Hermite interpolation theorems for band-limited functions of the linear canonical transforms with equidistant samples

We establish convergence analysis for Hermite-type interpolations for L2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2} ( \mathbb {R})$$\end{document}-entire functions of exponential type whose linear canonical transforms (LCT) are compactly supported. The results bridges the theoretical gap in implementing the derivative sampling theorems for band-limited signals in the LCT domain. Both complex analysis and real analysis techniques are established to derive the convergence analysis. The truncation error is also investigated and rigorous estimates for it are given. Nevertheless, the convergence rate is O(1/N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(1/\sqrt{N})$$\end{document}, which is slow. Consequently the work on regularization techniques is required.

the definition of the LCT, cf. [9,18,21,25], is a four-parameter transform where a, b, c, d are real numbers satisfying ad − cb = 1. We exclude the case b = 0 in our study as it is trivial. When For more details see [26,33,35,36]. The sampling theorem expansions for LCT and FRFT have been derived in many works, cf., e.g., [5,18,19,27,[30][31][32]34]. Let σ > 0 be fixed and let B 2 σ be the class of bandlimited signals in the LCT domain with band-width σ , i.e., [18], If PW 2 σ denotes the classical Paley-Wiener space of band-limited functions with bandwidth σ , cf. [11], then f ∈ B 2 σ if and only if there is f o ∈ PW 2 σ/b , b > 0, such that f (z) = e −i(a/2b)z 2 f o (z). The interpolation sampling theorem associated with the LCT states that, see, e.g., [18,23,24,30,32,34], if f ∈ B 2 σ then where z n := nπ b σ , n ∈ Z, are equidistant samples and Series (1.2) converges absolutely on C and uniformly on compact subsets of C and on R as well. The error analysis for truncation, amplitude and jitter errors associated with the sampling representation (1.2) are considered in [4,29,32]. If we take (a, b, d) = (0, 1, 0) in (1.2), series (1.2) turns to be the classical sampling theorem of Whittaker, Kotelnikov and Shannon (WKS), cf. [6,11]. If f ∈ PW 2 2σ , then f can reconstructed via the following Hermite interpolation sampling theorem Expansion (1.5) is also called the derivative sampling theorem, see [11]. Series (1.5) converges locally uniformly on C. The Hermite sampling representation was first established by Jagerman and Fogel (1956) in [14]. Li and Fang (2006) discussed the aliasing error associated with (1.5) in [17]. In [3], truncation, amplitude and jitter errors associated with (1.5) are investigated. The Hermite sampling theorem or derivative sampling theorem associated with the LCT is given in [18], see also [19], but nothing is said about the types of convergence of this theorem. Therefore the main outcomes of this paper are itemized as follows.
• The convergence analysis for the Hermite sampling theorem associated with the LCT is established in Section 2. • We employ complex analysis techniques, which allow us to derive new important cases in different spaces of entire functions with prescribed growth. • The truncation error is estimated in both local (pointwise) and global (uniform) manners in Section 3. Moreover, in Section 4 we estimate the error when the truncation error is accompanied with other types of errors.
The experimental results are presented to show the accuracy and usefulness of derived results in Section 5.

LCT-Hermite interpolations
Let f ∈ B 2 2σ . Define the Hermite interpolation series associated with the LCT, to be: z ∈ C, z n = nπ b σ , n ∈ Z, and a, b ∈ R, b > 0 are arbitrary. We give sufficient conditions for which the H[ f ](z) converges to f . We will also derive estimates for the remainder where In the following D N denotes the closed square We also denote by C N = ∂D N to the boundary of D N . All contours are taken to be positively oriented. We denote by D o N to the interior of D N . The following lemma gives a Cauchy-type representation for the remainder R N ( f (·)).
As a function of ζ , K z (ζ ) is a meromorphic function of ζ with a simple pole at ζ = z and with double poles at ζ = z n , n ∈ Z. By the residue theorem, we obtain where Res(K z ; w) denotes the residue of K z at w. Obviously (2.9) Substituting from (2.8) and (2.9) into (2.7), we obtain (2.5).
Before proving the main convergence results of this section, we introduce the following important estimate.
Denote the integrals in (2.5) coming from the two horizontal legs of C N by I ± H , where + and − refer to the upper and the lower line segments, respectively, and the integrals coming from the two vertical legs of C N by I ± V , where + and − refer to the right and the left line segments, respectively. Then and direct substitutions leads immediately to (2.15) and (2.16) Inequality (2.10), implies Also similar to (2.13), we have which approaches zero as N → ∞ without depending on z. Now, we estimate I ± V . Inequality (2.10), implies Using (2.23), (2.24), and (2.13) leads us to the estimate and, for z = ± N + 1 which tends to zero as N → ∞ without depending on z. From (2.5) we conclude that R N [ f ](·) ⇒ 0 uniformly on Ω and the proof is complete.
As we have seen above the order of convergence of This order is very slow to establish uniform convergence on R. However if f (·) satisfies a faster decay than that of B 2 2σ -functions, we can prove uniform convergence on R as in the next theorem. The proof is established using real analysis techniques.
, for some k ∈ Z + , then the series (2.1) converges absolutely and uniformly on R to f .

Truncation error estimates
In this section, we will investigate the truncation error estimates associated with the Hermite sampling series (2.1). For N ∈ Z + , z ∈ R, the truncated series of (2.1) is and the associated truncation error is
In the following we establish a global error estimate for the truncation error on R.

Combining various types of errors
Several types of errors arise when (2.1) is used other than the truncation error, among which are amplitude and jitter errors. The amplitude error associated with (2.1) arises when the exact samples f (z n ) , f (z n ) , n ∈ Z are replaced by approximate closer ones f (z n ) , f (z n ) in the sampling series (2.1). Let ε n := f (z n ) − f (z n ), ε n := f (z n ) − f (z n ) be uniformly bounded by ε, i.e., |ε n | , ε n < ε for a sufficiently small ε > 0. The amplitude error is defined for z ∈ R by If f ∈ B 2 2σ satisfies the condition (3.14) and then, for 0 < ε ≤ min{π b/σ, σ/πb, 1/ √ e}, (4.1) is estimated in [1] by where M 1 , M 2 , ρ(x) and (x) are defined in (3.16) and (3.18). The jitter error associated with (2.1) arises when the nodes z n , n ∈ Z of (2.1) are deviated from the exact nodes. Let δ n , δ n , n ∈ Z denote the set of deviation values, such that |δ n |, |δ n | ≤ δ for a sufficiently small δ > 0. For z ∈ R, the jitter error associated with (2.1) is defined by Let f ∈ B 2 2σ satisfy the condition (3.14). For 0 and (x) are defined in (3.16) and (3.18).
Since only a finite number of samples are available in particular applications, the amplitude and jitter errors are the truncated ones and (4.8) where

Numerical examples
In this section, we introduce several numerical examples and comparisons between Hermite sampling approximations with LCT which introduced in this paper and the classical sampling with LCT. We also give tables illustrating the error for some numerical values. Furthermore, the results are depicted in various figures to demonstrate the accuracy of the approximations. All of the results in the following examples confirm the correctness and accuracy of the conclusions we reached in this paper. We have used Mathematica to derive these examples. It is convenient to introduce the following notations. Let f ∈ B 2 σ ⊆ B 2 2σ . For z ∈ R, N ∈ Z + , we denote by f C N (z) and f H N (z) to the truncated expansions: This function has both Lagrange-type and Hermite-type sampling expansions. Table 1 exhibits comparison between f C N (z) and f H N (z) at some points z i ∈ R, when N = 20, 30, 50. We apply Theorem 2.1 of [4] and (3.3) to compute the error bounds Table 1 indicates, the precision improves when N increases for both techniques. However, the Hermite interpolation approximations are better than the classical sampling representation of f (z) in terms of the absolute errors. In addition, the error bound for the Hermite interpolations is also better than the error bound for the Shannon-type interpolations. The real and imaginary parts of the complex signal f (z) and reconstructed signals f C N (z) and f H N (z) are plotted in Figs. 1 and 2 when |z| ≤ 2 ⊂ C, N = 15. Here a = b = d = 1/ √ 2.

Conclusions
In this paper we investigated the convergence and the error analysis of the Hermite sampling theorem associated with the LCT. We first established a rigorous convergence analysis on C and R. Both harmonic analysis and complex analysis approaches are utilized. The estimation of the remainder is rigorously derived as well, in both pointwise and uniform settings. It is worthwhile to emphasis that the convergence rate is as slow as O(1/ √ N ) as N → ∞ for band-limited functions. However, as in the classical case it will be improved if the band-limited function satisfies better smoothness conditions. Therefore there is a theoretical demand to derive regularized techniques to fasten the convergence rate. This can be established by incorporating the sampling representation with smoothing convergence kernels. The paper closes a gap in the convergence analysis associated with Hermite sampling representation for entire functions of exponential type.
Funding Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).
Data availability Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Declarations
Competing interests The authors have no relevant financial or non-financial interests to disclose.
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