Wavelet frames: Spectral techniques and extension principles

TL;DR

This paper characterizes dyadic wavelet frames for L^2(R) using spectral techniques based on the decomposability of the frame operator, connecting to Fourier domain methods and extension principles.

Contribution

It introduces spectral techniques for characterizing wavelet frames, enabling the calculation of all tight frames with minimal support generators.

Findings

Spectral formulas for wavelet frame characterization

Connection to Fourier domain fiberization techniques

Framework for computing tight wavelet frames with minimal support

Abstract

This work characterizes (dyadic) wavelet frames for $L^2({\mathbb R})$ by means of spectral techniques. These techniques use decomposability properties of the frame operator in spectral representations associated to the dilation operator. The approach is closely related to usual Fourier domain fiberization techniques, dual Gramian analysis and extension principles, which are described here on the basis of the periodized Fourier transform. In a second paper of this series, we shall show how the spectral formulas obtained here permit us to calculate all the tight wavelet frames for $L^2({\mathbb R})$ with a fixed number of generators of minimal support.