Variation diminishing-type properties for multivariate sampling Kantorovich operators
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Variation diminishing-type properties for multivariate sampling Kantorovich operators Laura Angeloni1 · Danilo Costarelli1 Luca Zampogni1
· Marco Seracini1 · Gianluca Vinti1 ·
Received: 25 June 2020 / Accepted: 13 August 2020 © The Author(s) 2020
Abstract In this paper we establish a variation-diminishing type estimate for the multivariate Kantorovich sampling operators with respect to the concept of multidimensional variation introduced by Tonelli. A sharper estimate can be achieved when step functions with compact support (digital images) are considered. Several examples of kernels have been presented. Keywords Multivariate generalized sampling Kantorovich series · Variation-diminishing type property · Averaged type kernel · Smoothing in digital image processing · Product kernel Mathematics Subject Classification 41A30 · 41A05
1 Introduction The concept of variation and also of multidimensional variation of the type introduced by Tonelli, was implicitly one of the first topics addressed by Prof. Domenico Candeloro (Mimmo, for friends). He did so in his first two papers [20,21] where he studied the Geöcze area as a Burkill–Cesari integral and the link between the absolute continuity and the Burkill– Cesari integral, respectively. Subsequently, in the last years he has worked for a long time on the variational integrals in the sense of Henstock–Kurzweil; in particular, in his papers
To Mimmo, the Mentor, the Scientist and the Man, with deep esteem, gratitude, friendship and affection The authors are partially supported by the “Department of Mathematics and Computer Science” of the University of Perugia (Italy) and are members of the “Research Italian Network on Approximation”. The first, the fourth and and the fifth author have been partially supported within the projects “Metodi di Teoria dell’Approssimazione, Analisi Reale, Analisi Nonlineare e loro applicazioni” and “Integrazione, Approssimazione, Analisi Nonlineare e loro Applicazioni”, funded by the 2018 and 2019 basic research fund of the University of Perugia. Finally, the first and the second author of the paper have been partially supported within the 2020 GNAMPA-INdAM Project: “Metodi di analisi reale e di teoria della misura per l’approssimazione attraverso operatori discreti e applicazioni”.
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Danilo Costarelli [email protected]
Extended author information available on the last page of the article
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[22,23], he managed to obtain, together with his co-authors, the existence of integrable selections in the variational sense, which was an open problem in Measure Theory for several years. It therefore seemed natural to us to dedicate the results of this paper to Mimmo, considering that these issues represented part of his scientific training in the early years and beyond. Each of us, in a different form, owes much to Mimmo; but surely together, we can express our gratitude for what he has left us, both from a scientific and a human point of view, and this contribution wants to be one of the many ways to tell him ... T
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