Mathematical Sciences: Harmonic Analysis and Fractal Spectral Asymptotics

该项目将涉及“分形”问题的研究 谱渐近”，即，谱渐近性 分形几何中物体的扩展。 这项工作 结合了数学研究的两个重要领域：谐波 分析谱理论和分形几何。 校长 研究人员发现了一个原型分形Plancherel公式 对于欧几里德空间中的傅立叶变换，他给出了一个更 精细分析自相似分形，他推广了 黎曼流形上热方程展开式的结果。 该提案的研究将把这项工作扩展到一些领域， 方向 特别是，首席研究员将获得更多 傅立叶展开的精确信息，他将概括 他以前的结果分层幂零李群，小波 由边值问题引起的扩张和扩张。

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.