Suppose that η_1,...,η_n are measurable functions in L^2(R).We call the n-tuple(η_1,...,η_n) a Parseval super frame wavelet of length n if {2^(k/2) η_1(2~kt-l) ⊕···⊕2^(k/2) η_n(2~kt-l):k,l∈Z} is a Parseval frame for L^2(R)⊕n.In high dimensional case,there exists a similar notion of Parseval super frame wavelet with some expansive dilation matrix.In this paper,we will study the Parseval super frame wavelets of length n,and will focus on the path-connectedness of the set of all s-elementary Parseval super frame wavelets in one-dimensional and high dimensional cases.We will prove the corresponding path-connectedness theorems.
Let A be a d×d real expansive matrix.An A-dilation Parseval frame wavelet is a function ψ∈L2(Rd),such that the set {|det A|n/2ψ(Ant-l):n∈Z,l∈Zd} forms a Parseval frame for L2 (R^d).A measurable function f is called an A-dilation Parseval frame wavelet multiplier if the inverse Fourier transform of f■ is an A-dilation Parseval frame wavelet whenever ψ is an A-dilation Parseval frame wavelet,where ■ denotes the Fourier transform of ψ.In this paper,the authors completely characterize all A-dilation Parseval frame wavelet multipliers for any integral expansive matrix A with |det(A)|=2.As an application,the path-connectivity of the set of all A-dilation Parseval frame wavelets with a frame MRA in L2(Rd) is discussed.
The single 2 dilation orthogonal wavelet multipliers in one dimensional case and single A-dilation(where A is any expansive matrix with integer entries and|det A|=2) wavelet multipliers in high dimensional case were completely characterized by the Wutam Consortium(1998) and Z. Y. Li, et al.(2010). But there exist no more results on orthogonal multivariate wavelet matrix multipliers corresponding integer expansive dilation matrix with the absolute value of determinant not 2 in L^2(R^2). In this paper, we choose 2I2=(_0~2 _2~0)as the dilation matrix and consider the 2 I2-dilation orthogonal multivariate waveletΨ = {ψ_1, ψ_2, ψ_3},(which is called a dyadic bivariate wavelet) multipliers. We call the3 × 3 matrix-valued function A(s) = [ f_(i, j)(s)]_(3×3), where fi, jare measurable functions, a dyadic bivariate matrix Fourier wavelet multiplier if the inverse Fourier transform of A(s)( ψ_1(s), ψ_2(s), ψ_3(s)) ~T=( g_1(s), g_2(s), g_3(s))~ T is a dyadic bivariate wavelet whenever(ψ_1, ψ_2, ψ_3) is any dyadic bivariate wavelet. We give some conditions for dyadic matrix bivariate wavelet multipliers. The results extended that of Z. Y. Li and X. L.Shi(2011). As an application, we construct some useful dyadic bivariate wavelets by using dyadic Fourier matrix wavelet multipliers and use them to image denoising.
