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Session III.2 - Approximation Theory

Wednesday, June 21, 15:30 ~ 16:00

Gaussian bounds for the heat kernel associated to prolate spheroidal wave functions with applications

Pencho Petrushev

University of South Carolina, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Gaussian upper and lower bounds are established for the heat kernel associated to the prolate spheroidal wave functions (PSWFs) of order zero. This result follows from a general principle that relates semigroups associated to a self-adjoint operator and its perturbation and their kernels. As an application of this general result we also establish the Gaussian bounds for the heat kernels associated to generalized univariate PSWFs and PSWFs on the unit ball in $\mathbb{R}^d$. Further, we develop the related to the PSWFs of order zero smooth functional calculus, which in turn is the necessary ground work in developing the theoryof Besov and Triebel- Lizorkin spaces associated with the PSWFs. One of our main results on Besov and Triebel-Lizorkin spaces associated to the PSWFs asserts that they are the same as the Besov and Triebel-Lizorkin spaces generated by the Legendre operator.

Joint work with Aline Bonami (Institut Denis Poisson, Universit\'e d’Orl\'eans, France) and Gerard Kerkyacharian (University Paris Diderot-Paris 7, LPMA, France).

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