A probabilistic proof for Fourier inversion formula

Tak Kwong Wong and Sheung Chi Phillip Yam

Statistics & Probability Letters, 2018, vol. 141, issue C, 135-142

Abstract: The 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.

Keywords: Fourier transform; Gamma distribution; Harmonic analysis; Law of large numbers; Saddle-point approximation; Solid angle (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1016/j.spl.2018.05.028

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