Topics in Fourier Analysis

摘要：本文拟研究傅立叶分析中与奇异积分算子、具有退化正则关系的傅立叶积分算子、Radon变换、曲线和曲面上的平均值以及相关的极大函数、波传播的规律性、非紧流形上的谱乘子以及这些主题之间的相互关系有关的许多主题。傅里叶分析的主题为数学及其应用的各个领域提供了强大的工具。P.I.建议进一步发展和完善其中的一些工具。这将导致对数学物理中偏微分方程研究中出现的积分算子，以及积分几何和层析成像研究中出现的积分算子有更好的定性理解。提高这种认识的潜在具体影响是巨大的。例如，断层扫描不仅为医学成像中经常使用的重建方法提供了数学基础，而且为解释地震数据提供了关键，使其具有不可夸大的“现实世界”意义。

Proposal: DMS-9970042Principal Investigator: Andreas SeegerAbstract: The proposed research will be concerned with a number of topics in Fourier analysis related to singular integral operators, Fourier integral operators with degenerate canonical relations, Radon transforms, averages over curves and surfaces and associated maximal functions, regularity of wave propagation, spectral multipliers on noncompact manifolds, and the multitude of interrelations between these topics.The subject of Fourier analysis has provided powerful tools for various areas of mathematics and its applications. The P.I. proposes to develop further and refine some of these tools. This should lead to a better qualitative understanding of integral operators that arise in the study of partial differential equations from mathematical physics, as well as in the study of integral geometry and tomography. The potential concrete implications of improving this understanding are enormous. Tomography, for instance, not only provides the mathematical foundation for the reconstruction methods that are employed constantly in medical imaging but also furnishes the key to interpreting seismic data, lending it a "real world" significance that cannot be overstated.