Multilinear Operators, Discrete Decompositions, and Spectral Resolution of Nanostructures

摘要：本文的研究内容包括与多线性奇异积分相关的算子分析和离散函数空间的研究。调查将基于与Littlewood-Paley理论，分子分解和时频分析工具相关的技术。在算子分析中提出的具体方面是继续合作发展线性calderon - zygmund理论的多线性对应物，并研究各种类型的多线性伪微分算子。离散函数空间分析中的问题包括控制均值振荡函数的采样问题和带限信号的逼近问题。该提案还包含一个跨学科的部分。特别地，在生物体的结构有色组织的光散射的解析公式中产生的理论问题将被考虑。将研究准有序几何的光谱分辨率和数学性质。研究的最后一部分将由数值计算和数据可视化辅助。与奇异积分相关的算子作为分析中的技术工具出现，也作为某些物理现象的数学建模中遇到的转换。这样的转换可以用来描述函数属性的变化，或者从系统的输入到输出的转换。函数或信号的性质通常需要从数据样本中编码的信息来理解。这种信息可以通过函数空间进行量化，并通过傅里叶分析和相关时频技术进行解码。傅里叶分析是衍射物理棱镜的数学版本。它把信号分解成不同振幅和振荡的波谱，就像棱镜把一束光衍射成不同波长的彩虹一样。分析中的现代分解技术为处理复杂信息提供了一种通用语言。这一分析领域的进步总是在科学问题上产生重要的进展，在这些问题上，需要分析大量复杂的数据集，以寻找有序的模式，减少不必要的信息，或将复杂的结构可视化。

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.