Wavelets and Quantum Algebras

TL;DR

This paper reveals that wavelets possess a q-deformed algebraic structure, linking them to quantum groups and offering a duality between scaling functions and their algebraic counterparts.

Contribution

It introduces a novel algebraic framework for wavelets using q-deformation, connecting wavelet theory with quantum algebra structures.

Findings

Wavelets have an associated non-linear, two-parameter algebra.

This algebra maps onto the quantum group $su_{q}(2)$ in a certain limit.

Examples include detailed analysis of Haar and B-wavelets.

Abstract

Wavelets, known to be useful in non-linear multi-scale processes and in multi-resolution analysis, are shown to have a q-deformed algebraic structure. The translation and dilation operators of the theory associate with any scaling equation a non-linear, two parameter algebra. This structure can be mapped onto the quantum group $su_{q}(2)$ in one limit, and approaches a Fourier series generating algebra, in another limit. A duality between any scaling function and its corresponding non-linear algebra is obtained. Examples for the Haar and B-wavelets are worked out in detail.