Wavelets provide a new class of orthogonal expansions in L 2 (R d) w i t h g o o d time/frequency localization and regularity/approximation properties Da2]. They have been successfully applied to signal processing, numerical analysis, and quantum mechanics Ru]. We study pointwise convergence properties of wavelet expansions and show that such expansions (and more generally, multiscale expansions) of L p functions (1 p 1) converge pointwise almost everywhere, and more precisely everywhere on the Lebesgue set of the function being expanded. We show that such convergence is partially insensitive to the order of summation of the expansion. It is shown that unlike Fourier series, a wavelet expansion has a summation kernel which is absolutely bounded by dilations of a radial decreasing L 1 convolution kernel H(jx ; yj). This fact provides another proof of L p convergence. These results hold in all dimensions, and apply to related multiscale expansions, including best approximations using spline functions.
Abstract.In this note we announce that under general hypotheses, wavelettype expansions (of functions in LP , 1 < p < oo , in one or more dimensions) converge pointwise almost everywhere, and identify the Lebesgue set of a function as a set of full measure on which they converge. It is shown that unlike the Fourier summation kernel, wavelet summation kernels P¡ are bounded by radial decreasing Lx convolution kernels. As a corollary it follows that best L2 spline approximations on uniform meshes converge pointwise almost everywhere. Moreover, summation of wavelet expansions is partially insensitive to order of summation.We also give necessary and sufficient conditions for given rates of convergence of wavelet expansions in the sup norm. Such expansions have order of convergence j if and only if the basic wavelet i// is in the homogeneous Sobolev space H7S~ ' . We also present equivalent necessary and sufficient conditions on the scaling function. The above results hold in one and in multiple dimensions.
When a Fourier series is used to approximate a function with a jump discontinuity, an overshoot at the discontinuity occurs. This phenomenon was noticed by Michelson [6] and explained by Gibbs [3] in 1899. This phenomenon is known as the Gibbs effect. In this paper, possible Gibbs effects will be looked at for wavelet expansions of functions at points with jump discontinuities. Certain conditions on the size of the wavelet kernel will be examined to determine if a Gibbs effect occurs and what magnitude it is. An if and only if condition for the existence of a Gibbs effect is presented, and this condition is used to prove existence of Gibbs effects for some compactly supported wavelets. Since wavelets are not translation invariant, effects of a discontinuity will depend on its location. Also, computer estimates on the sizes of the overshoots and undershoots were computed for some compactly supported wavelets with small support. c 1996 Academic Press, Inc. GIBBS PHENOMENON FOR FOURIER SERIESTo illustrate what is happening in the Gibbs effect, let us examine the partial sums of a Fourier series. Let g(x) be a periodic, piecewise smooth function with a jump discontinuity at x 0 . For any fixed x 1 , not equal to x 0 , the partial sums of g(x) at x 1 approach g(x 1 ). That is, if s n is the partial sum of g, thenHowever, if x is allowed to approach the discontinuity as the partial sums are taken to the limit, an overshoot, or undershoot, may occur. That is,and1 E-mail: kelly@math.uwlax.edu.are possible. This overshoot, or undershoot, is called the Gibbs phenomenon.Proposition 1.1 Let f be a function of bounded variation, 2π-periodic function. At each jump discontinuity x 0 of f, the Fourier series for f will overshoot (undershoot). The overshoot and undershoot will be approximately 9% of the magnitude of the jump |f(xFor further details for the Fourier series, see [8]. GENERAL WAVELET STRUCTURE AND COMPACTLY SUPPORTED WAVELETSA general structure, called a multiresolution analysis, for wavelet bases in L 2 (R) was described by Mallat [5]. Letand there is a φ ∈ V 0 such that {φ m,n } n∈Z is an orthonormal basis of V m , whereDefine W m such that V m+1 = V m W m . Thus, L 2 (R) = W m . Then there exists a ψ ∈ W 0 such that {ψ m,n } n∈Z is an orthonormal basis of W m , and {ψ m,n } m,n∈Z is a wavelet basis of L 2 (R), where ψ m,n (x) = 2 m/2 ψ(2 m x − n).The function φ is called the scaling function, and ψ is called the mother function.
A brief biographical account of Guido Weiss' life; his accomplishments, mathematics, and diverse interests; and several anecdotes of his interactions with family, friends, colleagues, and students.
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