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Filbir, F. ; Mhaskar, H.N.*

Marcinkiewicz-Zygmund measures on manifolds.

J. Complex. 27, 568-596 (2011)
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Let X be a compact, connected, Riemannian manifold (without boundary), rho be the geodesic distance on X, mu be a probability measure on X, and {phi(k)} be an worthonormal (with respect to mu) system of continuous functions, phi(0)(x) = 1 for all x is an element of X, {l(k)}(k=0)(infinity) be a nondecreasing sequence of real numbers with l(0) = 1, l(k) up arrow infinity as k -> infinity, Pi(L) := span {phi(j) : l(j) <= L}, L >= 0. We describe conditions to ensure an equivalence between the L(p) norms of elements of Pi(L). with their suitably discretized versions. We also give intrinsic criteria to determine if any system of weights and nodes allows such inequalities. The results are stated in a very general form, applicable for example, when the discretization of the integrals is based on weighted averages of the elements of Pi(L) on geodesic balls rather than point evaluations.
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Publikationstyp Artikel: Journalartikel
Dokumenttyp Wissenschaftlicher Artikel
Schlagwörter Data defined manifolds; Marcinkiewicz-Zygmund inequalities; Quadrature formulas; Discretization inequalities
ISSN (print) / ISBN 0885-064X
e-ISSN 0885-064X
Zeitschrift Journal of Complexity
Quellenangaben Band: 27, Heft: 6, Seiten: 568-596 Artikelnummer: , Supplement: ,
Verlag Elsevier
Begutachtungsstatus Peer reviewed