Journal of Fourier Analysis and Applications

, Volume 16, Issue 5, pp 629–657 | Cite as

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

Article

Abstract

Let {φk} be an orthonormal system on a quasi-metric measure space \({\mathbb{X}}\), {k} be a nondecreasing sequence of numbers with lim k→∞k=∞. A diffusion polynomial of degree L is an element of the span of {φk:kL}. 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 Kt and TyKt have Gaussian upper bounds, we prove the Bernstein inequality: for every p, 1≤p≤∞ and diffusion polynomial P of degree L, ‖TPpc1LcPp. 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.

Keywords

Approximation on manifolds Bernstein inequalities Marcinkiewicz-Zygmund inequalities Quadrature formulas 

Mathematics Subject Classification (2000)

41A17 41A55 58J35 58J90 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institute of Biomathematics and BiometryHelmholtz Center MunichNeuherbergGermany
  2. 2.Department of MathematicsCalifornia State UniversityLos AngelesUSA

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