3. Chebyshev Expansions

Abstract

Previous chapter Next chapter Other Titles in Applied Mathematics Numerical Methods for Special Functions3. Chebyshev Expansionspp.51 - 86Chapter DOI:https://doi.org/10.1137/1.9780898717822.ch3PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutExcerpt The best is the cheapest. —Benjamin Franklin 3.1 Introduction In Chapter 2, approximations were considered consisting of expansions around a specific value of the variable (finite or infinite); both convergent and divergent series were described. These are the preferred approaches when values around these points (either in ℝ or ℂ) are needed. In this chapter, approximations in real intervals are considered. The idea is to approximate a function ƒ(x) by a polynomial p(x) that gives a uniform and accurate description in an interval [a, b]. Let us denote by ℙn the set of polynomials of degree at most n and let g be a bounded function defined on [a, b]. Then the uniform norm ‖g‖ on [a, b] is given by ‖g‖ = max x∈ [a,b] ⁡ |g (x) | . 3.1 For approximating a continuous function ƒ on an interval [a, b], it is reasonable to consider that the best option consists in finding the minimax approximation, defined as follows. Definition 3.1. q ∈ ℙn is the best (or minimax) polynomial approximation to ƒ on [a, b] if ‖ƒ−q‖≤ ‖ƒ−p‖ ∀p⁢ ∈ ℙn . 3.2 Minimax polynomial approximations exist and are unique (see [152]) when ƒ is continuous, although they are not easy to compute in general. Instead, it is a more effective approach to consider near-minimax approximations, based on Chebyshev polynomials. Previous chapter Next chapter RelatedDetails Published:2007ISBN:978-0-89871-634-4eISBN:978-0-89871-782-2 https://doi.org/10.1137/1.9780898717822Book Series Name:Other Titles in Applied MathematicsBook Code:OT99Book Pages:xiv + 405Key words:Computation of special functions, Chebyshev expansions, Numerical quadrature, Recurrence relations and continued fractions, Asymptotic analysis, Zeros of special functions