Recent contributions to the theory of non-absolutely convergent fourier transforms
Miklós Mikolás · Integral Transforms and Special Functions · 1993
It's an old idea which goes back essentially to Fourier's pioneering works on the suject, to establish a direct connection between Fourier's double integral and the ordinary fourier series of the pertinent function. But this point of view remained a heuristic expedient up to now, and the rigorous foundation of Fourier transforms was built on the whole independently of the convergence theory of Fourier series-though naturally on the analogy of this field. The situation is somewhat similar also about the elegant results of Plancherel concerning square integrable functions and convergence in mean. In th paper [4] written in German, the author has dealed with some exact results of this kind, which will appear in more detailed form in the chapter on Fourier integrals of the monograph [5]. The present paper aims to summing up the most general devlopments found recently about the problem, together with certain applications. (See Theorems 1.-3).