Characterization of Generators for Multiresolution Analyses with Composite Dilations

and Applied Analysis 3 Proposition 1.3 see 4, 10 . A closed subspace X of L2 n is a reducing subspace if and only if X { f ∈ L2 n : supp ( f̂ ) ⊆ S } 1.5 for some measurable set S ⊆ n with ãS S. So, to be specific, one denotes a reducing subspace by L2 S ∨ instead of X. In particular, L2 n is a reducing subspace of L2 n . Definition 1.4 see 5–7 . Let B n be a subgroup of the integral affine group S̃Ln n the semidirect product of S̃Ln and n . The closed subspace V of L2 n is called a B n invariant subspace if BV V for any b, k ∈ B . Definition 1.5 see 2–4 . Let B be a countable subset of S̃Ln Z and A {ai : i ∈ }where a ∈ GLn . We say that a sequence {Vj}j∈ of closed subspaces of L2 n is an AB-MRA if the following holds: 1 V0 is a B n invariant space; 2 for each j ∈ ,Vj ⊂ Vj 1, and Vj D aV0; 3 ⋃ j∈ Vj L 2 n ; 4 ⋂ j∈ Vj {0}; 5 there exists φ ∈ V0 such that ΦB {DbTkφ : b ∈ B, k ∈ n} is a semiorthogonal PF for V0. The space V0 is called an AB scaling space, and the function φ is an AB scaling function for V0 or a generator of AB-MRA. Similarly, we say that a sequence {Vj}j∈ is an AB-RMRA if it is an AB-MRA on L2 S ∨, that is, conditions 1 , 2 , 4 , 5 , and 3 ′ ⋃ j∈ Vj L 2 S ∨ are satisfied. The fact that an AB-MRA can induce an AB-RMRA will be demonstrated by the obvious following results. Proposition 1.6. Let I be a countable index set and P the orthogonal projection operator from a Hilbert space H to its proper subspace K. If Ψ {ψi : i ∈ I} is a Parseval frame on H , then P Ψ {P ψi : i ∈ I} is a Parseval frame onK. Proposition 1.7. Let P be the orthogonal projection operator from a Hilbert space H to its reducing subspace K. Then P can commutate with the shift and dilation operators Tk and Da, respectively. Theorem 1.8. Suppose that {φ;Vj} is an AB-MRA, then {φ̃; Ṽj} is an AB-RMRA for L2 S ∨, where φ̃ : Pφ, Ṽ0 : span{DbTkφ̃ : b ∈ B, k ∈ n}, Ṽj : span{DajDbTkφ̃ : b ∈ B, k ∈ n}, and P is the orthogonal projection operator from L2 n to L2 S ∨. The rest of this paper is organized as follows. Theorem 1.8 and some properties of an AB-RMRA will be proved in Section 2. In Section 3, the characterization of the generator for an AB-RMRA will be established, which is the main purpose of this paper. Finally, some examples are provided to illustrate the general theory. 4 Abstract and Applied Analysis 2. Preliminaries In this section, we will firstly prove Theorem 1.8 as follows. We can easily prove that {DbTkφ̃ : b ∈ B, k ∈ n} is a Parseval frame sequence by Propositions 1.6 and 1.7. Naturally, {DbTkφ̃ : b ∈ B, k ∈ n} is a semi-Parseval frame for Ṽ0. Let φ ∈ V0 ⊂ L2 n . Thus Pφ φ1. For any f ∈ Ṽ0, we have