A Quadrature Formula for Diffusion Polynomials Corresponding to a Generalized Heat Kernel
By Frank Filbir, Hrushikesh Narhar Mhaskar. 2010
87 citations recorded by the source corpus.
Let {φk} be an orthonormal system on a quasi-metric measure space ${\mathbb{X}}$, {ℓk} be a nondecreasing sequence of numbers with lim k→∞ℓk=∞. A diffusion polynomial of degree L is an element of the span of {φk:ℓk≤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 Kt and TyKt have Gaussian upper bounds, we prove the Bernstein inequality: for every p, 1≤p≤∞ and diffusion polynomial P of degree L, ‖TP‖p≤c1Lc‖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.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/1f6c5f27942510b6481ac17c5a45e94ad6aae448
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In Applied Mathematics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Quadrature rules and distribution of points on manifolds (2010, background): “We would like to thank the referee for some useful suggestions and especially for bringing to our attention the papers [14] and [27], which have led us to improve the original draft, in particular Corollary 2.”
- Applications of classical approximation theory to periodic basis function networks and computational harmonic analysis (2013, background): “The lemma is proved in much greater generality in [20].”