Discrete operators of sampling type and approximation in ‐variation

Abstract

The paper deals with approximation results with respect to the φ‐variation by means of a family of discrete operators for φ‐absolutely continuous functions. In particular, for the considered family of operators and for the error of approximation, we first obtain some estimates which are important in order to prove the main result of convergence in φ‐variation. The problem of the rate of approximation is also studied. The discrete operators that we consider are deeply connected to some problems of linear prediction from samples in the past, and therefore have important applications in several fields, such as, for example, in speech processing. Moreover such family of operators coincides, in a particular case, with the generalized sampling‐type series on a subset of the space of the φ‐absolutely continuous functions: therefore we are able to obtain a result of convergence in variation also for the generalized sampling‐type series. Some examples are also discussed.

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

“…, which follows by simple calculation (or by applying Lemma 4.3 in Coroianu and Gal 32 ). Concluding, all the results in this paper are valid for the max-product Kantorovich sampling operators based on this kernel ( , ) and given by (4).…”

“…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).…”

Approximation by sampling type discrete operators

“…(see [2,11,12,13,14,15,16,25]). In this paper, we mainly inspired from the paper [1], where Angeloni and Vinti had some convergence results using discrete operators. The authors utilized from convergence in '-variation to get some convergence results in that work.…”

Approximation properties of mixed sampling-Kantorovich operators

“…where χ is a kernel, have been widely studied with respect to several notions of convergence, such as pointwise, uniform, L p , modular convergence [23][24][25] and also, recently [5,7], convergence in variation (for other approximation results in BV-spaces see, e.g., [9][10][11][12][13]). The interest of such operators is also due to their deep connections with problems of Signal and Image Processing: indeed they furnish an approximate version of the Shannon Sampling Theorem (see, e.g., [46,47]) that allows to reconstruct not necessarily band-limited signals (images).…”