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Multiwavelet sampling theorem in Sobolev spaces

Science China Mathematics 2010年第12期53卷 3197-3214页

作者： LI YouFa 1,2 & YANG ShouZhi 1, 1 Department of Mathematics, Shantou University, Shantou 515063, China 2 College of Mathematics and Information Sciences, Guangxi University, Nanning 530004, China 1. Department of Mathematics Shantou University Shantou 515063 China2. College of Mathematics and Information Sciences Guangxi University Nanning 530004 China

This paper is to establish the multiwavelet sampling theorem in Sobolev spaces. Sampling theorem plays a very important role in digital signal communication. The most classical sampling theorem is Shannon sampling theorem, which works for bandlimited signals. Recently, sampling theorems in wavelets or multiwavelets subspaces are extensively studied in the literature. In this paper, we firstly propose the concept of dual multiwavelet frames in dual Sobolev spaces (H s (R) , H-s (R)). Then we construct a special class of dual multiwavelet frames, from which the multiwavelet sampling theorem in Sobolev spaces is obtained. That is, for any f ∈ H s (R) with s > 1/2, it can be exactly recovered by its samples. Especially, the sampling theorem works for continuous signals in L 2 (R), whose Sobolev exponents are greater than 1 /2.

关键词： Sobolev spaces refinable function vectors dual multiwavelet frames delta function sampling theorem

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Study on the generator f of the operators I_n and I_t

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Science China Mathematics 2010年第10期53卷 2589-2603页

作者： CHEN HanLin Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,China 1. Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China

Define two operators In and It,the inner product operator In(g)(x) := j∈Zs(g,f(·-j))f(x-j) and the interpolation operator It(g)(x) := j∈Zs g(j)f(x-j),where f belongs to some space and integer s *** call f the generator of the operators In and *** is well known that there are many results on operators In and *** there remain some important problems to be further *** application we first need to find the available generators (that can recover polynomials as It(p) = p or In(p) = p,p ∈Πm-1) for constructing the relative *** this paper,we focus on the available generator in the class of spline *** shall see that not all spline functions can be used to construct available ***,we do find a spline function in S,of degree m-1,where m is even and S is a class of *** for odd m the problem is still *** on spline functions in this paper are new.

关键词： generator available generator inner product operator spline interpolation operator two-closed refinable

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