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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Best $L^ 2$-approximation of convergent moment series
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by Gerhard Baur and Bruce Shawyer PDF
Math. Comp. 54 (1990), 661-669 Request permission


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 \[ {a_n} = \int _0^1 { {t^n} d\phi (t)\quad {\text {with}}\quad } \int _0^1 { |d\phi (t)|} = 1,\] we have \[ {a_n} = \int _0^1 {{\mkern 1mu} {t^n}{{(1 - t)}^\delta } \psi (t) dt\quad {\text {with}}\quad } \int _0^1 {{\mkern 1mu} |\psi (t){|^2}} dt = 1.\] 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 \[ \frac {1}{{2\delta \sqrt {2\delta - 1} }}\frac {{n + 1}}{{\left ( {\begin {array}{*{20}{c}} {n + 2\delta } \\ n \\ \end {array} } \right )}} \sim \frac {{\Gamma (2\delta )}}{{\sqrt {2\delta - 1} }}\frac {1}{{{n^{2\delta - 1}}}}\quad {\text {as}}\;n \to \infty .\]
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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Math. Comp. 54 (1990), 661-669
  • MSC: Primary 40A25; Secondary 41A10, 65B10
  • DOI:
  • MathSciNet review: 1011437