A Quadrature Formula for Diffusion Polynomials Corresponding to a Generalized Heat Kernel

Abstract

Let {φ _k } be an orthonormal system on a quasi-metric measure space ${\mathbb{X}}$, {ℓ _k } be a nondecreasing sequence of numbers with lim _k→∞ ℓ _k =∞. A diffusion polynomial of degree L is an element of the span of {φ _k :ℓ _k ≤L}. The heat kernel is defined formally by $K_{t}(x,y)=\sum_{k=0}^{\infty}\exp(-\ell _{k}^{2}t)\phi_{k}(x)\overline{\phi_{k}(y)}$. If T is a (differential) operator, and both K _t and T _y K _t have Gaussian upper bounds, we prove the Bernstein inequality: for every p, 1≤p≤∞ and diffusion polynomial P of degree L, ‖TP‖ _p ≤c ₁ L ^c‖P‖ _p . In particular, we are interested in the case when ${\mathbb{X}}$ is a Riemannian manifold, T is a derivative operator, and $p\not=2$. In the case when ${\mathbb{X}}$ is a compact Riemannian manifold without boundary and the measure is finite, we use the Bernstein inequality to prove the existence of quadrature formulas exact for integrating diffusion polynomials, based on an arbitrary data. The degree of the diffusion polynomials for which this formula is exact depends upon the mesh norm of the data. The results are stated in greater generality. In particular, when T is the identity operator, we recover the earlier results of Maggioni and Mhaskar on the summability of certain diffusion polynomial valued operators.