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Full Description
There is a recent and increasing interest in harmonic analysis of non-smooth geometries. Real-world examples where these types of geometry appear include large computer networks, relationships in datasets, and fractal structures such as those found in crystalline substances, light scattering, and other natural phenomena where dynamical systems are present. Notions of harmonic analysis focus on transforms and expansions and involve dual variables. In this book on smooth and non-smooth harmonic analysis, the notion of dual variables will be adapted to fractals. In addition to harmonic analysis via Fourier duality, the author also covers multiresolution wavelet approaches as well as a third tool, namely, $L^2$ spaces derived from appropriate Gaussian processes. The book is based on a series of ten lectures delivered in June 2018 at a CBMS conference held at Iowa State University.
Contents
Introduction. Smooth vs the non-smooth categories
Spectral pair analysis for IFSs
Harmonic analyses on fractals, with an emphasis on iterated function systems (IFS) measures
Four kinds of harmonic analysis
Harmonic analysis via representations of the Cuntz relations
$\textit{ Positive definite functions }$ and kernel analysis
Representations of $\textit{Lie groups}$. Non-commutative harmonic analysis
Bibliography
Index.
