HAAR - TYPE MULTIWAVELET BASES AND SELF - AFFINE MULTI - TILES 389 Theorem 1
نویسنده
چکیده
Grr ochenig and Madych showed that a Haar-type wavelet basis of L 2 (R n) can be constructed from the characteristic function of a compact set if and only if is an integral self-aane 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 (x) := 1 (x); : : : ; r (x)] T , where = ((1 ; : : : ; r) is an r-tuple of compact sets in R n. We call a self-aane multi-tile because i 's tile R n by translation and have the property that each aane image A((i) is the union of translates of some j 's. We also construct associated Haar-type multiwavelets , and present examples using various dilation matrices A. 1. Introduction. Let A be an expanding matrix in M n (Z), that is, one with integer entries and all eigenvalues j i (A)j > 1. A compactly supported nonzero function f(x) 2 L 2 (R n) is called a scaling function of a multiresolution analysis with dilation
منابع مشابه
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), . . . ...
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