The modified \({\mathbf P}\)-integral and \({\mathbf P}\)-derivative and their applications
type: Item
instance of: scholarly article
zbMATH Open document ID: 1261.43002
published in: Sbornik: Mathematics
publication date: 2012-09-19T00:00:00Z
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 28A15
Mathematics Subject Classification ID: 43A70
Mathematics Subject Classification ID: 43A25
Mathematics Subject Classification ID: 28C05
Mathematics Subject Classification ID: 35S30
Mathematics Subject Classification ID: 46F12
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 6084148
zbMATH Keywords: direct product of finite cyclic groups
zbMATH Keywords: Fourier transform
zbMATH Keywords: \({\mathbf P}\)-derivative
zbMATH Keywords: \({\mathbf P}\)-integral
zbMATH Keywords: approximation theorems
author: S. S. Volosivets
MaRDI profile type: Publication
full work available at URL: sm2012v203n05abeh004237
OpenAlex ID: W1999564819
DOI: 10.1070/SM2012V203N05ABEH004237
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title: The modified \({\mathbf P}\)-integral and \({\mathbf P}\)-derivative and their applications
review text: Let \({\mathbf P}:= (p_j)_{j\in\mathbb{Z}}\) be a symmetric sequence of integers with \(p_0:= 0\) and \(P\setminus\{p_0\}\subset [2,N]\). Let \(G_{\mathbf P}\) be a subring of \(\bigoplus_j \mathbb{Z}_{p_j}\) consisting of series \((x_j)\) with \(\operatorname{card}\{x_j; j< 0\}<\infty\). Set \(m_j:= p_1\dotsm p_j\) for \(j> 0\) and \(m_j:= {1\over m_{-j+1}}\) for \(j<0\) (with \(m_0:= 1\)) and define a map \(\varphi: G_{\mathbf P}\to\mathbb{R}_+\) given by NEWLINE\[NEWLINE\varphi(x):= \sum_j {x_j\over m_j}.NEWLINE\]NEWLINE This is a bijection up to a countable subset of \(G_p\) which allows to equip \(\mathbb{R}_+\) with a ring structure and define a family \(\chi_y: G_{\mathbf P}\to \mathbb{C}\), \(y\in\mathbb{R}_+\), of characters by NEWLINE\[NEWLINE\chi_y(x):= \exp\left(2\pi i\sum_{j> 0} {x_j\varphi_{-j}+ x_{-j} \varphi_j\over p_j}.\right)NEWLINE\]NEWLINE In turn, the family defines a Walsh-Fourier transform given for \(f\in L_1(\mathbb{R}_+)\) by NEWLINE\[NEWLINE\widehat f(x):= \int_{\mathbb{R}_+} f(y)\overline{\chi_y(x)}\,dyNEWLINE\]NEWLINE and then the authors define (\({\mathbf P}\)-) derivative and integral using the corresponding multipliers.NEWLINENEWLINE Using these one can define analogs of Lipschitz, Sobolev and BMO spaces on \(G_p\). They contain the following results. ``Criteria for a function to have a representation as the \({\mathbf P}\)-integral or \({\mathbf P}\)-derivate 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 definitions of the \({\mathbf P}\)-derivative and Hölder-Besov classes are established. Some theorems on embedding into BMO, Lipschitz and Morrey spaces are proved.'' (From the author's abstract)
reviewed by: Yuri A. Brudnyi
description: scientific article; zbMATH DE number 6084148