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A note on uniform approximation of functions having a double pole

Part of: Other special functions Approximations and expansions Computational aspects and applications Numerical approximation and computational geometry Algorithms - Computer Science

Published online by Cambridge University Press:  01 May 2014

Ionela Moale  and
Veronika Pillwein
Ionela Moale
Affiliation:
Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz (JKU), Altenbergerstr. 69, 4040 Linz, Austria email [email protected]
Veronika Pillwein
Affiliation:
Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz (JKU), Altenbergerstr. 69, 4040 Linz, Austria email [email protected]

Abstract

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We consider the classical problem of finding the best uniform approximation by polynomials of $1/(x-a)^2,$ where $a>1$ is given, on the interval $[-\! 1,1]$. First, using symbolic computation tools we derive the explicit expressions of the polynomials of best approximation of low degrees and then give a parametric solution of the problem in terms of elliptic functions. Symbolic computation is invoked then once more to derive a recurrence relation for the coefficients of the polynomials of best uniform approximation based on a Pell-type equation satisfied by the solutions.

MSC classification

Secondary: 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 33E05: Elliptic functions and integrals 41A10: Approximation by polynomials 65D20: Computation of special functions, construction of tables 68W30: Symbolic computation and algebraic computation
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Issue 1 , 2014 , pp. 233 - 244
Copyright
© The Author(s) 2014 

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