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Filbir, F. ; Mhaskar, H.N.*

A quadrature formula for diffusion polynomials corresponding to a generalized heat kernel.

J. Fourier Anal. Appl. 16, 629-657 (2010)
DOI
Open Access Green as soon as Postprint is submitted to ZB.
Let {phi(k)} be an orthonormal system on a quasi-metric measure space X, {l(k)} be a nondecreasing sequence of numbers with lim(k ->infinity)l(k) = infinity. A diffusion polynomial of degree L is an element of the span of {phi(k) : l(k) <= L}. The heat kernel is defined formally by K-t (x, y) = Sigma(infinity)(k=0) exp(-l(k)(2)t)phi(k)(x)phi(k)(y). If T is a (differential) operator, and both K-t and TyKt have Gaussian upper bounds, we prove the Bernstein inequality: for every p, 1 <= p <= infinity and diffusion polynomial P of degree L, parallel to TP parallel to(p) <= c(1)L(c)parallel to P parallel to(p). In particular, we are interested in the case when X is a Riemannian manifold, T is a derivative operator, and p not equal 2. In the case when 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.
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Publication type Article: Journal article
Document type Scientific Article
Corresponding Author
Keywords Approximation on manifolds; Bernstein inequalities; Marcinkiewicz; Zygmund inequalities; Quadrature formulas; ELLIPTIC DIFFERENTIAL-OPERATORS; SCATTERED DATA; MANIFOLDS; SPHERE; WAVELETS; BOUNDS
ISSN (print) / ISBN 1069-5869
Journal Journal of Fourier Analysis and Applications, The
Quellenangaben Volume: 16, Issue: 5, Pages: 629-657 Article Number: , Supplement: ,
Publisher Birkhäuser
Publishing Place Boston, Inc.
Non-patent literature Publications
Reviewing status Peer reviewed
Institute(s) Institute of Biomathematics and Biometry (IBB)