Intrinsic s-elementary Parseval frame multiwavelets in L2(Rd)

A countable family {xj}, j∈J, in a separable Hilbert space H, is a Parseval frame for H if ‖f‖2=∑j∈J2|〈f,xj〉| holds for all f∈H. In the case that H=L2(Rd) and the affine system obtained from a finite subset Ψ={ψ1,ψ2,…,ψn} of L2(Rd) is a Parseval frame, Ψ is called an A-dilation Parseval frame multiwavelet (of length n). Here A stands for a d×d expansive matrix, and T, DA are the translation and A-dilation unitary operators acting on L2(Rd) defined by (Tℓf)(t)=f(t−ℓ), , ∀f∈L2(Rd), ℓ∈Zd, t∈Rd. In the special case that there exist disjoint measurable sets {E1,E2,…,En} such that for each i, Ψ is called an A-dilation s-elementary Parseval frame multiwavelet. A measurable set E is called a frame multiwavelet set of multiplicity m (under A-dilation) if E can be written as a disjoint union of measurable sets {E1,E2,…,Em} such that defines a Parseval frame multiwavelet Ψ={ψ1,ψ2,…,ψm}, and that this cannot be done for any integer less than m. An A-dilation s-elementary Parseval frame multiwavelet with length m that is defined on a frame multiwavelet set of multiplicity m is said to be intrinsic. It is known that single A-dilation wavelets exist in L2(Rd) for any expansive matrix A. In this paper, we show that for any d×d expansive matrix A and any given m∈N, the family of intrinsic A-dilation s-elementary Parseval frame multiwavelet with length m is not empty, and is path-connected under the norm topology of (L2m(Rd)). The same result holds for the family of all intrinsic A-dilation s-elementary Parseval frame multiwavelets of length m.