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Littlewood-Paley theory and the study of function spaces. (English) Zbl 0757.42006
Regional Conference Series in Mathematics. 79. Providence, RI: American Mathematical Society (AMS). vii, 132 p. (1991).
The Littlewood-Paley theorem provides a unified perspective onto $$L_ p$$-spaces, Hardy spaces, Lipschitz spaces, and BMO as well as the link between those function spaces and wavelet theory. Essentially it is Calderón’s formula which gives suitable decompositions of those function spaces and allows to describe Besov spaces and the Triebel- Lizorkin spaces in a suitable way. The authors discuss the general $$\varphi$$-transform (elsewhere called continuous wavelet transform), give a detailed proof of the Meyer-Lemarie existence theorem for orthogonal wavelets, make the connections to those function spaces and round off their presentation by three applications: numerical approximation of integral operators, the fractal dimension of the graph of functions in Besov spaces, and the use of wavelets in image compression. The booklet appears to be useful for graduate students and researchers with interest in function spaces, approximation theory or wavelet theory.
Reviewer: M.Takev (Sofia)

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 42C15 General harmonic expansions, frames 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems