p-Adic Multiresolution Analysis and Wavelet Frames

We study p-adic multiresolution analyses (MRAs). A complete characterization of test functions generating an MRA (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions and that all such scaling functions generate the Haar MRA. We also suggest a method for constructing sets of wavelet functions and prove that any set of wavelet functions generates a p-adic wavelet frame.     

Some Results in the Theory Modulated Filter Banks and Modulated Wavelet Tight Frames

Perfect reconstruction (PR) FIR filter banks, obtained by modulation of a linear-phase, lowpass, prototype filter and of length 2Mm are well known. Recently, PR modulated filter banks (MFBs) with the analysis and synthesis banks obtained from different prototypes have been reported. This paper describes a general form of modulation that includes modulations used in the literature. This modulation depends on an integer parameter, the modulation phase. The PR property is characterized for MFBs with finite and infinite impulse response filters. The MFB PR problem reduces to roughly M/2 two-channel PR problems. A natural dichotomy in the PR conditions leads us to the concepts of Type 1 and Type 2 MFBs. Unitary MFBs are characterized by the M/2 two-channel PR filter banks also being unitary (for FIR filters of length N = 2Mm, these results are given in (Malvar, Electr Lett. 26, June 1990, 906–907; Koilpillai and Vaidyanathan, IEEE Trans. SP40, No. 4, Apr. 1992, 770–783)). We also give a necessary and sufficient condition for a large class (including FIR) unitary MFB prototypes to have symmetric (even or odd) prototype filters, and exhibit unitary MFBs without symmetric prototypes. A parameterization of all FIR unitary MFBs is also given. An efficient design procedure for FIR unitary MFBs is developed. It turns out that MFBs can be implemented efficiently using Type III and Type IV DCTs. Compactly supported modulated wavelet tight frames are shown to exist and completely parameterized. K-regular modulated WTFs are designed numerically and analytically by solving a set of non-linear equations over the parameters. Design of optimal modulated WTFs for the representation of any given signal is described with examples, and this is used to design smooth modulated WTFs.     

N维M进制广义多尺度分析及其应用

In this paper,we extend the generalized multiresolution analysis(GMRA)to higher dimensional spaces.The GMRA is generalized from each ladder spaceexpanded by a different scaling function with positive integer dilation factor m≥2.The n-d GMRA is discussed in orthogonal and bi-orthogonal cases.Then the optimalm-band wavelets are applied in processing the image datasets of the human bodyslices.The efficiency and superiority of the algorithm can be seen from the processingresults.     

Analysis in Theory and Applications

正Aims and Scope:Analysis in Theory and Applications(ATA) began its publication in 1984.It publishes original papers in the following fields:Approximation Theory, Dynamical Systems,Function Spaces, Fractals, Partial Differential Equations,Geometric Analysis,     

Analysis in Theory and Applications

正Aims and Scope:Analysis in Theory and Applications(ATA)began its publication in 1984.It publishes original papers in the following fields:Approximation Theory,Dynamical Systems,Function Spaces,Fractals,Partial Differential Equations,Geometric Analysis,Harmonic Analysis,Signal Analysis,Transform Analysis,Numerical Approximation,Special Functions and Wavelets,as well as their applications.ATA is supervised by Department of Education of Jiangsu Province and sponsored by Nanjing University.     

Analysis in Theory and Applications

<正>Aims and Scope:Analysis in Theory and Applications(ATA)began its publication in 1984.It publishes original papers in the following fields:Approximation Theory,Dynamical Systems,Function Spaces,Fractals,Partial Differential Equations,Geometric Analysis,     

Analysis in Theory and Applications

正Aims and Scope:Analysis in Theory and Applications(ATA)began its publication in 1984.It publishes original papers in the following fields:Approximation Theory,Dynamical Systems,Function Spaces,Fractals,Partial Differential Equations,Geometric Analysis,Harmonic Analysis,Signal Analysis,Transform Analysis,Numerical Approximation,Special Functions and Wavelets,as well as their applications.     

Analysis in Theory and Applications

正Aims and Scope:Analysis in Theory and Applications(ATA)began its publication in 1984.It publishes original papers in the following fields:Approximation Theory,Dynamical Systems,Function Spaces,Fractals,Partial Differential Equations,Geometric Analysis,Harmonic Analysis,Signal Analysis,Transform Analysis,Numerical Approximation,Special Functions and Wavelets,as well as their applications.     

Analysis in Theory and Applications

正Aims and Scope:Analysis in Theory and Applications(ATA)began its publication in 1984.It publishes original papers in the following fields:Approximation Theory,Dynamical Systems,Function Spaces,Fractals,Partial Differential     

Analysis in Theory and Applications

<正>Aims and Scope:Analysis in Theory and Applications(ATA) began its publication in 1984.It publishes original papers in the following fields:Approximation Theory, Computational Analysis,Dynamical Systems,     

Analysis in Theory and Applications

<正>Aims and Scope: Analysis in Theory and Applications (ATA) began its publication in 1984. It publishes original papers in the following fields: Approximation Theory, Computational Analysis, Dynamical Systems, Fractals, Function Spaces, Geometric Analysis, Harmonic Analysis, Numerical     

Analysis in Theory and Applications

<正>Aims and Scope:Analysis in Theory and Applications(ATA)began its publication in 1984.It publishes original papers in the following fields:Approximation Theory,Dynamical Systems,Function Spaces,Fractals,Partial Differential Equations,Geometric Analysis,Harmonic Analysis,Signal Analysis,     

Analysis in Theory and Applications

<正>Aims and Scope:Analysis in Theory and Applications(ATA)began its publication in 1984.It publishes original papers in the following fields:Apiroximation Theory,Dynamical Systems,Function Spaces,Fractals,Partial Differential     

Analysis in Theory and Applications

正Aims and Scope:Analysis in Theory and Applications(ATA)began its publication in 1984.It publishes original papers in the following fields:Approximation Theory,Dynamical Systems,Function Spaces,Fractals,Partial Differential Equations,Geometric Analysis,Harmonic Analysis,Signal Analysis,Transform Analysis,Numerical Approximation,Special Functions and Wavelets,as well as their applications.     

Approximation of the Inverse Frame Operator and Applications to Gabor Frames

A frame allows every element in a Hilbert space to be written as a linear combination of the frame elements, with coefficients called frame coefficients. Calculation of the frame coefficients requires inversion of an operator S on . We show how the inverse of S can be approximated as close as we like using finite-dimensional linear algebra. In contrast with previous methods, our approximation can be used for any frame. Various consequences for approximation of the frame coefficients or approximation of the solution to a moment problem are discussed. We also apply the results to Gabor frames and frames consisting of translates of a single function.     

The Diagram Approach in Knot Theory and Applications to Graph Theory

The paper is a survey of a series of publications of the authors awarded by I. I Shuvalov First Prize of Lomonosov Moscow State University for scientific activity, 2017.     

A Theory of Generalized Multiresolution Structure and Pseudoframes of Translates

The notion of a Generalized Multiresolution Structure (GMS) in LL is introduced. Basically, it consists of an increasing sequence of closed subspaces of LL, with a textit{pseudoframe of translates} at each level. Using their shift-invariance a frame-like decomposition is derived based on such a GMS. As a major new contribution the construction of affine frames for LtwoRLtwoR based on an GMS is presented. A fast algorithm for the GMS-based affine frame decomposition and reconstruction using filter banks is presented as well.     

About Nonstationary Multiresolution Analysis and Wavelets

The characterization of orthonormal bases of wavelets by means of convergent series involving only the mother wavelet is known, as well as the characterization of wavelets which can be constructed from a stationary multiresolution analysis or a scaling function (see for example [11] and references therein). Here we show that under some asymptotic condition, these results remain true in the nonstationary case.     

Coorbit Spaces and Banach Frames on Homogeneous Spaces with Applications to the Sphere

This paper is concerned with the construction of generalized Banach frames on homogeneous spaces. The major tool is a unitary group representation which is square integrable modulo a certain subgroup. By means of this representation, generalized coorbit spaces can be defined. Moreover, we can construct a specific reproducing kernel which, after a judicious discretization, gives rise to atomic decompositions for these coorbit spaces. Furthermore, we show that under certain additional conditions our discretization method generates Banach frames. We also discuss nonlinear approximation schemes based on the atomic decomposition. As a classical example, we apply our construction to the problem of analyzing and approximating functions on the spheres.