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A general method of approximation. I


Authors: Staffan Wrigge and Arne Fransén
Journal: Math. Comp. 38 (1982), 567-588
MSC: Primary 41A50; Secondary 15A57, 65D15
MathSciNet review: 645672
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Abstract: In this paper we study two families of functions, viz. F and H, and show how to approximate the functions considered in the interval [0,1 ]. The functions are assumed to be real when the argument is real.

We define

$\displaystyle F = \{ f;({\text{i}})\,f\left( {\frac{1}{2} + x} \right) = f\left... ...\;f(x)\;{\text{is analytic in a sufficiently large neighborhood of}}\;x = 0\}, $

$\displaystyle H = \{ h;({\text{j}})\;h\left( {\frac{1}{2} + x} \right) = - h\le... ...\;h(x)\;{\text{is analytic in a sufficiently large neighborhood of}}\;x = 0\}. $

The approximations are defined in the interval [0,1 ] by

$\displaystyle \min \int_0^1 {{{\left( {f(x) - \sum\limits_{n = 1}^k {{c_{n,k}}{{[x(1 - x)]}^n}} } \right)}^2}{x^q}{{(1 - x)}^q}\;dx} $

and

$\displaystyle \min \int_0^1 {{{\left( {h(x) - (1 - 2x)\sum\limits_{n = 1}^k {{c_{n,k}}{{[x(1 - x)]}^n}} } \right)}^2}{x^q}{{(1 - x)}^q}\;dx} ,$

where $ q \in \{ 0,1,2, \ldots \} $.

The associated matrices are analyzed using the theory of orthogonal polynomials, especially the Jacobi polynomials $ {G_n}(p,q,x)$. We apply the general theory to the basic trigonometric functions $ \sin (x)$ and $ \cos (x)$.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1982-0645672-9
Keywords: Approximation theory, inverse matrices, Jacobi polynomials
Article copyright: © Copyright 1982 American Mathematical Society