Asymptotic expansions of Legendre series coefficients for functions with interior and endpoint singularities

Author:
Avram Sidi

Journal:
Math. Comp. **80** (2011), 1663-1684

MSC (2000):
Primary 40A05, 40A10, 41A58, 41A60, 42C10

DOI:
https://doi.org/10.1090/S0025-5718-2010-02454-8

Published electronically:
December 30, 2010

MathSciNet review:
2785473

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the Legendre expansion of a function on . In an earlier work [A. Sidi, *Asymptot. Anal.*, 65 (2009), pp. 175-190], we derived asymptotic expansions as for , assuming that , but may have arbitrary algebraic-logarithmic singularities at one or both endpoints . In the present work, we extend this study to functions that are infinitely differentiable on , except at finitely many points in and possibly at one or both of the endpoints and , where they may have arbitrary algebraic singularities, including finite jump discontinuities. Specifically, we assume that, for each , has asymptotic expansions of the form

where , , and , , , and are constants independent of . In the course of this study, we also derive a full asymptotic expansion as for integrals of the form where and or but may have arbitrary algebraic singularities at and/or .

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Additional Information

**Avram Sidi**

Affiliation:
Computer Science Department, Technion, Israel Institute of Technology, Haifa 32000, Israel

Email:
asidi@cs.technion.ac.il

DOI:
https://doi.org/10.1090/S0025-5718-2010-02454-8

Keywords:
Legendre polynomials,
Legendre series,
interior singularities,
endpoint singularities,
asymptotic expansions.

Received by editor(s):
March 17, 2010

Received by editor(s) in revised form:
May 25, 2010

Published electronically:
December 30, 2010

Additional Notes:
This research was supported in part by the United States–Israel Binational Science Foundation grant no. 2008399.

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.