Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation

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 coeff...

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Bibliographic Details
Main Authors: Liu, Wenjie, Wang, Li-Lian, Wu, Boying
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2022
Subjects:
Online Access:https://hdl.handle.net/10356/160560
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Institution: Nanyang Technological University
Language: English
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Summary: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∞-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.