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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Best $ L\sp 2$-approximation of convergent moment series

Authors: Gerhard Baur and Bruce Shawyer
Journal: Math. Comp. 54 (1990), 661-669
MSC: Primary 40A25; Secondary 41A10, 65B10
MathSciNet review: 1011437
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Abstract: The authors continue the investigation into the problem of finding the best approximation to the sum of a convergent series, $ \sum\nolimits_{n = 0}^\infty {{x^n}{a_n}} $, where $ \{ {a_n}\} $ is a moment sequence.

The case considered is where $ x = 1$. This requires a proper subset of the set of all moment series. Instead of having

$\displaystyle {a_n} = \int_0^1 {\,{t^n}\,d\phi (t)\quad {\text{with}}\quad } \int_0^1 {\,\vert d\phi (t)\vert} = 1,$

we have

$\displaystyle {a_n} = \int_0^1 {{\mkern 1mu} {t^n}{{(1 - t)}^\delta }\,\psi (t)... ...d {\text{with}}\quad } \int_0^1 {{\mkern 1mu} \vert\psi (t){\vert^2}} \,dt = 1.$

With this subset, the authors find the best sequence-to-sequence transformation and show that the error in this transformation of $ (n + 1)$ terms of the series is

$\displaystyle \frac{1}{{2\delta \sqrt {2\delta - 1} }}\frac{{n + 1}}{{\left( {\... ... {2\delta - 1} }}\frac{1}{{{n^{2\delta - 1}}}}\quad {\text{as}}\;n \to \infty .$

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Article copyright: © Copyright 1990 American Mathematical Society