Harmonic analysis, wavelets and applications. (English) Zbl 0931.42017

Caffarelli, Luis (ed.) et al., Hyperbolic equations and frequency interactions. Providence, RI: American Mathematical Society. IAS/ Park City Math. Ser. 5, 159-226 (1999).
This is an eight lecture tutorial in which the basics of wavelet theory and its applications to the solution of differential equations are elegantly covered. The text is sprinkled with insights that will be appreciated by expert and novice alike, and with judiciously chosen exercises and bibliographic references. The first lecture is introductory. Lecture 2 treats multiresolution analysis in a one variable setting. Lectures 3 and 4 discuss various examples, Lecture 5 deals with other functional spaces, and Lecture 6 deals with pointwise convergence. Two–dimensional wavelets and operators are introduced in the seventh lecture, and the work culminates in the eighth lecture, where all the pieces are put together to show how wavelets can be used in the numerical solution of a parabolic equation, namely Burger’s equation \( \partial _t u = \nu \partial_{xx}u + u \partial_x u\) on the interval \([0, 1]\), with periodic boundary conditions and with the initial condition \(u(x,0)=\sin(2 \pi x)\).
For the entire collection see [Zbl 0906.00018].


42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
43-02 Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis
42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems