Journal of Geometric Analysis

Volume 17, Issue 1, March 2007 pp. 87-96.

Some Simple Haar-Type Wavelets in Higher Dimensions

Authors: Ilya A. Krishtal, Benjamin D. Robinson, Guido L. Weiss, and Edward N. Wilson
Author institution: Northern Illinois University, Mathematical Sciences, Watson Hall 320, DeKalb, IL 60115-2888 Department of Mathematics, Washington University, St. Louis, Missouri 63130 Department of Mathematics, Washington University, St. Louis, Missouri 63130 Department of Mathematics, Washington University, St. Louis, Missouri 63130

Summary: An orthonormal wavelet system in $\R^d$, $d\in\N$, is a countable collection of functions $\{\psi^\ell_{j,k}\}$, $j\in\Z$, $k\in\Z^d$, $\ell = 1,\dots,L$, of the form $$ \psi^\ell_{j,k}(x) = |\det a|^{-j/2} \psi^\ell\big(a^{-j}x-k\big) \equiv \big(D_{a^j} T_k \psi^\ell\big)(x) $$ that is an orthonormal basis for $L^2(\R^d)$, where $a\in{\rm GL}_d(\mathbb R)$ is an expanding matrix. The first such system to be discovered (almost 100 years ago) is the Haar system for which $L=d=1$, $\psi^1(x) = \psi(x) =\chi_{[0,1/2)}(x)-\chi_{[1/2,1)}(x)$, $a = 2$. It is a natural problem to extend these systems to higher dimensions. A simple solution is found by taking appropriate products $\Phi(x_1,x_2,\dots,x_d) = \varphi_1(x_1)\varphi_2(x_2)\dots\varphi_d(x_d)$ of functions of one variable. The obtained wavelet system is not always convenient for applications. It is desirable to find ``nonseparable'' examples. One encounters certain difficulties, however, when one tries to construct such MRA wavelet systems. For example, if $ a = \big( \begin{smallmatrix} 1 & -1 \\ 1 & \ \ 1 \end{smallmatrix} \big) $ is the quincunx dilation matrix, it is well-known (see, e.g., \cite{kr.GM}) that one can construct nonseparable Haar--type scaling functions which are characteristic functions of rather complicated fractal--like compact sets. In this work we shall construct considerably simpler Haar--type wavelets if we use the ideas arising from ``composite dilation'' wavelets. These were developed in \cite{kr.GLLWW2} and involve dilations by matrices that are products of the form $a^jb$, $j\in\Z$, where $a\in{\rm GL}_d(\R)$ has some ``expanding'' property and $b$ belongs to a group of matrices in $\GL_d(\R)$ having $|\det b| = 1$.

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