Best $L^ 2$approximation of convergent moment series
HTML articles powered by AMS MathViewer
 by Gerhard Baur and Bruce Shawyer PDF
 Math. Comp. 54 (1990), 661669 Request permission
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 \[ {a_n} = \int _0^1 { {t^n} d\phi (t)\quad {\text {with}}\quad } \int _0^1 { d\phi (t)} = 1,\] 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 sequencetosequence transformation and show that the error in this transformation of $(n + 1)$ terms of the series is \[ \frac {1}{{2\delta \sqrt {2\delta  1} }}\frac {{n + 1}}{{\left ( {\begin {array}{*{20}{c}} {n + 2\delta } \\ n \\ \end {array} } \right )}} \sim \frac {{\Gamma (2\delta )}}{{\sqrt {2\delta  1} }}\frac {1}{{{n^{2\delta  1}}}}\quad {\text {as}}\;n \to \infty .\]References

N. I. Achieser, Theroy of approximation, Ungar, New York, 1956.
 Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. III, McGrawHill Book Co., Inc., New YorkTorontoLondon, 1955. Based, in part, on notes left by Harry Bateman. MR 0066496
 W. B. Jurkat and B. L. R. Shawyer, Best approximations of alternating series, J. Approx. Theory 34 (1982), no. 4, 397–422. MR 656640, DOI 10.1016/00219045(82)90082X
 W. B. Jurkat and B. L. R. Shawyer, Best approximations of alternating series, J. Approx. Theory 34 (1982), no. 4, 397–422. MR 656640, DOI 10.1016/00219045(82)90082X
 B. L. R. Shawyer, Best approximation of alternating power series, J. Math. Anal. Appl. 114 (1986), no. 2, 360–375. MR 833592, DOI 10.1016/0022247X(86)900892
 B. L. R. Shawyer, Best approximation of positive power series, Math. Comp. 43 (1984), no. 168, 529–534. MR 758199, DOI 10.1090/S00255718198407581992
 Jet Wimp, New methods for accelerating the convergence of sequences arising in Laplace transform theory, SIAM J. Numer. Anal. 14 (1977), no. 2, 194–204. MR 433798, DOI 10.1137/0714013
Additional Information
 © Copyright 1990 American Mathematical Society
 Journal: Math. Comp. 54 (1990), 661669
 MSC: Primary 40A25; Secondary 41A10, 65B10
 DOI: https://doi.org/10.1090/S00255718199010114376
 MathSciNet review: 1011437