alpha N})\) with \(\alpha >0\).
In Sect. 4 we overcome this limitation. Inspired by results for the WKS sampling series, we use a multiplier that is based on a Gaussian function. It is not Mellin bandlimited and the corresponding generalized exponential sampling series cannot reproduce Mellin bandlimited functions f, but approximates them with a socalled aliasing error which adds to the truncation error. Nevertheless, one has the amazing phenomenon that the sum of these two errors converges to zero with a higher rate than the truncation error of an exact formula (Theorem 11, Corollaries 2 and 3).
Finally, in Sect. 5, we carry out numerical experiments for three examples of admissible functions f in order to illustrate some of the results of Sects. 3 and 4. An algorithm based on the results of Sect. 4 may be the method of choice in computational exponential sampling.
As to the methods employed in this paper, we mention that we use only tools form Mellin analysis which are based on the theory of polaranalytic functions, introduced in [5] and further developed in [6,7,8,9] (also see the forthcoming book [10]).
2 Basic notations and preliminary results
In what follows, we denote by \(\mathbb {N},\) and \(\mathbb {Z}\) the sets of positive integers and integers, respectively, by \(\mathbb {R}\) and \(\mathbb {R}^+\) the sets of real and positive real numbers, respectively, and by \(\mathbb {C}\) the set of complex numbers.
Let \(C(\mathbb {R}^+)\) be the space of all continuous functions defined on \(\mathbb {R}^+.\)
For \(1\le p < +\infty ,\) let \(L^p(\mathbb {R}^+)\) be the space of all Lebesgue measurable and pintegrable complexvalued functions defined on \(\mathbb {R}^+\) endowed with the usual norm \(\Vert f\Vert _p.\) Analogous notations hold for functions defined on \(\mathbb {R}.\)
For \(p=1\) and \(c \in \mathbb {R},\) we introduce the space (see [14])
endowed with the norm
More generally, let \(X^p_c\) denote the space of all functions \(f: \mathbb {R}^+\rightarrow \mathbb {C}\) such that \(f(\cdot ) (\cdot )^{c1/p}\in L^p (\mathbb {R}^+)\), where \(1<p< \infty .\) Finally, for \(p=\infty \), we define \(X^\infty _c\) as the space comprising all measurable functions \(f : \mathbb {R}^+\rightarrow \mathbb {C}\) such that \(\Vert f\Vert _{X^\infty _c}:= \sup _{x>0}x^{c}f(x) < \infty ;\) see [16] for \(p=2\) and [10] for general p.
For \(h \in \mathbb {R}^+\) and \(c \in \mathbb {R}\), the Mellin translation operator \(\tau _h^c\), applying to functions \(f: \mathbb {R}^+ \rightarrow \mathbb {C},\) is defined by
Setting \(\tau _h:= \tau ^0_h,\) we have \((\tau _h^cf)(x) = h^c (\tau _hf)(x)\) and \(\Vert \tau _h^c f\Vert _{X_c} = \Vert f\Vert _{X_c}.\)
The Mellin transform of a function \(f\in X_c\) is the linear and bounded operator defined by (see, e.g., [14])
More generally, for \(1<p \le 2,\) the Mellin transform \(M_c^p\) of \(f \in X^p_c\) is given by (see [16] for \(p=2\) and [10] for general p)
for \(s=c+it,\) in the sense that
where \(p'\) is the conjugate exponent of p, that is, \(1/p+ 1/p'=1\).
The basic tool for a selfcontained and independent treatment of Mellin analysis is the theory of polaranalytic functions. The notion of polar analyticity was first introduced in [5], and subsequently the theory of these functions was developed in the papers [6,7,8,9]; also see the forthcoming book [10]. Here we recall the definition of this notion. Let \(\mathbb {H}:=\{(r, \theta ): r>0,\,\theta \in \mathbb {R}\}\) be the polar plane. By \(\mathscr {D}\) we denote a domain in \(\mathbb {H}\), that is, a nonempty, open and connected subset of \(\mathbb {H}.\)
Definition 1
We say that \(f:\mathscr {D} \rightarrow \mathbb {C}\) is polaranalytic on \(\mathscr {D}\) if for every \((r_0, \theta _0) \in \mathscr {D}\) the limit
exists and is the same howsoever \((r, \theta )\) approaches \((r_0, \theta _0)\) within \(\mathscr {D}.\)
It is easy to see that, writing \(f(r, \theta ) = u(r, \theta ) + i v(r, \theta )\) with u, v being the real and imaginary parts of f, the function f is polaranalytic on \(\mathscr {D}\) if and only if u and v have continuous partial derivatives on \(\mathscr {D}\) which satisfy the Cauchy–Riemann equations in polar form (see, e.g., [7, 10]).
Next, we recall the definitions of two fundamental function spaces.
Definition 2
For \(c \in \mathbb {R},\) \(T>0\) and \(p \in [1, +\infty ]\) the Mellin–Bernstein space \(\mathscr {B}^p_{c,T}\) comprises all functions \(f: \mathbb {H}\rightarrow \mathbb {C}\) with the following properties:

(i)
f is polaranalytic on \(\mathbb {H};\)

(ii)
\(f(\cdot , 0) \in X^p_{c};\)

(iii)
there exists a positive constant \(C_f\) such that
$$\begin{aligned} \left f(r,\theta )\right \le C_f r^{c}e^{T\theta }\qquad ((r, \theta ) \in \mathbb {H}). \end{aligned}$$
It is easily seen that the following inclusions hold:
and
Furthermore, the Mellin–Bernstein spaces are translation invariant in the following sense. If \(f\in \mathscr {B}_{c,T}^p\) and \((r_0, \theta _0)\) is any point of \(\mathbb {H}\), then the function
also belongs to \(\mathscr {B}_{c,T}^p\). Note that \(f_{(r_0,\theta _0)}\) is obtained from f by the Mellin translation \(\tau _{r_0}^c\) with respect to the first variable and an ordinary translation by \(\theta _0\) with respect to the second variable. For a proof one has to verify that \(f_{(r_0,\theta _0)}\) satisfies conditions (i)–(iii) of Defintion 2. For (i) one can use the Cauchy–Riemann equations in polar form. For (ii) one may employ a result in [5, Theorem 4.2], which yields that
and it is easily seen that (iii) holds with \(C_{f_{(r_0,\theta _0)}}=e^{T\left \theta _0\right }C_f.\)
Definition 3
For \(c \in \mathbb {R},\,T>0\) and \(p\in [1, 2]\), the Mellin–Paley–Wiener space \(B^p_{c, T}\) comprises all functions \(f\in C(\mathbb {R}^+)\cap X^p_c\) which are Mellin bandlimited to \([T, T]\), that is, \([f]^\wedge _{M_c^p}(c+it) = 0\) a.e. for \(t >T.\)
The following basic result is a Mellin version of the classical Paley–Wiener theorem (see [5, 10]).
Theorem 1
A function \(\varphi \in X^2_c\) belongs to the Mellin–Paley–Wiener space \(B^2_{c,T}\) if and only if there exists a function \(f \in \mathscr {B}^2_{c,T}\) such that \(f(\cdot , 0) = \varphi (\cdot ).\)
In connection with sampling formulas, the second index of the Mellin–Bernstein spaces and the Mellin–Paley–Wiener spaces is often expressed as a multiple of \(\pi \) to the benefit that the sample points will contain no \(\pi \). Subsequently we want to stick to this custom.
A second basic result is the following version of the Parseval formula in Mellin analysis (see [3]).
Theorem 2
Let \(f,g \in B^2_{c,\pi T}.\) Then
We conclude this section by recalling the \({{\,\textrm{sinc}\,}}\) function, defined on \(\mathbb {C}\) by
It can be used to introduce the \({{\,\textrm{lin}\,}}\) function by
We agree that \({{\,\textrm{lin}\,}}(r):= {{\,\textrm{lin}\,}}_0(r)\). By \(r^{c}e^{ic\theta }{{\,\textrm{lin}\,}}(re^{i\theta })\) for \((r,\theta )\in \mathbb {H}\), we obtain a polaranalytic continuation of \({{\,\textrm{lin}\,}}_c\) to the whole of \(\mathbb {H}\).
3 A Mellin bandlimited multiplier
We start with a lemma that will be useful for studying the convergence of the arising exponential sampling series.
Lemma 1
Let \(\Psi \in \mathscr {B}_{0,\pi \delta }^2\), where \(\delta \in \, ]0, 1]\). Then, for \(w>0\), we have
The series converges uniformly on compact subsets of \(\mathbb {H}\).
Proof
Using the notation (3), we may write
By the translation invariance of the Mellin–Bernstein spaces and the MellinPaley–Wiener theorem, we have
Therefore the Mellin–Parseval formula (see [3, Theorem 4]) and (4) yield
Analogously we conclude that
Now (5) is an immediate consequence of the Cauchy–Schwarz inequality.
It remains to verify the assertion on uniform convergence. For \(N\in \mathbb {N}\), we have again by the Cauchy–Schwarz inequality and (6) that
This shows that it suffices to prove the assertion on uniform convergence for \(\sum _{k\in \mathbb {Z}} \left {{\,\textrm{lin}\,}}(e^{k}r^w e^{iw\theta })\right ^2\) only.
Now let C be a compact subset of \(\mathbb {H}\). There exists an integer \(m\in \mathbb {N}\) such that \(\sup _{(r,\theta )\in C} w\left \log r +i\theta \right \le m\). Then for \(\left k\right >m\) we have
and so for \(N>m\),
Hence, given \(\varepsilon >0\), there exists an \(N\in \mathbb {N}\) such that
for \((r,\theta )\in C.\) This completes the proof. \(\square \)
Theorem 3
For \(\delta \in ]0,1[\), let \(\psi \in B_{0,\pi \delta }^2\) such that \(\psi (1)=1\). Suppose that \(f\in \mathscr {B}_{c,\pi T}^\infty \), where \(c\in \mathbb {R}\) and \(T>0\). Then for \(w=T/(1\delta )\), we have
The series converges absolutely and uniformly on compact subsets of \(\mathbb {R}^+.\)
Proof
For any \(\rho \in \mathbb {R}^+\) serving as a parameter, consider the function \(\psi (\rho ^w/(\cdot )^w)\). It is easily verified that it belongs to \(B_{0,\pi \delta w}^2\). By Theorem 1, there exists a function \(\Psi _\rho \in \mathscr {B}_{0,\pi \delta w}^2\) such that \(\Psi _\rho (r,0)=\psi (\rho ^w/r^w)\) for all \(r\in \mathbb {R}^+.\) Now we readily see from the definition of Mellin–Bernstein spaces that \(f\Psi _\rho \in \mathscr {B}_{c,\pi w}^2.\) Again by the MellinPaley–Wiener theorem,
Therefore Theorem A with T replaced by w applies to this function and yields
This holds for any \(\rho \in \mathbb {R}^+\). Substituting \(\rho =r\), we obtain (7).
It remains to verify the assertion on convergence. Since
we may rewrite the righthand side of (7) as
and so
Now the proof is completed by employing Lemma 1 with \(\theta =0\). \(\square \)
Remark 1
Note that formula (7) of Theorem 3 remains true for all \(w> T/(1\delta )\). This is a consequence of the inclusions between Mellin–Bernstein spaces. Indeed, if \(f\in \mathscr {B}_{c,\pi T}^\infty \), then \(f\in \mathscr {B}_{c,\pi T_1}^\infty \) for all \(T_1 >T\) and so Theorem 3 allows us to set \(w=T_1/(1\delta )\). A corresponding observation holds for some of the subsequent statements.
In the following we shall show that Theorem 3 can be extended in various ways. It will suffice to consider the case \(c=0\) only. For if \(f\in \mathscr {B}_{c,\pi T}^\infty \), then
belongs to \(\mathscr {B}_{0,\pi T}^\infty \). Hence the reconstruction of f can be obtained by applying Theorem 3 with \(c=0\) to g and simply multiplying both sides of the corresponding equation (7) by \(r^{c}\).
First we show that Theorem 3 can be extended to a reconstruction of f on the whole of \(\mathbb {H}\). For this we need a corresponding extension of the multiplier \(\psi \) as guaranteed by Theorem 1. We denote it by \(\Psi \).
Theorem 4
For \(\delta \in \, ]0,1[\), let \(\Psi \in \mathscr {B}_{0,\pi \delta }^2\) such that \(\Psi (1,0)=1.\) Suppose that \(f\in \mathscr {B}_{0,\pi T}^\infty \), where \(T>0\). Then for \(w=T/(1\delta )\), we have
The series converges absolutely and uniformly on compact subsets of \(\mathbb {H}\).
Proof
Since
the assertion on the convergence follows from Lemma 1. Hence
exists for all \((r,\theta )\in \mathbb {H}\) and defines a continuous function g. We also note that, as a function of \((r,\theta )\), each term of the series defining g is polaranalytic on \(\mathbb {H}\).
Now let \(\mathcal {R}\) be any rectangle in \(\mathbb {H}\) with its edges parallel to the axes of the \((r,\theta )\) coordinate system and denote by \(\partial \mathcal {R}\) its positively oriented boundary. Then, for any \(N\in \mathbb {N}\),
as a consequence of Cauchy’s theorem for polaranalytic functions (see [6, Theorem 4.1]). On the other hand, given \(\varepsilon >0\), the uniform convergence on compact subsets of \(\mathbb {H}\) guarantees the existence of an integer \(N\in \mathbb {N}\) such that
This allows us to conclude that
Now a Morera type theorem (see [6, Theorem 4.2]) implies that g is polaranalytic on \(\mathbb {H}\). Then \(h:=fg\) is also polaranalytic on \(\mathbb {H}\) and, as a consequence of Theorem 3, we have \(h(r,0)=0\) for all \(r\in \mathbb {R}^+\). Hence the identity theorem for polaranalytic functions (see [7, Theorem 2]) implies that h is identically zero on \(\mathbb {H}\), as was to be shown. \(\square \)
3.1 Generalized exponential sampling
Now we want to generalize Theorem 3 in another way. Setting \(\varphi := \psi {{\,\textrm{lin}\,}}\) and denoting \(f(\cdot , 0)\) simply by f, we may write the righthand side of (7) for \(c=0\) as
Series of this form have been studied under the name generalized exponential sampling series (see [11, 12]). In order that \((S_w^\varphi f)(r)\) exists and approximates f(r) in some sense, it was required that \(\varphi \) belongs to a class \(\Phi \) which is defined as follows.
Definition 4
The class \(\Phi \) comprises all continuous functions \(\varphi \,:\,\mathbb {R}^+ \rightarrow \mathbb {C}\) with the following properties: For every \(u\in \mathbb {R}^+\), we have

(i)
\( \sum _{k\in \mathbb {Z}} \varphi (e^{k}u) = 1,\)

(ii)
\( \sup _{u\in \mathbb {R}^+}\sum _{k\in \mathbb {Z}} \left \varphi (e^{k}u)\right <\infty ,\)

(iii)
\( \lim _{\rho \rightarrow \infty }\sum _{\left k\log u\right >\rho } \left \varphi (e^{k}u)\right = 0,\) uniformly with respect to \(u\in \mathbb {R}^+\).
For example, if \(\varphi \in \Phi \), then \((S_w^\varphi f)(r)\) exists for every bounded function \(f\,:\, \mathbb {R}^+ \rightarrow \mathbb {C}\) and \(\lim _{w\rightarrow \infty }(S_w^\varphi f)(r)=f(r)\) at each continuity point of f (see [11, Theorem 3.2]). In view of generalized exponential sampling, we can extend Theorem 3 as follows.
Theorem 5
For \(\delta \in \, ]0,1[\), let \(\psi \in B_{0,\pi \delta }^2\) such that \(\psi (1)=1\). Set \(\varphi := \psi {{\,\textrm{lin}\,}}\). Then the following statements hold:

(a)
\(\; \varphi \in \Phi \);

(b)
\(\; \varphi \log \in B_{0,\pi (1+\delta )}^2\);

(c)
for each bounded function \(f\,:\,\mathbb {R}^+\rightarrow \mathbb {C}\) the series (8) converges absolutely and uniformly on compact subsets of \(\mathbb {R}^+\) and
$$\begin{aligned} \sup _{r\in \mathbb {R}^+}\left \left(S_w^\varphi f\right)(r)\right \le \Vert \psi \Vert _{X_0^2} \sup _{r\in \mathbb {R}^+}\left f(r)\right ; \end{aligned}$$ 
(d)
for each bounded function \(f\,:\,\mathbb {R}^+\rightarrow \mathbb {C}\) the series (8) is interpolating with respect to the sample points, that is,
$$\begin{aligned} \left(S_w^\varphi f\right)(e^{\ell /w})= f(e^{\ell /w}) \quad (\ell \in \mathbb {Z}); \end{aligned}$$ 
(e)
for each \(f\in \mathscr {B}_{0,\pi w(1\delta )}^\infty \), the restriction to \(\mathbb {R}^+\) is a fixpoint of \(S_w^\varphi \), that is,
$$\begin{aligned} S_w^\varphi f(\cdot ,0) = f(\cdot , 0). \end{aligned}$$
Proof
Clearly \(\varphi \) is continuous. Since the constant function \(g(r, \theta ) \equiv 1\) on \(\mathbb {H}\) belongs to any of the spaces \(\mathscr {B}^\infty _{0,T},\) for every \(T>0,\) it follows from Theorem 3 applied to the function g that
for all \(u\in \mathbb {R}^+\). As a consequence of Lemma 1 with \(w=1\) and \(\theta =0\), we have
Next, let \(\rho \ge 1\) and denote by \(\lfloor \rho \rfloor \) the largest integer not exceeding \(\rho \). Then, by the Cauchy–Schwarz inequality and Theorem 2,
This shows that the lefthand side converges uniformly to zero as \(\rho \rightarrow \infty .\) Altogether, we see that \(\varphi \in \Phi \).
Next we note that
Since \(\psi \in B_{0, \pi \delta }^2,\) there exists by Theorem 1 a function \(\Psi \in \mathscr {B}^2_{0, \pi \delta }\) such that \(\psi (r) = \Psi (r,0)\) and so the function \(\Theta (r, \theta ) :=\Psi (r,\theta ) \frac{\sin (\pi (\log r + i\theta ))}{\pi }\) belongs to \(\mathscr {B}^2_{0, \pi (\delta +1)}\). Again by Theorem 1, one has that \(\varphi (r) \log r= \Theta (r,0)\) belongs to \(B^2_{0, \pi (\delta +1)}\), and so (b) holds.
Assertion (c) is verified with the help of Lemma 1.
As regards (d), it suffices to note that if \(\ell \in \mathbb {Z}\) and \(r=e^{\ell /w}\), then
with Kronecker’s delta.
Finally assertion (e) is contained in Theorem 3. \(\square \)
It is remarkable that Theorem 5 has a converse. For this we need statements (b) and (d) only. In fact, any generalized exponential sampling series (8) that is interpolating and has a Mellin bandlimited \(\varphi \) satisfying (b) is a multiplier version of the sampling series of Theorem A.
Theorem 6
A generalized exponential sampling series (8) has the properties (b) and (d) of Theorem 5 if and only if there exists a \(\psi \in B_{0,\pi \delta }^2\) such that \(\psi (1)=1\) and \(\varphi =\psi {{\,\textrm{lin}\,}}\).
Proof
The sufficiency is already contained in Theorem 5.
Necessity. Suppose that statements (b) and (d) of Theorem 5 hold. Then, by Theorem 1, there exists a function \(F\in \mathscr {B}_{0,\pi (1+\delta )}^2\) such that \(F(r,0)=\varphi (r) \log r\) for all \(r\in \mathbb {R}^+\). Now consider
The denominator on the righthand side has simple zeros at the points \((e^\ell ,0)\) for \(\ell \in \mathbb {Z}\) and is different from zero at all other points of \(\mathbb {H}\). Hence G is polaranalytic on \(\mathbb {H}\setminus \{(e^\ell ,0)\,:\,\ell \in \mathbb {Z}\}\). Since \({{\,\textrm{lin}\,}}\) is bounded on \(\mathbb {R}^+\), we conclude from statement (d) with \(w=1\) that
Thus
for all \(\ell \in \mathbb {Z}\). Hence the points \((e^\ell ,0)\) for \(\ell \in \mathbb {Z}\) are all removable isolated singularities of G, and so G has a continuation that is polaranalytic on the whole of \(\mathbb {H}\).
It is easily verified that
As a consequence of statement (b), we have
with a constant \(C_F\) depending on F only. Hence
Next, let \(N\in \mathbb {N}\) and let \(\mathcal {R}_N\) be a rectangle with vertices at \((e^{\pm (N+1/2)}, \pm 1)\). On its vertical edges, we have \(\left \sin (\pi (\log r +i\theta ))\right \ge \cosh (\pi \theta )\), and so (12) yields that on these line segments
On the horizontal edges, it follows from (13) that
The bounds (14) and (15) do not depend on N. By the maximum principle for polaranalytic functions (see [9, Theorem 12]), it follows that
Combined with (13), we see that
where
We would like to show that \(G( \cdot , 0)\in X_0^2\) which is a little delicate because of the removable singularities. Therefore, we first show that \(G(\cdot ,1)\in X_0^2\). By the translation invariance of MellinBernstein spaces, we know that \(F(\cdot , 1)\in X_0^2\) and by (4),
From this we conclude with the help of (11) that \(G(\cdot , 1)\in X_0^2\) and
Now, by employing the Cauchy–Riemann equations in polar form and manipulating (16) slightly, we find that \(G_1\,:\,(r,\theta ) \mapsto G(r,\theta +1)\) belongs to \(\mathscr {B}_{0,\pi \delta }^2\). But then, by the translation invariance, G itself belongs to \(\mathscr {B}_{0,\pi \delta }^2\).
Finally, setting \(\psi := G(\cdot , 0)\), we have by the Theorem 1 that \(\psi \in B_{0,\pi \delta }^2\) and
Furthermore \(\psi (1) = \varphi (1) = {{\,\textrm{lin}\,}}(1) =1 \) as a consequence of (10) for \(\ell =0\). This completes the proof. \(\square \)
3.2 Approximation of bounded, continuous functions
The approximation power of \(S_w^{\psi {{\,\textrm{lin}\,}}}f\) can be expressed in terms of the best approximation of f by functions from \(\mathscr {B}_{0,\pi T}^\infty \), where \(T=w(1\delta )\). For this, we introduce
The announced result can now be stated as follows:
Proposition 7
For \(\delta \in ]0,1[\), let \(\psi \in B_{0,\pi \delta }^2\) such that \(\psi (1)=1.\) Suppose that \(f\in C(\mathbb {R}^+)\cap X_0^\infty \). Then for \(w>0\) and \(T=w(1\delta )\), we have
Proof
Given \(\varepsilon >0\), there exists a function \(g_\varepsilon \in \mathscr {B}_{0,\pi T}^\infty \) such that \(\Vert fg_\varepsilon (\cdot ,0)\Vert _{X_0^\infty } \le E_{\pi T}^\infty [f]+\varepsilon .\) By Theorem 3, we have \(g_\varepsilon (r,0)=(S_w^{\psi {{\,\textrm{lin}\,}}}g_\varepsilon (\cdot , 0))(r)\). Now employing statement (c) of Theorem 5, we obtain
The lefthand side does not depend on \(\varepsilon \). Letting \(\varepsilon \rightarrow 0\), we arrive at (17). \(\square \)
Note that \(S_w^{\psi {{\,\textrm{lin}\,}}}f\) comes close to a best approximation by functions from \(\mathscr {B}_{0,\pi T}^\infty \) but misses it in two ways. First \(S_w^{\psi {{\,\textrm{lin}\,}}}f\) is the restriction to \(\mathbb {R}^+\) of a function from \(\mathscr {B}_{0,\pi w(1+\delta )}^\infty \) while \(T=w(1\delta )\). Secondly, there is a factor \(1+\Vert \psi \Vert _{X_0^2}\) on the righthand side of (17), which is at least as big as \(1+\delta ^{1/2}\). Indeed, by Theorem 2, we have
where equality is attained for \(\psi (r)={{\,\textrm{lin}\,}}(r^\delta )\).
3.3 The truncation error
Next we want to estimate the truncation error of the series \((S_w^{\psi {{\,\textrm{lin}\,}}}f)(r)\). It will depend on the decay of \(\left \psi (r)\right \) as \(r\rightarrow 0+\) and \(r\rightarrow +\infty \). Therefore we will classify multipliers \(\psi \) according to their decay.
Definition 5
Let \(\delta \in \, ]0,1[\) and let \(\mu \) be a positive, decreasing function on \([0, +\infty [\) such that \(\mu (t)\rightarrow 0\) as \(t\rightarrow +\infty .\) Then the class \(B_{0,\pi \delta }^2(\mu )\) comprises all functions \(\psi \in B_{0,\pi \delta }^2\) such that \(\psi (1)=1\) and
Example 1
For an integer \(m\ge 2\), let \(\psi (r):={{\,\textrm{lin}\,}}^m(r^{\delta /m})\) and let \(\kappa :=\pi \delta /m\). Then \(\psi \in B_{0,\pi \delta }^2\), \(\psi (1)=1\) and
We also have
The first bound does not extend to \(r=1\), the second is not decreasing, but the harmonic mean of these two bounds leads to
Hence
defines an admissible function \(\mu \) that yields (18).
When \(r\in \mathbb {R}^+\) is given and we want to approximate \((S_w^{\psi {{\,\textrm{lin}\,}}}f)(r)\) by \(2N+1\) terms of this series, it seems reasonable to take those terms whose sample points \(e^{k/w}\) are closest to r. This means that we associate with (w, r) the integer \(N_{w,r}:=\lfloor w\log r + 1/2\rfloor \) and keep only those terms of the series for which
The truncated series obtained in this way shall be denoted by
We also use the notation
The following proposition provides an estimate of the truncation error in terms of \(\mu \) and N.
Proposition 8
Let \(\delta \in \, ]0,1[\), \(\psi \in B_{0,\pi \delta }^2(\mu )\) and \(w>0\). Suppose that f is a function defined on \(\mathbb {R}^+\) such that . Then
for \(N\in \mathbb {N}\) and \(r\in \mathbb {R}^+.\)
Proof
Obviously,
Since \(\log r^w N_{w,r}\in \, ]{\textstyle \frac{1}{2}}, {\textstyle \frac{1}{2}}]\), we find that \(\left \log r^w N_{w,r} j\right \ge \left j\right {\textstyle \frac{1}{2}}\) for \(\left j\right >N\) and so (18) implies
Hence
Finally, noting that
we obtain
This completes the proof. \(\square \)
Remark 2
The bound on the righthand side of (19) can be simplified by replacing it with
However, the sine in (20) shows that the truncated series is still interpolating.
Proposition 8 combined with Theorem 3 implies the following result.
Corollary 1
Let \(\delta \in \,]0,1[\) and \(\psi \in B_{0,\pi \delta }^2(\mu )\). Suppose that \(f\in \mathscr {B}_{0,\pi T}^\infty \), where \(T>0\). Then, for \(w=T/(1\delta )\), we have
for \(N\in \mathbb {N}\) and \( r\in \mathbb {R}^+\).
Now we want to show that Proposition 8 has a generalization that admits unbounded functions f. Roughly speaking, this is achieved by taking a portion of the decaying multiplier \(\psi \) for compensating the growth of f while the remaining portion serves for the decay of the truncation error as \(N\rightarrow \infty \).
Theorem 9
For \(\delta \in \,]0,1[\), let \(\psi \in B_{0,\pi \delta }^2(\mu )\) and let f be a function defined on \(\mathbb {R}^+\). Suppose that
where K is a nondecreasing, positive function defined on \([0, +\infty [\) such that
for some \(\lambda \in [0, 1]\). Then for \(w>1\) the generalized exponential sampling series \((S_w^{\psi {{\,\textrm{lin}\,}}}f)(r)\) converges absolutely for each \(r\in \mathbb {R}^+\) and
for \(N\in \mathbb {N}\).
Proof
It suffices to prove (22). We have
Next we estimate the term
Noting again that \(\xi := w\log r  N_{w,r} \in \,]{\textstyle \frac{1}{2}}, {\textstyle \frac{1}{2}}]\), we find that
We distinguish two cases: If
then
and so
If
then
Therefore
and so
Comparing the estimates (23) and (24), we see that in any case
The proof is completed by employing inequality (20) for the summation over j. \(\square \)
The choice of \(\lambda \) allows us some flexibility. For \(\lambda =0\), we see from condition (21) that only bounded functions f are admissible and then we essentially recover Proposition 8 with the rate of convergence to zero of the truncation error being \(\mathcal {O}(\mu (N)/N)\) as \(N\rightarrow \infty \). Note that inequality (23) remains true if \(\lambda \) is replaced by any \(\lambda '\in \,]\lambda , 1]\). We may even raise both sides of (21) to the power \(\lambda '/\lambda \) and obtain
This means that, if \(K(0)\ge 1\), we can enlarge the class of admissible functions by requiring that
but for the functions added in this way the guaranteed rate of convergence of the trunction error will be \(\mathcal {O}(\mu (N)^{1\lambda '}/N)\) only. When we choose \(\lambda '=1\), we fix a largest class of admissible functions but the speed of convergence in the worst case reduces to \(\mathcal {O}(1/N)\).
As a suitable majorant \(K(\cdot )\) for the admissible functions we can always take \(K(t):= c \mu (t)^{\lambda }\) with a constant \(c>0\), which satisfies (23) trivially.
In view of Proposition 8 and Theorem 9 it is desirable to have a rapidly decaying multiplier \(\psi \). This raises the nontrivial question as to how fast \(\psi \) can decay. Similarly, prescribing \(\mu \) as a rapidly decaying function, we may ask if the class \(B_{0,\pi \delta }^2(\mu )\) specified in Definition 5 will be nonempty? In the case of entire functions of exponential type an answer to the correponding questions can be found in [21, p. 101], which reads as follows: Let \(S(r)\ge 1\) be increasing. A necessary and sufficient condition for there to exist entire functions \(\varphi \not \equiv 0\) of exponential type with
is that
If that condition is met, there are entire functions \(\varphi \not \equiv 0\) of arbitrarily small exponential type satisfying the inequality in question.
We recall that there is a correspondence between Mellin bandlimited functions and entire functions of exponential type. If \(\psi \in B_{0, \pi \delta }^2\) and \(f(x):= \psi (e^x)\), then f is the restriction to \(\mathbb {R}\) of an entire function of exponential type \(\pi \delta \) that belongs to the Lebesgue class \(L^2\) on the real line. The converse is also true. Given f of the latter kind, then \(\psi \) defined by \(\psi (r):= f(\log r)\) belongs to \(B_{0, \pi \delta }^2\). Therefore the cited result implies the following criterion.
Proposition 10
The class \(B_{0, \pi \delta }^2(\mu )\) specified in Definition 4 is nonempty if and only if
This shows that in Proposition 8 and Theorem 9 we can have \(\mathcal {O}(e^{N/(\log N)^\gamma })\) with \(\gamma >1\) for the convergence of the trunction error as \(N\rightarrow \infty \) but it is not possible to have such a rate of convergence with \(\gamma \le 1\). In particular, we cannot have \(\mathcal {O}(e^{\alpha N})\) for some positive \(\alpha \).
4 A multiplier based on a Gaussian function
Unfortunately, multipliers \(\psi \in B_{0,\pi \delta }(\mu )\) with
are not easily available. They can be constructed by an infinite product of \({{\,\textrm{lin}\,}}\) functions (see [18] for corresponding products of \({{\,\textrm{sinc}\,}}\) functions) which is quite inconvenient for computations. But there is a striking observation. When we analyse the proofs of Proposition 8 and Theorem 9, we find that only the estimate (18) is needed but we nowhere use that \(\psi \in B_{0,\pi \delta }^2\). For example, we could take
Then inequality (18) will hold with \(\mu (t)= e^{t^2}\) and (19) will be valid with this function \(\mu \). Thus we obtain an exceptionally fast converging truncation error by using a very convenient multiplier. However, since this multiplier \(\psi \) is not Mellin bandlimited, Theorem 3 and Corollary 1 are not applicable. We produce a socalled aliasing error which may prevent \(S_w^{\psi {{\,\textrm{lin}\,}}}f\) from being a good approximation to f.
The same dilemma arises when one equips the classical sampling formula of Whittaker–Kotel’nikov–Shannon with a multiplier. Nevertheless some scientists experimented with Gaussian multipliers, that is with functions of the form \(t \mapsto c_1e^{c_2t^2}\), where \(c_1\) and \(c_2\) are positive constants, despite the fact that these functions are not bandlimited. It was reported that the Chinese chemist G.W. Wei and his collaborators tackled diverse dynamical problems arising in physics and chemistry by using such multipliers. Inspired by their excellent numerical results, Qian [25] (also see Creamer–Qian [26]) provided a mathematical justification for this approach with rigorous error estimates. Using methods of Fourier analysis, these authors showed that a bandlimited function can be approximated by \(2N+1\) terms of its classical sampling series equipped with a cleverly scaled Gaussian multiplier with an error that decays like
where \(\alpha \) is a positive constant. This is an amazing result since it shows that, by allowing in addition an aliasing error, one can achieve a better approximation by \(2N+1\) samples of f than with the truncated series of an exact formula. This result was considerably extended in [29] by using methods of complex analysis. Multidimensional versions were presented in [2] and in [1]. Micchelli et al. [23] considered the sequence of samples of a bandlimited function f occurring in the classical sampling formula and asked for an optimal algorithm that approximates f(x) by using \(2N+1\) of these samples. They found that such an algorithm converges for the worst case of the function f as fast as (26) with some \(\alpha >0\) but not faster. Hence sampling with a Gaussian multiplier as initiated by Wei and Qian has the rate of convergence of an optimal algorithm and is therefore the method of choice for computations. A very early use of Gaussian multipliers in a somewhat different context can be found in E.T. Whittaker’s construction of “cotabular” functions; see [30].
The distinguished role of Gaussian functions can be made plausible by the following reflection: For the truncation error to converge fast one should have a rapidly decaying multiplier \(\psi \). For the aliasing error to converge fast one should have a multiplier \(\psi \) whose Fourier transform decays rapidly. This leads us naturally to Gaussian functions since they decay rapidly and their Fourier transforms are again Gaussian functions.
In the case of exponential sampling, appropriately scaled functions of the form (25) may be a reasonable choice. We set
with a positive \(\alpha \) to be fixed later, consider the truncated generalized exponential sampling series \(\bigl (S_{w,N}^{\psi {{\,\textrm{lin}\,}}}f\bigr )(r)\) and call it
with G referring to Gauss. The following theorem includes an efficient algorithm for approximating the restriction to \(\mathbb {R}^+\) of a polaranalytic function from its exponential samples.
Theorem 11
Let f be polaranalytic on \(\mathbb {H}\) such that
where K is a nondecreasing, nonnegative function defined on \([0, +\infty [\) and \(T\ge 0\). Then, for \(w>T\), \(\alpha =\pi (1T/w)/2\), \(N\in \mathbb {N}\) and \(r\in \mathbb {R}^+\), we have
where
Proof
First we note that for \(r\in \{e^{k/w} \,:\, k\in \mathbb {Z}\}\) the assertion is trivially true. For \(r\in \mathbb {R}^+\setminus \{e^{k/w} \,:\, k\in \mathbb {Z}\}\), \((\rho , \vartheta )\in \mathbb {H}\) and \(\lambda :=\alpha /N\), consider
This function F is polaranalytic on \(\mathbb {H}\) except for isolated singularities at the points (r, 0) and \((e^{k/w},0)\) with \(k\in \mathbb {Z}\), which are all poles of order 1. We want to show by employing the residue theorem for polaranalytic functions (see [7, Sect. 4]) that the error of the approximation of f(r, 0) by \((G_{w,N}^\alpha f(\cdot ,0))(r)\) can be represented by a contour integral of F. For this, we have to calculate the residues \({{\,\textrm{res}\,}}_0\) of the poles of F. It is easily seen that
For calculating the residues at \((e^{k/w},0)\), we first factor the sine in the denominator of F as follow:
With this decomposition, a basic formula for the residues of poles of order 1 yields
Now write \(N':= N+{\textstyle \frac{1}{2}}\) and denote by \(\mathcal {R}_N\) the rectangle with vertices at
and by \(\partial \mathcal {R}_N\) its positively oriented boundary. By the residue theorem for polaranalytic functions, we have
Next we denote by \(I_{\textrm{hor}}^\pm \) the contributions to the integral coming from the two horizontal parts of \(\partial \mathcal {R}_N\), where \(+\) and − refer to the upper and lower line segment, respectively. Similarly, we denote by \(I_{\textrm{vert}}^\pm \) the contributions to the integral coming from the two vertical parts of \(\partial \mathcal {R}_N\), where \(+\) and − refer to the right and left line segment, respectively. Then
We note that for \((\rho ,\vartheta )\in \partial \mathcal {R}_N\), we have
and
Therefore
Since
and
we conclude that
For the contributions coming from the vertical line segments, we have
Here
and
Substituting \(\theta =w\vartheta \) and noting that
for \(\theta \in [N, N]\), we arrive at
Employing the estimates (28) and (29) in conjunction with (27), we readily complete the proof. \(\square \)
Note that in Theorem 11 the speed of convergence may be lower than \(\mathcal {O}(e^{\alpha N})\) since N also occurs in the argument of the nondecreasing function K. However, when the restriction of f to \(\mathbb {R}^+\) is bounded, such a deterioration is not possible. The following statement is an obvious consequence of Theorem 11.
Corollary 2
Let \(f\in \mathscr {B}_{0,\pi T}^\infty \), where \(T>0\). Then, for \(w>T\), \(\alpha =\frac{\pi }{2}(1\frac{T}{w})\), \(N\in \mathbb {N}\) and \(r\in \mathbb {R}^+\), we have
where \(C_f\) is the constant introduced in Definition 2 for \(c=0\) and \(\beta _N\) is as in Theorem 11.
When we proved the assertion of Theorem 11 for an arbitrarily chosen \(N\in \mathbb {N}\), we did not really need that the function f is polaranalytic on the whole of \(\mathbb {H}\). It will suffice that it is polaranalytic in a strip
with \(d> N/w\). This condition can always be satisfied by choosing \(w= (N+ \varepsilon )/d\) with \(\varepsilon >0\). This leads us to the following statement in which the hypothesis is designed such that \(T=0\) and the function K is a constant \(K_f\) which allows us to let \(\varepsilon \) approach zero on both sides of the error estimate.
Corollary 3
Let f be polaranalytic in a strip \(S_d\) with \(d>0\). Suppose that
Then for \(N\in \mathbb {N}\) and \(r\in \mathbb {R}^+\), we have
where
5 Numerical experiments
As a first example, we consider the function
which belongs to \(\mathscr {B}^\infty _{0,\pi }\). It qualifies for Corollaries 1 and 2. In the case of Corollary 1, we use the multiplier \(\psi \) of Example 1 with \(\delta =1/2\) and \(m=8\). Then the constructed function \(\mu \) becomes
In view of Remark 1, we may choose \(w=16\) in both corollaries. This enables us to compare the two statements by computing approximations of \(f(r_j,0)\) at \(r_j= e^{(j+1/2)/w}\) in both cases. Results for the true absolute errors are shown in the columns of Table 1. For the chosen points \(r_j\) the error bounds of the two corollaries do not depend on j. They are presented in the last line. We see that the approximation by the operator \(G^\alpha _{w,N}\) of Corollary 2 is much better than the approximation by \(S^{\psi {{\,\textrm{lin}\,}}}_{w,N}\) of Corollary 1. Moreover, the error bound of Corollary 2 is closer to the true errors.
Our second example is the function
which is polaranalytic on \(\mathbb {H}\) but unbounded on \(\mathbb {R}^+\). It satisfies the hypotheses of Theorem 11 with \(K(t)=\cosh (t)\), \(T=0\) and \(\alpha =\pi /2\). We choose \(w=16/\pi \) and compute approximations of \(f(r_j,0)\) for \(r_j=e^{(j+1/2)/w}\). The true absolute errors and their bounds are shown in Tables 2 and 3 for four different values of N. The absolute errors grow with increasing \(\left \log r_j\right \) as suggested by the bound of Theorem 11 which contains \(\left \log r\right \) in the argument of the increasing function K. However, the relative errors
and their bounds are nearly constant. Taking into account the contribution of K, the dependence of the error bounds on N is \(\mathcal {O}\bigl (\frac{e^{7\pi N/16}}{\sqrt{N}}\bigr )\) as \(N\rightarrow \infty \).
As a third example, we consider the function
It is polaranalytic and bounded on the strip \(S_1\) with
Therefore it may serve for illustrating Corollary 3. The columns of Table 4 show the index j and the true absolute errors at the points \(r_j=e^{(j+1/2)/N}\) for four choices of N. For these points, the error bounds provided by Corollary 3 do not depend on j. They are presented in the last line. It turns out that they are very realistic. The factor of overestimation of the largest absolute error in a column decreases with increasing N. For \(N=5\) it is 1.53; for \(N=30\) it has decreased to 1.18.
References
Asharabi, R.M., AlHaddad, F.H.: On multidimensional sincGauss sampling formulas for analytic functions. Electron. Trans. Numer. Anal. 55, 242–262 (2022)
Asharabi, R.M., Prestin, J.: On twodimensional classical and Hermite sampling. IMA J. Numer. Anal. 36, 851–871 (2016)
Bardaro, C., Butzer, P.L., Mantellini, I.: The MellinParseval formula and its interconnections with the exponential sampling theorem of optical physics. Integral Transf. Spec. Funct. 27(1), 17–29 (2016)
Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: On the PaleyWiener theorem in the Mellin transform setting. J. Approx. Theory 207, 60–75 (2016)
Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: A fresh approach to the PaleyWiener theorem for Mellin transforms and the MellinHardy spaces. Math. Nachr. 290, 2759–2774 (2017)
Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: Development of a new concept of polar analytic functions useful in Mellin analysis. Complex Var. Elliptic Equ. 64(12), 2040–2062 (2019)
Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: Integration of polaranalytic functions and applications to Boas’ differentiation formula and Bernstein’s inequality in Mellin setting. Boll. Unione Mat. It. 13, 503–514 (2020)
Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: Valiron’s interpolation formula and a derivative sampling formula in the Mellin setting acquired via polaranalytic functions. Comput. Methods Funct. Theory. 20, 629–652 (2020)
Bardaro, C., Butzer, P.L., Mantellini, I.: Schmeisser, G: Polaranalytic functions: Old and new results, applications. Results Math. 77, 64 (2022)
Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G: An Introduction to Mellin Analysis and its Applications, forthcoming book
Bardaro, C., Faina, L., Mantellini, I.: A generalization of the exponential sampling series and its approximation properties. Math. Slovaca 67(6), 1481–1496 (2017)
Bardaro, C., Mantellini, I., Schmeisser, G.: Exponential sampling Series: Convergence in MellinLebesgue Spaces. Results Math. 74, 119 (2019)
Bertero, M., Pike, E.R.: Exponential sampling method for Laplace and other dilationally invariant transforms I. Singularsystem analysis. II. Examples in photon correction spectroscopy and Fraunhofer diffraction. Inverse Problems 7, 1–20; 21–41 (1991)
Butzer, P.L., Jansche, S.: A direct approach to the Mellin transform. J. Fourier Anal. Appl. 3, 325–375 (1997)
Butzer, P.L., Jansche, S.: The exponential sampling theorem of signal analysis. Atti Sem. Mat. Fis. Univ. Modena, Suppl. Vol. 46, 99–122 (1998), special issue dedicated to Professor Calogero Vinti
Butzer, P.L., Jansche, S.: A selfcontained approach to Mellin transform analysis for square integrable functions, applications. Integral Transf. Spec. Funct. 8, 175–198 (1999)
Campbell, L.L.: Sampling theorem for the Fourier transform of a distribution with bounded support. SIAM J. Appl. Math. 16, 626–636 (1968)
Gervais, R, Rahman, Q.I., Schmeisser, G.: A bandlimited function simulating a durationlimited one, in: P.L. Butzer, R.L. Stens and B. Sz.Nagy (eds.) Anniversery Volume on Approximation Theory and Functional Analysis, Vol. 65 of ISNM, Birkhäuser Verlag, Basel, 355–362 (1984)
Helms, D., Thomas, J.B.: Truncation error of samplingtheorem expansions. Proc. IRE 50, 179–184 (1962)
Jagerman, D.: Bounds for truncation error of the sampling expansion. SIAM J. Appl. Math. 14, 714–723 (1966)
Koosis, P.: The Logarithmic Integral, vol. I. Cambridge University Press, Cambridge (1988)
Lee, J.: Approximate interpolation and sampling theory. SIAM J. Appl. Math. 32, 731–744 (1977)
Micchelli, C.A., Xu, Y., Zhang, H.: Optimal learning of bandlimited functions from localized sampling. J. Complex. 25, 85–114 (2009)
Ostrowsky, N., Sornette, D., Parker, P., Pike, E.R.: Exponential sampling method for light scattering polydispersity analysis. Opt. Acta 28, 1059–1070 (1994)
Qian, L.W.: On the regularized WhittakerKotel’nikovShannon sampling formula. Proc. Am. Math. Soc. 131, 1169–1176 (2003)
Qian, L.W., Creamer, D.B.: A modification of the sampling series with a Gaussian multiplier. Sampl. Theory Signal Image Process. 5, 307–325 (2006)
Rahman, Q.I., Schmeisser G.: Reconstruction and approximation of functions from samples, in: G. Meinardus and G. Nürnberger (eds.) Delay Equations, Approximation and Application, Vol. 74 of ISNM, Birkhäuser Verlag, Basel, 213–233 (1985)
Schmeisser, G.: Interconnections between the multiplier methods and the window methods in generalized sampling. Sampl. Theory Signal Image Process. 9, 1–24 (2010)
Schmeisser, G., Stenger, F.: Sinc approximation with a Gaussian multiplier. Sampl. Theory Signal Image Process. 6, 199–221 (2007)
Whittaker, E.T.: On the functions which are represented by the expansion of the interpolationtheory. Proc. Roy. Soc. Edinburgh Sect. A 35, 181–194 (1915)
Acknowledgements
Carlo Bardaro and Ilaria Mantellini have been partially supported by the “Gruppo Nazionale per l’Analisi Matematica e Applicazioni (GNAMPA)” of the “Istituto di Alta Matematica (INDAM)” as well as by the projects “Ricerca di Base 2019 of University of Perugia (title: Misura, Integrazione, Approssimazione e loro Applicazioni)” and “Progetto Fondazione Cassa di Risparmio cod. nr. 2018.0419.021 (title: Metodi e Processi di Intelligenza artificiale per lo sviluppo di una banca di immagini mediche per fini diagnostici (B.I.M.))”.
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no relevant financial or nonfinancial interests to declare.
Additional information
Communicated by Rudolf L. Stens.
Dedicated to Paul L. Butzer, a master and close friend, in high esteem of his great work in approximation and sampling theory.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bardaro, C., Mantellini, I. & Schmeisser, G. Exponential sampling with a multiplier. Sampl. Theory Signal Process. Data Anal. 21, 8 (2023). https://doi.org/10.1007/s43670023000488
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s43670023000488
Keywords
 Polaranalytic functions
 Mellin–Paley–Wiener spaces
 Mellin–Bernstein spaces
 Multipliers
 Exponential sampling