Convergence acceleration of modified Fourier series in one or more dimensions

Adcock, Ben

doi:10.1090/S0025-5718-2010-02393-2

Convergence acceleration of modified Fourier series in one or more dimensions
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Math. Comp. 80 (2011), 225-261 Request permission

Abstract:

Modified Fourier series have recently been introduced as an adjustment of classical Fourier series for the approximation of nonperiodic functions defined on $d$-variate cubes. Such approximations offer a number of advantages, including uniform convergence. However, like Fourier series, the rate of convergence is typically slow.

In this paper we extend Eckhoff’s method to the convergence acceleration of multivariate modified Fourier series. By suitable augmentation of the approximation basis we demonstrate how to increase the convergence rate to an arbitrary algebraic order. Moreover, we illustrate how numerical stability of the method can be improved by utilising appropriate auxiliary functions.

In the univariate setting it is known that Eckhoff’s method exhibits an auto-correction phenomenon. We extend this result to the multivariate case. Finally, we demonstrate how a significant reduction in the number of approximation coefficients can be achieved by using a hyperbolic cross index set.

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