Existence of Uncertainty Minimizers for the Continuous Wavelet Transform

TL;DR

This paper proves the existence of mother wavelets that minimize uncertainty functionals in the continuous wavelet transform, ensuring optimal localization properties for signal analysis.

Contribution

It establishes the theoretical existence of minimizers for recently derived wavelet uncertainty functionals, advancing wavelet design theory.

Findings

Existence of minimizers for two wavelet uncertainty functionals proven.

Provides theoretical foundation for optimal wavelet design.

Enhances understanding of wavelet localization properties.

Abstract

Continuous wavelet design is the endeavor to construct mother wavelets with desirable properties for the continuous wavelet transform (CWT). One class of methods for choosing a mother wavelet involves minimizing a functional, called the wavelet uncertainty functional. Recently, two new wavelet uncertainty functionals were derived from theoretical foundations. In both approaches, the uncertainty of a mother wavelet describes its concentration, or accuracy, as a time-scale probe. While an uncertainty minimizing mother wavelet can be proven to have desirable localization properties, the existence of such a minimizer was never studied. In this paper, we prove the existence of minimizers for the two uncertainty functionals.