Intrinsic s-elementary Parseval frame multiwavelets in

A countable family { x j } , j ∈ J , in a separable Hilbert space H, is a Parseval frame for H if ‖ f ‖ 2 = ∑ j ∈ J | 〈 f , x j 〉 | 2 holds for all f ∈ H . In the case that H = L 2 ( R d ) and the affine system { D A j T k ψ i | j ∈ Z , k ∈ Z d , 1 ⩽ i ⩽ n } obtained from a finite subset Ψ = { ψ 1 , ψ 2 , … , ψ n } of L 2 ( R d ) 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, D A are the translation and A-dilation unitary operators acting on L 2 ( R d ) defined by ( T ℓ f ) ( t ) = f ( t − ℓ ) , ( D A f ) ( t ) = | det A | 1 2 f ( A t ) , ∀ f ∈ L 2 ( R d ) , ℓ ∈ Z d , t ∈ R d . In the special case that there exist disjoint measurable sets { E 1 , E 2 , … , E n } such that ψ ˆ i = 1 ( 2 π ) d / 2 χ E i 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 { E 1 , E 2 , … , E m } such that ψ ˆ i = 1 ( 2 π ) d / 2 χ E i defines a Parseval frame multiwavelet Ψ = { ψ 1 , ψ 2 , … , ψ m } , 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 L 2 ( R d ) 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 ( L 2 ( R d ) ) m . The same result holds for the family of all intrinsic A-dilation s-elementary Parseval frame multiwavelets of length m.