Mathematical Sciences: Harmonic Analysis on Spaces of       Homogeneous Type
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{'SignBlockName': 'William Y. Velez', 'PO_EMAI': None, 'PO_PHON': None}
Historically, classical harmonic analysis originates with               the problem of modeling such physical phenomena as the                          transmission of sound and heat, and in particular the                           superposition of fundamental modes of transmission.                             Mathematicaly, this translates into the more general idea of the                representation of functions in terms of simple building blocks,                 and in the modern theory one studies generalized functions and                  operators as well.  Classically, the building blocks would                      typically be related to the algebraic structure of the underlying               space; however, in current "robust" theory they may be chosen in                relation to a class of operators, and the underlying domain may                 be less well structured.  Most research is understandable in                    terms of operators that carry one space into another.  The                      underlying domains include classical Euclidean spaces and                       generalizations such as space-time, Lie transformation groups,                  local fields, and discrete structures.  Applications are                        pervasive throughout mathematics, and in applications as wide                   ranging as antenna design, CAT scans, tomography, seismology,                   particle physics, economic forecasting, pattern recognition, and                coding theory.  Seemingly abstract domains such as local fields                 and "trees" are, in fact, natural settings for applications in                  pattern recognition and numerical approximation.                                        Professors Weiss and Taibleson conduct research in the                  following areas: Hary space theory, in terms of which the                       building blocks are atoms and molecules; convergence of Fourier                 series by means of spaces generated by "blocks"; interpolation                  theory, which is concerned with mappings of families of                         functions; and the development of harmonic analysis on trees.                   Professor Weiss is a leader in the study of harmonic analysis                   that treats building blocks that do not derive from algebraic                   structure, while Professor Taibleson is a leader in the extension               of classical harmonic analysis techniques to local fields and                   trees.  Their proposed research includes the extension of                       methods, used in the study of blocks, to related function spaces                and properties that are not yet well understood; extension of                   interpolation methods; and the cexpansion of harmonic analysis on               trees to more general structures such as graphs.  The latter                    would bring this methodology to bear in the context of                          probability and combinatorics.
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[{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}, {'Code': '1262', 'Text': 'MODERN ANALYSIS'}]
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