Haar bases for $L^2(\mathbb{Q}_2^2)$ generated by one wavelet function

TL;DR

This paper provides an explicit description of all wavelet functions generating orthogonal bases in the $p$-adic quincunx Haar MRA on $L^2(Q_2^2)$, revealing their connection to separable Haar MRA.

Contribution

It explicitly characterizes all wavelet functions for the $p$-adic quincunx Haar MRA and links these bases to two-dimensional separable Haar MRA.

Findings

All wavelet functions for the $p$-adic quincunx Haar MRA are described explicitly.

Each wavelet function generates an orthogonal basis in $L^2(Q_2^2)$.

A connection between quincunx Haar bases and separable Haar MRA is established.

Abstract

The concept of $p$-adic quincunx Haar MRA was introduced and studied in~\cite{KS10}. In contrast to the real setting, infinitely many different wavelet bases are generated by a $p$-adic MRA. We give an explicit description for all wavelet functions corresponding to the quincunx Haar MRA. Each one generates an orthogonal basis, one of them was presented in~\cite{KS10}. A connection between quincunx Haar bases and two-dimensional separable Haar MRA is also found.