Mathematical Sciences: Harmonic Analysis and Fractal Spectral Asymptotics

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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.