On the analytical summation of Fourier series and its relation to the asymptotic behaviour of Fourier transforms

Abstract

Forms of the Poisson summation formula (PSF) appropriate for the summation of semi-infinite and infinite Fourier series are derived. Application of these results to the acceleration of convergence of various types of series with monotonically decreasing coefficient functions yields transformed series with terms that decay either exponentially or with the inverse first or second power of the index variable. These two very different convergence properties are explained in terms of the asymptotic properties of the relevant Fourier transforms, which are in turn related to the power series expansions of the summand functions in the original Fourier series. The result is that the Poisson summation formula works best for Fourier cosine series in which the summand functions are expansible in even powers, and for Fourier sine series in which the summand functions have power series with odd powers. Here, application of the PSF produces series of terms that decay exponentially with increasing argument x. In contrast, application of the semi-infinite version of the PSF to Fourier cosine series of terms with odd-power expansions, or to Fourier sine series of terms with even-power expansions yields transformed series involving functions of the form , which decay approximately as . If the summand function in the Fourier series has a power series with both even and odd powers, the transformed series involves sine and cosine integral functions, which decay approximately as . Fourier series of these last three types in general require additional acceleration, for example, by application of the Kummer transformation.