Haar-type multiwavelet bases and self-affine multi-tiles

Grochenig and Madych showed that a Haar-type wavelet basis of L 2 (R n ) can be constructed from the characteristic function xn of a compact set O if and only if fl is an integral self-affine tile of Lebesgue measure one.In this paper we generalize their result to the multiwavelet settings.We give a complete characterization of Haar-type scaling function vectors Xn(x) := [XQi ( x )i • • • J XQr ( X )] T i where ft = (fii, ..., ft r ) is an r-tuple of compact sets in R n .We call Q a self-affine multi-tile because Q^'s tile R n by translation and have the property that each affine image A(Qi) is the union of translates of some fij's.We also construct associated Haar-type multiwavelets , and present examples using various dilation matrices A.