Embedding Schramm spaces into Chanturiya classes

被引：0
作者：
Milad Moazami Goodarzi
机构：
[1] Shiraz University,Department of Mathematics, Faculty of Sciences
来源：
Banach Journal of Mathematical Analysis | 2021年 / 15卷
关键词：
Fourier series; Uniform convergence; Fourier coefficients; Generalized bounded variation; Embedding; 42A20; 42A16; 46E35; 46E30; 26A45;
D O I：
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摘要：
The main theorem of this paper establishes a necessary and sufficient condition for embedding Schramm spaces ΦBV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi BV$$\end{document} into Chanturiya classes V[ν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V[\nu ]$$\end{document}. This result is new even for the classical spaces in the theory of Fourier series, namely, for the Wiener and the Salem classes. Furthermore, it provides a characterization of the embedding of Waterman classes ΛBV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda BV$$\end{document} into V[ν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V[\nu ]$$\end{document}. As a by-product of the main result, we establish a convergence criterion for the Fourier series of functions of ΦBV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi BV$$\end{document}; this is an extension of a well-known result due to Salem. An estimate on the magnitude of the Fourier coefficients in the space ΦBV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi BV$$\end{document} is also given, and finally it is shown that some of these results can be extended to a more general setting.
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