Haar - Type Multiwavelet Bases and Self - Affine Multi - Tiles

Abstract. Gröchenig and Madych showed that a Haar-type wavelet basis of L2(Rn) can be constructed from the characteristic function χΩ of a compact set Ω if and only if Ω 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 χΩ(x) := [χΩ1 (x), . . . , χΩr (x)] T , where Ω = (Ω1, . . . , Ωr) is an r-tuple of compact sets in Rn. We call Ω a self-affine multi-tile because Ωi’s tile R n by translation and have the property that each affine image A(Ωi) is the union of translates of some Ωj ’s. We also construct associated Haar-type multiwavelets , and present examples using various dilation matrices A.