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Mat. Sb., 2012, Volume 203, Number 5, Pages 3–32 (Mi msb7804)  

This article is cited in 2 scientific papers (total in 2 papers)

The modified $\mathbf P$-integral and $\mathbf P$-derivative and their applications

S. S. Volosivets

Saratov State University named after N. G. Chernyshevsky, Faculty of Mathematics and Mechanics

Abstract: The paper is concerned with properties of the modified $\mathbf P$-integral and $\mathbf P$-derivative, which are defined as multipliers with respect to the generalized Walsh-Fourier transform. Criteria for a function to have a representation as the $\mathbf P$-integral or $\mathbf P$-derivative of an $L^p$-function are given, and direct and inverse approximation theorems for $\mathbf P$-differentiable functions are established. A relation between the approximation properties of a function and the behaviour of $\mathbf P$-derivatives of the appropriate approximate identity is obtained. Analogues of Lizorkin and Taibleson's results on embeddings between the domain of definition of the $\mathbf P$-derivative and Hölder-Besov classes are established. Some theorems on embeddings into $\operatorname{BMO}$, Lipschitz and Morrey spaces are proved.
Bibliography: 40 titles.

Keywords: modified $\mathbf P$-integral, modified $\mathbf P$-derivative, multiplicative Fourier transform, direct and inverse approximation theorems, Hölder-Besov spaces.

DOI: https://doi.org/10.4213/sm7804

Full text: PDF file (699 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2012, 203:5, 613–644

Bibliographic databases:

UDC: 517.51
MSC: Primary 26A33, 28A15; Secondary 42C10, 43A15, 43A70, 43A25, 28C05, 35S30, 46F12, 46E35, 41A30
Received: 22.10.2010 and 06.02.2012

Citation: S. S. Volosivets, “The modified $\mathbf P$-integral and $\mathbf P$-derivative and their applications”, Mat. Sb., 203:5 (2012), 3–32; Sb. Math., 203:5 (2012), 613–644

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. S. Platonov, “On spectral synthesis on zero-dimensional Abelian groups”, Sb. Math., 204:9 (2013), 1332–1346  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. S. S. Volosivets, “Modified Bessel ${\mathbf P}$-integrals and $\mathbf P$-derivatives and their properties”, Izv. Math., 78:5 (2014), 877–901  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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