We study the open question on determination of jumps for functions raised by Shi and Hu in 2009. An affirmative answer is given for the case that spline-wavelet series are used to approximate the functions.
Suppose that η1,...,η_n are measurable functions in L2（R）.We call the n-tuple（η1,...,ηn） a Parseval super frame wavelet of length n if {2^（k/2） η1（2~kt-l） ⊕···⊕2^（k/2） ηn（2kt-l）：k,l∈Z} is a Parseval frame for L2（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.
In this paper, we will introduce multiresolution analysis with composite dilations and give a characterization of generator for multiresolution analysis with composite dilations.
