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 x d real expansive matrix. An A-dilation Parseval frame wavelet is a function φ E n2 （Rd）, such that the set {｜det A｜n/2φ（Ant -l） ：n ∈ Z, l∈ Zd} forms a Parseval frame for L2 （Rd）. A measurable function f is called an A-dilation Parseval frame wavelet multiplier if the inverse Fourier transform of fφ is an A-dilation Parseval frame wavelet whenever φ is an A-dilation Parseval frame wavelet, where φ denotes the Fourier transform of φ. 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.
