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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Pointwise convergence of wavelet expansions
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by Susan E. Kelly, Mark A. Kon and Louise A. Raphael PDF
Bull. Amer. Math. Soc. 30 (1994), 87-94 Request permission


In this note we announce that under general hypotheses, wavelet-type expansions (of functions in ${L^p}$, $1 \leq p \leq \infty$, 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 ${P_j}$ are bounded by radial decreasing ${L^1}$ convolution kernels. As a corollary it follows that best ${L^2}$ 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 s if and only if the basic wavelet $\psi$ is in the homogeneous Sobolev space $H_h^{ - s - d/2}$. We also present equivalent necessary and sufficient conditions on the scaling function. The above results hold in one and in multiple dimensions.
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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 30 (1994), 87-94
  • MSC (2000): Primary 42C15; Secondary 40A30
  • DOI:
  • MathSciNet review: 1248218