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Discrete operators of sampling type and approximation in φ ‐variation
Author(s) -
Angeloni Laura,
Vinti Gianluca
Publication year - 2018
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201600508
Subject(s) - mathematics , variation (astronomy) , type (biology) , convergence (economics) , series (stratigraphy) , rate of convergence , sampling (signal processing) , space (punctuation) , computer science , ecology , paleontology , channel (broadcasting) , computer network , physics , filter (signal processing) , astrophysics , economics , computer vision , biology , economic growth , operating system
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.