Technical Report CS0778

Authors: A. Sidi
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In this paper we consider the efficient acceleration of the convergence of classical trigonometric Fourier series and their generalizations, such as Fourier-Legendre and Fourier-Bessel series, and many others as well. A common approach to this problem has been through the application of nonlinear sequence transformations to the series in question. Most well known sequence transformations, however, are either ineffective in general or lose their effectiveness near points of singularity of the corresponding limit functions. The recent d-transformation of Levin and Sidi, on the other hand, has been observed to be very effective on a large family of infinite series that includes the generalized Fourier series above when applied in the appropriate manner. In the present work we propose a new approach involving the d-transformation for the efficient summation of generalized Fourier series, by which we produce very accurate and stable approximations in an economical way. In this approach one first extends the given series in a suitable fashion by including the corresponding functions of the second kind, and then applying the d-transformation to the extended series with the few free parameters of the transformation properly adjusted. This new approach is economical in the sense that, for a given required level of accuracy, the number of terms of the series that it uses in the acceleration procedure is about half the number used when applying the d-transformation directly to the given series. In addition, it achieves high accuracy near points of singularity of the limit function. We give a convergence theorem pertaining to the new approach and demonstrate the effectiveness of it with numerical examples. \newpage

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