Best $L^ 2$-approximation of convergent moment series

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 sequence-to-sequence transformation and show that the error in this transformation of $(n + 1)$ terms of the series is \[ \frac {1}{{2\delta \sqrt {2\delta - 1} }}\frac {{n + 1}}{{\left ( {\begin {array}{*{20}{c}} {n + 2\delta } \\ n \\ \end {array} } \right )}} \sim \frac {{\Gamma (2\delta )}}{{\sqrt {2\delta - 1} }}\frac {1}{{{n^{2\delta - 1}}}}\quad {\text {as}}\;n \to \infty .\]