Multilinear Operators, Discrete Decompositions, and Spectral Resolution of Nanostructures

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ABSTRACT:The research to be conducted includes the analysis of operators associated withmultilinear singular integrals and the study of discrete functions spaces. Theinvestigations will be based on techniques related to Littlewood-Paley theory,molecular decompositions, and time-frequency analysis tools. Specific aspectsproposed in the analysis of operators are to continue collaborations in thedevelopment of a multilinear counterpart of the linear Calderon-Zygmundtheory, and to study various classes of multilinear pseudodifferentialoperators. Problems in the analysis of discrete function spaces include issuesabout the sampling of functions with controlled mean oscillations and theapproximation of band limited signals. The proposal also contains aninterdisciplinary component. In particular, theoretical problems arising in theanalytic formulation of the scattering of light by structurally colored tissuesof living organisms will be considered. The spectral resolution andmathematical properties of quasi-ordered geometries will be investigated. Thelast part of the research will be assisted by numerical computation and datavisualization.Operators associated with singular integrals arise as technical tools inanalysis and also as transformations encountered in the mathematical modeling ofcertain physical phenomena. Such transformations can be used to describe thechanges in the properties of a function or the transition from the input to theoutput of a system. Properties of functions or signals often need to beunderstood from the information encoded in samples of the data. Suchinformation can be quantified by function spaces and decoded by Fourier analysisand related time-frequency techniques. Fourier analysis is the mathematicalversion of a diffracting physical prism. It resolves a signal into a spectrumof waves of different amplitudes and oscillations in a similar way that a prismdiffracts a ray of light into a rainbow of colors of different wavelengths.Modern decomposition techniques in analysis provide a universal language for theprocessing of complicated information. Progress in this area of analysis alwaysproduces important advances in scientific problems where large and complicatedsets of data need to be analyzed to search for ordered patterns, reduceunnecessary information, or visualize intricate structures.