The Distributional Finite Fourier Transform
G. K. Warmbrod · SIAM Journal on Applied Mathematics · 1969
The finite Fourier transform for periodic distributions is generalized in such a way that for every distribution f and every open interval $( - a,a)$ there exists a Fourier series which agrees with f on $( - a,a)$, and the inverse Fourier transform of any ultradistribution is the limit of a family of Fourier series as the period tends to infinity. Examples are given to illustrate that in many instances it is practical to calculate the inverse Fourier transform by the above limiting process.