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Saturation by the Fourier transform method for the sampling Kantorovich series based on bandlimited kernels

  • Danilo CostarelliEmail author
  • Gianluca Vinti
Article

Abstract

In the present paper, we study the saturation order in the space \(L^1({\mathbb {R}})\) for the sampling Kantorovich series based upon bandlimited kernels. The above study is based on the so-called Fourier transform method, introduced in 1960 by P. L. Butzer. As a first result, the saturation order is derived in a Bernstein class; here, it is crucial to derive the Fourier transform of the above sampling-type series, which can be expressed in a suitable closed form. Subsequently, the saturation is reached in the whole space \(L^1({\mathbb {R}})\). At the end of the paper, several examples of bandlimited kernels, such as the Fejér’s and Bochner–Riesz’s kernel, have been recalled and the saturation order of the corresponding sampling Kantorovich series has been stated.

Keywords

Sampling Kantorovich series Fourier transform method Bandlimited kernel Generalized sampling operator Singular integral Saturation theorem 

Mathematics Subject Classification

42A38 42A68 42A85 41A25 41A05 47A58 

Notes

Acknowledgements

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The first author has been partially supported within the 2019 GNAMPA-INdAM Project “Metodi di analisi reale per l’approssimazione attraverso operatori discreti e applicazioni”, while the second author within the projects: (1) Ricerca di Base 2017 dell’Università degli Studi di Perugia - “Metodi di teoria degli operatori e di Analisi Reale per problemi di approssimazione ed applicazioni” , (2) Ricerca di Base 2018 dell’Università degli Studi di Perugia - “Metodi di Teoria dell’Approssimazione, Analisi Reale, Analisi Nonlineare e loro Applicazioni” , (3) “Metodi e processi innovativi per lo sviluppo di una banca di immagini mediche per fini diagnostici” funded by the Fondazione Cassa di Risparmio di Perugia, 2018. This research has been accomplished within RITA (Research ITalian network on Approximation).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interests.

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Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of PerugiaPerugiaItaly

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