Weyl-Heisenberg frame operators and Kohn-Nirenberg symbols

TL;DR

This paper derives an explicit formula for the Kohn-Nirenberg symbol of Weyl-Heisenberg frame operators using elementary Fourier analysis, applicable to new window functions and avoiding distribution theory.

Contribution

It introduces a novel elementary Fourier analysis approach to express Kohn-Nirenberg symbols of Weyl-Heisenberg frame operators directly from Gabor atoms.

Findings

Explicit Kohn-Nirenberg symbol formula derived

Applicable to new classes of window functions

Method avoids distribution theory and remains within L^2(R)

Abstract

An explicit expression for the Kohn-Nirenberg symbol of a Weyl- Heisenberg frame operator on $L^2(\mathbb{R})$ is obtained directly from the Gabor atom coming from new classes of window functions. This new approach, using only elementary Fourier analysis, is independent of the theory of distributions and works strictly inside $L^2(\mathbb{R})$. Kohn-Nirenberg operators are introduced and are shown to be Weyl-Heisenberg frame operators in suitable cases.