Mathematical Sciences: Harmonic Analysis and Fractal Spectral Asymptotics

该项目将涉及“分形谱渐近性”问题的研究，即在分形几何学中出现的物体的谱展开的渐近行为。这项工作结合了数学研究的两个重要领域：调和分析-谱理论和分形几何。主要研究人员发现了欧氏空间中傅里叶变换的原型分形普朗彻尔公式，对自相似分形给出了更精细的分析，并将结果推广到黎曼流形上的热方程展开式。提案研究将在多个方向上扩展这项工作。特别是，主要研究者将获得关于傅立叶展开的更精确的信息，并且他将把他以前的结果推广到分层幂零李群、小波展开和由边值问题引起的展开。

The project will involve the study of problems in "fractal spectral asymptotics", i.e., the asymptotic behavior of spectral expansions of objects arising in fractal geometry. This work unites two important areas of mathematical research: harmonic analysis-spectral theory and fractal geometry. The principal investigator has found a proto-type fractal Plancherel formula for Fourier transforms in Euclidean space, he has given a more refined analysis for self-similar fractals and he has generalized the results to heat equation expansions on Riemannian manifolds. The proposal research will extend this work in a number of directions. In particular, the principal investigator will obtain more precise information on Fourier expansions, and he will generalize his previous results to stratified nilpotent Lie groups, wavelet expansions and expansions arising from boundary value problems.