Multilinear Operators, Discrete Decompositions, and Spectral Resolution of Nanostructures

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

ABSTRACT: The research to be conducted includes the analysis of operators associated with multilinear singular integrals and the study of discrete functions spaces. The investigations will be based on techniques related to Littlewood-Paley theory, molecular decompositions, and time-frequency analysis tools. Specific aspects proposed in the analysis of operators are to continue collaborations in the development of a multilinear counterpart of the linear Calderon-Zygmund theory, and to study various classes of multilinear pseudodifferential operators. Problems in the analysis of discrete function spaces include issues about the sampling of functions with controlled mean oscillations and the approximation of band limited signals. The proposal also contains an interdisciplinary component. In particular, theoretical problems arising in the analytic formulation of the scattering of light by structurally colored tissues of living organisms will be considered. The spectral resolution and mathematical properties of quasi-ordered geometries will be investigated. The last part of the research will be assisted by numerical computation and data visualization. Operators associated with singular integrals arise as technical tools in analysis and also as transformations encountered in the mathematical modeling of certain physical phenomena. Such transformations can be used to describe the changes in the properties of a function or the transition from the input to the output of a system. Properties of functions or signals often need to be understood from the information encoded in samples of the data. Such information can be quantified by function spaces and decoded by Fourier analysis and related time-frequency techniques. Fourier analysis is the mathematical version of a diffracting physical prism. It resolves a signal into a spectrum of waves of different amplitudes and oscillations in a similar way that a prism diffracts a ray of light into a rainbow of colors of different wavelengths. Modern decomposition techniques in analysis provide a universal language for the processing of complicated information. Progress in this area of analysis always produces important advances in scientific problems where large and complicated sets of data need to be analyzed to search for ordered patterns, reduce unnecessary information, or visualize intricate structures.