Enhanced Symmetry of the $p$-adic Wavelets

TL;DR

This paper explores the extended symmetry properties of $p$-adic wavelets, showing they form a basis with larger symmetry groups due to their eigenfunction nature of certain operators.

Contribution

It demonstrates the enhanced symmetry group of $p$-adic wavelets, extending the understanding of their mathematical structure and properties.

Findings

$p$-adic wavelets form a basis with recursive construction.

They are eigenfunctions of a pseudo-differential operator.

The symmetry group of $p$-adic wavelets is larger than previously known.

Abstract

Wavelet analysis has been extended to the $p$-adic line $\mathbb{Q}_p$. The $p$-adic wavelets are complex valued functions with compact support. As in the case of real wavelets, the construction of the basis functions is recursive, employing scaling and translation. Consequently, wavelets form a representation of the affine group generated by scaling and translation. In addition, $p$-adic wavelets are eigenfunctions of a pseudo-differential operator, as a result of which they turn out to have a larger symmetry group. The enhanced symmetry of the $p$-adic wavelets is demonstrated.