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The dramatic changes that came about in analysis during the twentieth century are truly amazing. In the thirties, complex methods and Fourier series played a seminal role. After many improvements, mostly achieved by the Calderón-Zygmund school, the action today is taking place in spaces of homogeneous type. No group structure is available and the Fourier transform is missing, but a version of harmonic analysis is still available. Indeed the geometry is conducting the analysis. The authors succeed in generalizing the construction of wavelet bases to spaces of homogeneous type. However wavelet bases are replaced by frames, which in many applications serve the same purpose.

Harmonic analysis --- Spaces of homogeneous type --- Operations Research --- Mathematical Theory --- Mathematics --- Civil & Environmental Engineering --- Physical Sciences & Mathematics --- Engineering & Applied Sciences --- Harmonic analysis. --- Spaces of homogeneous type. --- Homogeneous type spaces --- Spaces, Homogeneous type --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Approximation theory. --- Fourier analysis. --- Functional analysis. --- Functions of complex variables. --- Functions of a Complex Variable. --- Analysis. --- Fourier Analysis. --- Abstract Harmonic Analysis. --- Functional Analysis. --- Approximations and Expansions. --- Banach algebras --- Calculus --- Mathematical analysis --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Complex variables --- Elliptic functions --- Functions of real variables --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Analysis, Fourier --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- 517.1 Mathematical analysis --- Math --- Science --- Quasi-metric spaces --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic