Mathematical Sciences: Some Problems in Harmonic Analysis Related to Wavelets

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The thrust of this mathematical research revolves around applications of recent developments in wavelet theory. The first application will focus on algebras of linear operators generated by those of Calderon-Zygmund type. These operators arise in the analysis of partial differential equations. The operators are too complex to study individually; the goal of current research is to choose wavelet bases in which the operators may be represented as almost-diagonal matrices. From this representation, one can read off many hidden properties of the operators. A second line of investigation concerns wavelet decompositions in higher dimensional space. This work will begin with the representation of radial functions in term of wavelets which somehow respect the symmetry. The fundamental difficulty is that one cannot symmetrize one-dimensional wavelets and maintain the desired translation properties. Some progress has been made in developing a new wavelet-type function with many desirable properties preserved. A third direction of this research is that of distinguishing the behavior of the different Riesz transforms on functions spaces using wavelet-type representations. Wavelet theory has introduced an extraordinary new tool into the field of harmonic analysis. It contains the elements of a highly effective theoretical method for the representation of functions along with natural algorithms for the computation of various expansions of functions which are simultaneously local in time and frequency. There applications are only beginning to be felt in the scientific and engineering community.