Spectacularly large coefficients in Müntz's theorem

Journal article

Abstract:: Müntz’s theorem asserts, for example, that the linear span of the even powers 1,x2,x4,… is dense in C([0,1]). We show that the associated expansions are so inefficient as to have no conceivable relevance to any actual computation. For example, approximating f(x)=x to accuracy ε=10−6 in this basis requires powers larger than x280,000 and coefficients larger than 10107,000. We present a theorem establishing exponential growth of coefficients with respect to 1/ε .

Files:: Trefethen_2023_Spectacularly_large_coefficientsV.pdf

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Language:: English
Keywords:: expansion coefficients

FFR

Müntz approximation theorem

polynomial approximation
Pubs id:: 1311475
Local pid:: pubs:1311475
Deposit date:: 2022-12-05

Copyright holder:: Lloyd N. Trefethen
Rights statement:: ©2023 The Author. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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