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.
