On the Properties of the Integer Translates of a Square Integrable Function in L(R)

(1.1) 〈ψ〉 = span{ψ(· − k) ≡ ψk : k ∈ Z} That is, 〈ψ〉 is the L2(R) closure of the space generated by the finite linear combinations of the integer translates, (Tkψ)(x) ≡ ψ(x − k) ≡ ψk(x), of ψ ∈ L2(R). Such a space is often called the principal shift invariant space generated by ψ. In general, a closed subspace V ⊂ L2(R) is shift invariant if and only if TkV ⊂ V for all translates Tk, k ∈ Z. The theory of wavelets is based on the properties of the principal shift invariant spaces. The scaling space V0 in a multiresolution analysis (MRA) is an example of such a space; the zero resolution wavelet spaceW0 is another such space. These two spaces enjoy very different features. The basic properties of 〈ψ〉 are determined by those of the generating system B = {ψk = Tkψ : k ∈ Z}. As we will show, those properties are reflected by simple properties of the periodization function