# Jackson’s inequalities in Mellin’s analysis

Jackson’s inequalities in Mellin’s analysis Let c∈R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$c\in {\mathbb {R}}$$\end{document} and Xc2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X_{c}^{2}$$\end{document} be the set of functions f:R+→C\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f: {\mathbb {R}}_{+}\rightarrow {\mathbb {C}}$$\end{document} such that f(·)(·)c-1/2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f(\cdot )(\cdot )^{c-1/2}$$\end{document} is square integrable in the Lebesgue’s sense over R+\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}_{+}$$\end{document}. The Mellin integral transform of f is given by M[f](c+it):=limρ→+∞∫1/ρρuc+it-1f(u)du,t∈R.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}\begin{aligned} {\mathcal {M}}[f](c+it):=\lim _{\rho \rightarrow +\infty }\int _{1/\rho }^{\rho }u^{c+it-1}f(u)du, \;\; t \in {\mathbb {R}}. \end{aligned}\end{document}The focus of this research is to prove analogs of Jackson’s direct and some inverse theorems in terms of best approximations of functions f∈Xc2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f \in X_{c}^{2}$$\end{document} with bounded spectrum and the Mellin moduli of smoothness of all orders constructed by the Mellin Steklov operators. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ANNALI DELL UNIVERSITA DI FERRARA Springer Journals

# Jackson’s inequalities in Mellin’s analysis

, Volume OnlineFirst – Apr 17, 2023
20 pages

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# References (24)

Publisher
Springer Journals
Copyright © The Author(s) under exclusive license to Università degli Studi di Ferrara 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
0430-3202
eISSN
1827-1510
DOI
10.1007/s11565-023-00462-9
Publisher site
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### Abstract

Let c∈R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$c\in {\mathbb {R}}$$\end{document} and Xc2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X_{c}^{2}$$\end{document} be the set of functions f:R+→C\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f: {\mathbb {R}}_{+}\rightarrow {\mathbb {C}}$$\end{document} such that f(·)(·)c-1/2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f(\cdot )(\cdot )^{c-1/2}$$\end{document} is square integrable in the Lebesgue’s sense over R+\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}_{+}$$\end{document}. The Mellin integral transform of f is given by M[f](c+it):=limρ→+∞∫1/ρρuc+it-1f(u)du,t∈R.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}\begin{aligned} {\mathcal {M}}[f](c+it):=\lim _{\rho \rightarrow +\infty }\int _{1/\rho }^{\rho }u^{c+it-1}f(u)du, \;\; t \in {\mathbb {R}}. \end{aligned}\end{document}The focus of this research is to prove analogs of Jackson’s direct and some inverse theorems in terms of best approximations of functions f∈Xc2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f \in X_{c}^{2}$$\end{document} with bounded spectrum and the Mellin moduli of smoothness of all orders constructed by the Mellin Steklov operators.

### Journal

ANNALI DELL UNIVERSITA DI FERRARASpringer Journals

Published: Apr 17, 2023

Keywords: Mellin transforms; Mellin Steklov operator; Jackson’s theorems; Best approximations; 33D15; 33E30; 44A05