A new computational method based on Wilson wavelets is proposed for solving a class of nonlinear stochastic It^{o}-Volterra integral equations. To do this a new stochastic operational matrix of It^{o} integration for Wilson wavelets is obtained. Block pulse functions (BPFs) and collocation method are used to generate a process to forming this matrix. Using these basis functions and their operat...

In 2006 Guo, Labate, Lim, Weiss, and Wilson introduced the theory of MRA composite dilation wavelets. We continue their work by studying the filter properties of such wavelets and present several important examples.

In this paper we compare different transmultiplexer structures with respect to the ISI/ICI occuring for typical time–invariant channels. In particular we consider wavelet–type, Gabor-type (the class containing OFDM and DMT) and Wilson-type (offset-QAM/OFDM) transmultiplexer. We present both theoretical results (based on a recently developed perturbation theory of coherent Riesz bases) and numer...

We consider the problem of recovering a structured sparse representation of a signal in an overcomplete time-frequency dictionary with a particular structure. For infinite dictionaries that are the union of a nice wavelet basis and a Wilson basis, sufficient conditions are given for the Basis Pursuit and (Orthogonal) Matching Pursuit algorithms to recover a structured representation of an admis...

coherent WaveBurst (cWB) is a highly configurable pipeline designed to detect broad range of gravitational-wave (GW) transients in the data worldwide network GW detectors. The algorithmic core cWB time–frequency analysis with Wilson–Daubechies–Meyer wavelets aimed at identification events without prior knowledge signal waveform. has been active development since 2003 and it used analyze all sci...

We compare three types of coherent Riesz families (Gabor systems, Wilson bases, and wavelets) with respect to their perturbation stability under convolution with elements of a family of typical channel functions. This problem is of key relevance in the design of modulation signal sets for digital communication over time–invariant channels. Upper and lower bounds on the orthogonal perturbation a...

We study the orthogonal perturbation of various coherent function systems (Gabor systems, Wilson bases, and wavelets) under convolution operators. This problem is of key relevance in the design of modulation signal sets for digital communication over time–invariant channels. Upper and lower bounds on the orthogonal perturbation are formulated in terms of spectral spread and temporal support of ...

This report aims to present my research updates on distance function wavelets (DFW) based on the fundamental solutions and general solutions of the Helmholtz, modified Helmholtz, and convection-diffusion equations, which include the isotropic HelmholtzFourier (HF) transform and series, the Helmholtz-Laplace (HL) transform, and the anisotropic convection-diffusion wavelets and ridgelets. The lat...

Searching for a general and technically simple multi-scale formalism to treat interacting fermions, we develop a (Wilson-Kadanoo) block renormalization group mechanism, which, due to the property of \orthogonality between scales", establishes a trivial link between the correlation functions and the eeective potential ow, leading to simple expressions for the generating and correlation functions...