## Some classical problems: The heat kernel point of view

- April 10, 2013
- 3:30 p.m.
- LeConte 312

## Abstract

It is a classical topic to look to orthonormal basis of polynomials on a compact set $X$ of $ R^d$, with respect to some Radon measure $\mu$. For example: the one dimensional interval (Jacobi), the unit sphere (Spherical harmonics), the ball and the simplex (work of Petrushev, Xu, ...). In this framework, one can be interested in the best approximation of functions by polynomials of fixed degree, in $L _ p(\mu)$, and to build a suitable frame for characterization of function spaces related to this approximation. This constructions have been carried using special functions estimates.

Actually, we will be interested by spaces where the polynomials give the spectral spaces of some positive selfadjoint operator. Under suitable conditions, a "natural" metric $\rho$ could be defined on $X$ such that $(X, \rho, \mu)$ is a homogeneous space, and if the associated semi-group has a good "Gaussian" behavior, then we could apply the procedure developed in recent works by P. Petrushev, T. Coulhon and G.K., to built such frames, and such function spaces.

Joint work with P. Petrushev and Y. Xu.