Classifying Tight Weyl-Heisenberg Frames

TL;DR

This paper characterizes when Weyl-Heisenberg frames are tight, providing explicit conditions and classifications for functions forming orthonormal bases and dual frames in L^2(R).

Contribution

It offers necessary and sufficient conditions for tight Weyl-Heisenberg frames and explicitly classifies functions generating orthonormal bases and dual frames.

Findings

Derived explicit conditions for tight WH-frames.

Classified functions forming orthonormal bases.

Provided a simple classification of dual frames.

Abstract

A Weyl-Heisenberg frame for L^2(R) is a frame consisting of translates and modulates of a fixed function. In this paper we give necessary and sufficient conditions for this family to form a tight WH-frame. This allows us to write down explicitly all functions g for which all translates and modulates of g form an orthonormal basis for L^2(R). There are a number of consequences of this classification, including a simple direct classification of the alternate dual frames to a WH-frame (A result originally due to Janssen).