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Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
- Publisher
- Springer Journals
- Copyright
- Copyright © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021
- ISSN
- 1019-7168
- eISSN
- 1572-9044
- DOI
- 10.1007/s10444-021-09905-3
- Publisher site
- See Article on Publisher Site
Abstract
We present a new fractional Taylor formula for singular functions whose Caputo fractional derivatives are of bounded variation. It bridges and “interpolates” the usual Taylor formulas with two consecutive integer orders. This enables us to obtain an analogous formula for the Legendre expansion coefficient of this type of singular functions, and further derive the optimal (weighted) L∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$L^{\infty }$\end{document}-estimates and L2-estimates of the Legendre polynomial approximations. This set of results can enrich the existing theory for p and hp methods for singular problems, and answer some open questions posed in some recent literature.
Journal
Advances in Computational Mathematics – Springer Journals
Published: Jan 1, 2021
Keywords: Approximation by Legendre polynomials; Functions with interior and endpoint singularities; Optimal estimates; Fractional Taylor formula; 41A10; 41A25; 41A44; 65N35