Numerical Methods for PDEs: In Occasion of Raytcho Lazarov's 70th Birthday
- Pencho Petrushev, University of South Carolina
- Gaussian Bounds for the Heat Kernel on the Interval, Ball, and Simplex
We establish Gaussian upper and lower bounds for the heat kernel associated with the Jacobi operator and polynomials on the interval. Gaussian bounds are also proved for the heat kernels associated with orthogonal polynomials and respective operators on the ball and simplex with weights. The general machinery of Dirichlet spaces is utilized in this development, where the local Poincare inequality plays a crucial role. These results are used for the construction of localized frames, which in turn provide a tool for decomposition of weighted Besov and Triebel-Lizorkin spaces in the settings of interest.