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A probabilistic proof for Fourier inversion formula

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  • Wong, Tak Kwong
  • Yam, Sheung Chi Phillip

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.

Suggested Citation

  • Wong, Tak Kwong & Yam, Sheung Chi Phillip, 2018. "A probabilistic proof for Fourier inversion formula," Statistics & Probability Letters, Elsevier, vol. 141(C), pages 135-142.
    • Handle: RePEc:eee:stapro:v:141:y:2018:i:c:p:135-142
    DOI: 10.1016/j.spl.2018.05.028
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    References listed on IDEAS

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    1. Walker, Stephen G., 2015. "A probabilistic proof of the Hardy inequality," Statistics & Probability Letters, Elsevier, vol. 103(C), pages 6-7.
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