A probabilistic proof for Fourier inversion formula

Publication date: October 2018Source: Statistics & Probability Letters, Volume 141Author(s): Tak Kwong Wong, Sheung Chi Phillip YamAbstractThe celebrated Fourier inversion formula provides a useful way to re-construct a regular enough, e.g. square-integrable, function via its own Fourier transform. In this article, we give the first probabilistic proof of this classical theorem, even for Euclidean spaces of arbitrary dimension. Particularly, our proof motivates why the one-half weight, for the one-dimensional case in Lemma 1, comes naturally to play due to the inherent spatial symmetry; another similar interpretation can be found in the higher dimensional analogue.

Source: Statistics and Probability Letters - July 10, 2018 Category: Statistics Source Type: research