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Harmonic analysis: commutative and non-commutative. Direct product and semi-direct product. Classification of groups. Harmonic analysis on the symmetric group.
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- Group Methods in Commutative Harmonic Analysis?
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Cycle notation and cycle type. Partial rankings. Subgroups and normal subgroups. Decomposition of the group algebra: isotypal components.
The Cooley-Tukey algorithm and its interpretation in terms of subgroups. The SnOB software library.
Commutative Harmonic Analysis II: Group Methods in Commutative Harmonic Analysis - Google Livros
Lie groups and invariance. Generators, the exponential map and Lie algebra. Connection to spherical harmonics. Johnson asserts that the group algebra is amenable if and only if the underlying group is amenable. An analogous result for the Fourier algebra has been proved only very recently by Z.
Ruan and it involves so-called operator amenability by viewing the Fourier algebra as an operator space. Modern abstract harmonic analysis has broad impact to related fields such as operator space theory, Banach space theory, operator algebras, Banach algebras, geometric group theory, non-commutative geometry and locally compact quantum groups. In fact operator space theory sometimes called a quantized functional analysis has played a very significant role in the development of non-commutative harmonic analysis including the study of the Fourier algebras, group C algebras and von-Neumann algebras in recent years.
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Recent advances in applied harmonic analysis include the development of a localized Fourier analysis, to provide decompositions of function spaces through simultaneous timefrequency or space-phase representations. The short-time Fourier transform is an early instance of this approach, which has been richly extended by the Wigner, wavelet, Gabor, and Stockwell transforms.
Deep questions in this area relate to the characterization of the classical Banach spaces Lp, Sobolev in terms of the coefficient space for the discrete and continuous transforms and related modulation spaces, as well as the factorization or approximate diagonalization of classical operator families such as the Calderon-Zygmund, pseudodifferential, and Fourier integral operators. Techniques in operator algebras also are applicable to the analysis of this localized version of Fourier analysis: for instance, the work of Larsen, Bratelli and Jorgenson, and others, gives an operator algebraic approach to analyzing wavelet frames, while the work of Balan, Feichtinger, Grochenig, and others demonstrates a deep connection between sampling theory for Gabor transforms and the structure of non commutative tori.
This local Fourier analysis is closely related to the phase space approach to partial differential equations, also known as micro local analysis, which leads to novel applications including in the numerical solutions of partial differential equations, analysis of stochastic PDEs, mathematical wavefield propagation, imaging technology medical, seismic, and other , and general signal processing.
Commercial applications include the development of cellphone technology, and seismic processing tools used in oil and gas exploration.
Ongoing research includes developing links between geometric ray theory in wave propagation and the more recent wavefield and path integral methods of time-frequency analysis. It investigates combinatorial and analytic properties of configurations of geometric objects.