228 research outputs found
Marcinkiewicz--Zygmund measures on manifolds
Let be a compact, connected, Riemannian manifold (without
boundary), be the geodesic distance on , be a
probability measure on , and be an orthonormal system
of continuous functions, for all ,
be an nondecreasing sequence of real numbers with
, as , , . We describe conditions to ensure an
equivalence between the norms of elements of 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 on geodesic
balls rather than point evaluations.Comment: 28 pages, submitted for publicatio
Metric entropy, n-widths, and sampling of functions on manifolds
We first investigate on the asymptotics of the Kolmogorov metric entropy and
nonlinear n-widths of approximation spaces on some function classes on
manifolds and quasi-metric measure spaces. Secondly, we develop constructive
algorithms to represent those functions within a prescribed accuracy. The
constructions can be based on either spectral information or scattered samples
of the target function. Our algorithmic scheme is asymptotically optimal in the
sense of nonlinear n-widths and asymptotically optimal up to a logarithmic
factor with respect to the metric entropy
Com repartir punts uniformement en esferes?
El teorema de WhittakerKotel'nikovShannon
permet reconstruir de manera
estable una funció de banda limitada i energia finita a partir dels valors de la funció als
enters. Així doncs, els enters són un exemple de conjunt de mostreig estable però són,
a més, un conjunt de punts ben distribuïts a la recta. En aquest article tractarem de la
relació entre l'existència de conjunts de punts ben distribuïts i el mostreig estable de
funcions a diferents dominis (la recta, el cercle i l'esfera).How to distribute points uniformly on spheres?
The Whittaker-Kotelnikov-Shannon theorem allows us to recover, in a stable
way, a bandlimited function of finite energy by using only its values on the
integers. Hence, the integers give an example of a set of stable sampling which
is evenly distributed on the real line. In this paper, we consider the relation
between the existence of evenly distributed sets of points and sets of stable
sampling in various settings (the real line, the circle and the sphere)
Asymptotically optimal designs on compact algebraic manifolds
We find -designs on compact algebraic manifolds with a number of points comparable to the dimension of the space of polynomials of degree on the manifold. This generalizes results on the sphere by Bondarenko, Radchenko and Viazovska. Of special interest is the particular case of the Grassmannians where our results improve the bounds that had been proved previously
Marcinkiewicz-Zygmund Inequalities for Polynomials in Bergmann and Hardy Spaces
We study the relationship between sampling sequences in infinite dimensional Hilbert spaces of analytic functions and Marcinkiewicz-Zygmund inequalities in subspaces of polynomials. We focus on the study of the Hardy space and the Bergman space in one variable because they provide two settings with a strikingly different behavior
Boundedness of maximal functions on non-doubling manifolds with ends
Let be a manifold with ends constructed in \cite{GS} and be the
Laplace-Beltrami operator on . In this note, we show the weak type
and boundedness of the Hardy-Littlewood maximal function and of the
maximal function associated with the heat semigroup \M_\Delta f(x)=\sup_{t> 0}
|\exp (-t\Delta)f(x)| on for . The significance of
these results comes from the fact that does not satisfies the doubling
condition
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