Bernstein-type operators diminish the φ-variation

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.

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“…In the main properties of the φ‐variation are presented, while for further papers about such concept we refer to, e.g., .…”

Section: Preliminariesmentioning

confidence: 99%

Discrete operators of sampling type and approximation in ‐variation

Angeloni

Vinti

2017

Mathematische Nachrichten

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“…The classical definition of Jordan variation [37] was extended in the literature in several directions: one of the first generalizations was the quadratic variation introduced by Wiener [38], extended to the -variation, ≥ 1 [39,40], and later to the concept of -variation. The -variation was first introduced by Young [41] and then extensively studied by Musielak and Orlicz and their school (see, e.g., [22,[42][43][44][45][46][47][48][49]).…”

Section: Journal Of Function Spacesmentioning

confidence: 99%

A Review on Approximation Results for Integral Operators in the Space of Functions of Bounded Variation

Angeloni

Vinti

2016

Journal of Function Spaces

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We present a review on recent approximation results in the space of functions of bounded variation for some classes of integral operators in the multidimensional setting. In particular, we present estimates and convergence in variation results for both convolution and Mellin integral operators with respect to the Tonelli variation. Results with respect to a multidimensional concept ofφ-variation in the sense of Tonelli are also presented.

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“…Further results can be found, for example, in –, , , , . We also recall that other concepts of φ‐variation were given in the literature: among them, for example, the fine φ‐variation (see, e.g., ), the φ‐variation in the sense of Riesz (see, e.g., , , , ) or the φ‐variation in the sense of Schramm , .…”

Section: Definitions and Notationsmentioning

confidence: 99%