The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. II. Piecewise continuous functions and functions with poles near the interval $[0, 1]$

Abstract:

In Part I, the MIPS method for calculating Fourier coefficients was introduced, and applied to functions $f \in {C^{(p)}}[0,1]$. In this part two extensions of the theory are described. One modification extends the theory to piecewise continuous functions, $f \in P{C^{(p)}}[0,1]$. Using these results the method may be used to calculate approximations to trigonometrical integrals (in which the length of the interval need not coincide with a period of the trigonometrical weighting function). The other modification treats functions which are analytic, but whose low-order derivatives vary rapidly due to poles in the complex plane near the interval of integration. Essentially these poles are ’subtracted out’ but this is done implicitly by the inclusion of additional terms in the standard series. The practical application of these modified methods requires that the nature and location of the discontinuities—or poles—be known at least approximately.