A generalization of the exponential sampling series and its approximation properties

Abstract

Here we introduce a generalization of the exponential sampling series of optical physics and establish pointwise and uniform convergence theorem, also in a quantitative form. Moreover we compare the error of approximation for Mellin band-limited functions using both classical and generalized exponential sampling series.

Approximation by truncated max‐product operators of Kantorovich‐type based on generalized (ϕ,ψ)‐kernels

“…It remains to check condition (iv). Firstly, by similar reasonings to those used for the proofs of relations (14) and (15), for all x ∈ [jb∕n, (j + 1)b∕n] we get |nx − kb| ≥ |nx − b| and |nx − kb| ≥ |nx − ( + 1)b|, for all k = 0, · · ·, n. Now, denote F(u) = u e u −1 , u ≥ 0. If we prove that F is nonincreasing on [0, + ∞), then we immediately get that condition (iv) in Definition 1.1 is satisfied.…”

“…The sinc-approximation operators were first introduced and studied in Borel 1 and Plana 2 and Whittaker 3 under the name of cardinal function and of truncated cardinal function. Later on, the properties of these linear approximation operators and their applications in signal theory were intensively studied in, eg, other studies [4][5][6][7][8][9][10][11]12,[13][14][15][16][17][18][19][20][21] (see also the references cited therein).…”

Bivariate Generalized Exponential Sampling Series and Applications to Seismic Waves

“…In [7] the one-dimensional generalized exponential sampling series was introduced, in which the generating (one-dimensional) kernel φ : R + → R satisfies the assumptions: We will denote by Ψ the set comprising all functions φ : R + → R satisfying (φ.1), (φ.2) and (φ.3). Using a product of two such functions, we can construct examples of two-dimensional kernel ϕ ∈ Φ.…”

“…Example 6.1. Denoting by r + the positive part of a number r ∈ R, for n ∈ N, we define the (one-dimensional) Mellin spline of order n, as (see [7,10] (6.8)…”

“…Now, as it happens for the Fourier band-limited functions, the assumption that f is Mellin band-limited is very restrictive, due to the Mellin-Paley-Wiener theorem (see [4], [5]). Therefore, in a recent paper [7] it was studied a generalization of the one dimensional exponential sampling theorem where the lin c function is replaced by an arbitrary function ϕ satisfying suitable assumptions in an analogous way as for the generalized sampling series of Fourier analysis (see [15,16,6,21,2,1]). In this way, we obtained an approximate reconstruction of a not necessarily Mellin band-limited function f.…”