Self-similarity and multiwavelets in higher dimensions

CABRELLI, CARLOS A.; HEIL, CHRISTOPHER; MOLTER, URSULA M.

MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY (AMS)

American Mathematical Society

Lugar: Providence, RI, US; Año: 2004 vol. 170 p. 1 - 82

0065-9266

Let $A$ be a dilation matrix, an $n imes n$ expansive matrixthat maps a full-rank lattice $Gamma subset R^n$ into itself.Let $Lambda$ be a finite subset of $Gamma$, and for $k in Lambda$let $c_k$ be $r imes r$ complex matrices.The refinement equation corresponding to $A$, $Gamma$, $Lambda$, and$c = set{c_k}_{k in Lambda}$ is$f(x) = sum_{k in Lambda} c_k , f(Ax-k)$.A solution $f colon R^n o C^r$, if one exists,is called a refinable vector function or a vector scaling functionof multiplicity $r$.In this manuscript we characterize the existence of compactly supported$L^p$ or continuous solutions of the refinement equation,in terms of the $p$-norm joint spectral radius of a finite set offinite matrices determined by the coefficients $c_k$.We obtain sufficient conditions for the $L^p$ convergence($1 le p le infty$) of the Cascade Algorithm$f^{(i+1)}(x) = sum_{k in Lambda} c_k , f^{(i)}(Ax-k)$,and necessary conditions for the uniform convergence of theCascade Algorithm to a continuous solution.We also characterize those compactly supported vector scaling functionswhich give rise to a multiresolution analysis for $L^2(R^n)$of multiplicity $r$, and provide conditions under which there existcorresponding multiwavelets whose dilations and translations form anorthonormal basis for $L^2(R^n)$.