Existence of uncertainty minimizers for the continuous wavelet transform

Author:

Halvdansson Simon1ORCID,Olsen Jan‐Fredrik2,Sochen Nir3,Levie Ron4

Affiliation:

1. Department of Mathematics NTNU Norwegian University of Science and Technology Trondheim Norway

2. Centre for Mathematical Sciences Lund University Lund Sweden

3. School of Mathematical Sciences Tel Aviv University Tel Aviv Israel

4. Faculty of Mathematics Technion ‐ Israel Institute of Technology Haifa Israel

Abstract

AbstractContinuous 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.

Funder

Deutsche Forschungsgemeinschaft

Ministry of Health, State of Israel

Publisher

Wiley

Subject

General Mathematics

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