论文标题:关于小波逼近的一些研究
Some Research on Wavelet Approximations
论文作者
论文导师 曹怀信,论文学位 硕士,论文专业 基础数学
论文单位 陕西师范大学,点击次数 243,论文页数 45页File Size1068K
2006-05-01论文网 http://www.lw23.com/lunwen_1163597/
Wavelet series;pointwise convergence;uniform convergence;rate of approximation;remainder;estimation;quasi-positive δ sequence
本文研究小波逼近及其应用。讨论小波级数的收敛性,其中包括一维小波级数与高维小波级数的部分和f_m对f的收敛性及其对其收敛速度的精确估计。本文分为三章,分别对一维小波,高维小波与Shannon小波的收敛性进行探讨。 第一章研究一维小波逼近,讨论小波级数的收敛性和收敛速度。首先,我们例出几种常见的小波。其次,我们给出当尺度函数φ满足条件 |φ(x)≤C/((1+|x|)~(1+β"))((?)x∈R C,β>0是常数)时,小波级数的部分和f_m对f的逐点收敛性与一致收敛性以及收敛速度的精确估计。最后,我们建立当小波函数φ满足条件时小波级数余项的两个估计,其中∑_(k=1)~(+∞)ak是收敛的正项级数。 第二章研究高维小波逼近,讨论高维小波级数的收敛性,通过引入拟正δ序列的概念,研究它的性质,建立一个一致逼近序列,并得到该序列的逐点收敛性;继而,我们证明了当尺度函数满足 |φ(x)≤(C/((1+||x||)~(1+β")))((?)x∈R~d C,β>0是常数)时,相应的再生核序列{qm}_(m∈z)是一个拟正δ序列,从而建立高维小波展开式的一致收敛定理。这一定理推广了G.G.Walter在文章Pointwise Convergence of Wavelet Expansions中建立的相应结果。 第三章研究研究Shannon小波逼近,讨论Shannon小波级数的部分和S_n(f,x)对f的收敛性。一般来说,即使是连续函数f∈L~2(R)的部分和S_n(f,x)也不一定处处收敛于f。但是,如果f在某个区间[a,b]上具有全变差,或者具有有界变差,或者具有∧-有界变差,或者f为单调型函数,那么我们可以得到相应的收敛性和逼近误差。
This paper studies the approximations of the partial sum f_m of wavelet series, concludiong one-dimensional wavelet series and high-dimensional wavelet series, to f, and the exact estimation of the approximation rate. This paper is devided into three chapters.Chapter 1 studies the convergence and approximation rate of one-dimensional wavelet series. First, we give several common wavelets. Second, we establish two exact estimations of the approximation rate of the partial sum f_m to f when the scaling funciton satisfies(contants).At last, we give several estimations of the remainder of the wavelet expansion when the wavelet function satisfieswhere the series converges.Chapter 2 is devoted to the study of the convergence of high-dimensional wavelet series. By introducing the concept of Quasi-positive δ Sequence, we discuss its properties, construct a uniformly approximation sequence, and then obtain the pointwise convergence of this sequence. At last, we prove that when the scaling function satisfies(constants),the reproducing kernel sequence {q_m}_(m∈Z) is a Quasi-positive δ Sequence, and then get a theorem of uniform convergence. This theorem generalizes a theorem due to G. G. Walter"s in paper "Pointwise Convergence of Wavelet Expansions".Chapter 3 discusses the convergence of Shannon wavelet series. We prove the convergence and give the estimations of the rate of approximation when the function f is of totally variation, or of bounded variation, or of ∧— bounded variation on some interval [a, b], or f is a monotonic-like function,
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