Asymptotic behavior of Eckhoff’s method for Fourier series convergence acceleration

Abstract

The current paper considers the problem of recovering a function using a limited number of its Fourier coefficients. Specifically, a method based on Bernoulli-like polynomials suggested and developed by Krylov, Lanczos, Gottlieb and Eckhoff is examined. Asymptotic behavior of approximate calculation of the so-called "jumps" is studied and asymptotic L 2 constants of the rate of convergence of the method are computed.

Convergence acceleration of modified Fourier series in one or more dimensions

“…Usually the r th cardinal function p [i] r is specified to be a polynomial of degree 2(r + 1) − i [4,10,22], in which case q [i] r = x 2(r+1)−i and we refer to {p [i] r } as cardinal polynomials. This explains the name "polynomial subtraction".…”

“…The standard construction of Eckhoff's approximation [4,10] uses the cardinal functions p [i] r and valuesĀ [i] r [f ] given by (1.11). Indeed, this is the most simple form to consider for analysis.…”

“…This approach is fairly standard [4,10]. For practical purposes, the least squares systems are solved by singular value decompositions, which can be computed to high accuracy for Cauchy matrices [14].…”

“…Analysis of the rate of convergence was carried out in [4]. The extension to bivariate functions was developed, without analysis, in [26,24,27].…”

“…In standard implementations [4,10,20,22] such a function is constructed from a certain set of polynomials (the possibility of using other functions was suggested in [10]). Although this is the most convenient choice, it leads to poor numerical stability.…”