Riesz Wavelets, Tiling and Spectral Sets in LCA Groups

摘要

Let G be a locally compact abelian (LCA) group equipped with a Haar measure. A collection of measurable subsets { i} m i= 1 in <^> G is called a Riesz wavelet collection if there are countable subsets A. Aut(G) and i. G such that for <^>. i := 1 i, the family W :=. m i= 1 { (a) 1/ 2.i (a(x) -.) :.. i, a. A} is a Riesz basis for L2(G). In this paper we show that if i ' s are Riesz spectral sets and =. i i is a multiplicative tiling set for <^> G, thenW is a Riesz basis for L2(G). The converse also holds if the unit element e. G belongs to all i. As a result, if i ' s multi- tile <^> G additively by lattice and is a multiplicative tiling, then W is a Riesz basis for L2(G). When m = 1, we show that the multiplicative tiling property of is equivalent to the Riesz spectral property of <^> a(), a. A, provided that W is a Riesz basis.