The summation of Fourier, Chebyshev, and Legendre series

摘要

In two previous papers by the author, closed expressions for the sums of Fourier, Chebyshev, and Legendre series were obtained in terms of integrals, and these are readily evaluated by numerical quadrature in cases where the coefficients of the series are moments of a function f(u) which is known, or which can readily be obtained from those moments—usually by Mellin- or Laplace-transform inversion. In many cases, however, f(u), although existing, is either unknown or not readily obtainable. This paper shows how the sums of such series may be approximated as accurately as we please in many cases, without explicit knowledge of f(u).