Short-time Fourier transform and superoscillations

TL;DR

This paper explores how the short-time Fourier transform (STFT) interacts with superoscillations, revealing preservation properties and connections to various time-frequency analysis tools and functions.

Contribution

It introduces new results on the behavior of superoscillations under STFT and links them to Gabor analysis, Hermite functions, and polyanalytic functions.

Findings

STFT preserves superoscillatory behavior in the limit

Connections established between superoscillations and Gabor spaces

Computed STFT action on sequences with Hermite windows

Abstract

In this paper we investigate new results on the theory of superoscillations using time-frequency analysis tools and techniques such as the short-time Fourier transform (STFT) and the Zak transform. We start by studying how the short-time Fourier transform acts on superoscillation sequences. We then apply the supershift property to prove that the short-time Fourier transform preserves the superoscillatory behavior by taking the limit. It turns out that these computations lead to interesting connections with various features of time-frequency analysis such as Gabor spaces, Gabor kernels, Gabor frames, 2D-complex Hermite polynomials, and polyanalytic functions. We treat different cases depending on the choice of the window function moving from the general case to more specific cases involving the Gaussian and the Hermite windows. We consider also an evolution problem with an initial datum given by superoscillation multiplied by the time-frequency shifts of a generic window function. Finally, we compute the action of STFT on the approximating sequences with a given Hermite window.