Dirac brushes (or, the fractional Fourier transform of Dirac combs)

TL;DR

This paper explores the fractional Fourier transform of Dirac combs, revealing conditions for resulting measures to be discretely supported and connecting these findings to the functional equations of Jacobi theta functions.

Contribution

It characterizes when the fractional Fourier transform of Dirac combs yields discrete measures and links these cases to classical theta function identities using advanced mathematical tools.

Findings

Fractional Fourier transform of Dirac combs can produce discrete measures under specific conditions.

The measures are related to the functional equations of Jacobi theta functions.

Small-angle limits reveal Euler spirals composed of Gauss sums.

Abstract

In analogy with the Poisson summation formula, we identify when the fractional Fourier transform, applied to a Dirac comb in dimension one, gives a discretely supported measure. We describe the resulting series of complex multiples of delta functions, and through either the metaplectic representation of $SL(2,\mathbb{Z})$ or the Bargmann transform, we see that the the identification of these measures is equivalent to the functional equation for the Jacobi theta functions. In tracing the values of the antiderivative in certain small-angle limits, we observe Euler spirals, and on a smaller scale, these spirals are made up of Gauss sums which give the coefficient in the aforementioned functional equation.