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Density methods and results in approximation theory

Allan Pinkus (2004)

Banach Center Publications

Approximation theory and functional analysis share many common problems and points of contact. One of the areas of mutual interest is that of density results. In this paper we briefly survey various methods and results in this area starting from work of Weierstrass and Riesz, and extending to more recent times.

Density questions in the classical theory of moments

Christian Berg, J. P. Reus Christensen (1981)

Annales de l'institut Fourier

Let μ be a positive Radon measure on the real line having moments of all orders. We prove that the set P of polynomials is note dense in L p ( R , μ ) for any p > 2 , if μ is indeterminate. If μ is determinate, then P is dense in L p ( R , μ ) for 1 p 2 , but not necessarily for p > 2 . The compact convex set of positive Radon measures with same moments as μ is studied in some details.

Difference functions of periodic measurable functions

Tamás Keleti (1998)

Fundamenta Mathematicae

We investigate some problems of the following type: For which sets H is it true that if f is in a given class ℱ of periodic functions and the difference functions Δ h f ( x ) = f ( x + h ) - f ( x ) are in a given smaller class G for every h ∈ H then f itself must be in G? Denoting the class of counter-example sets by ℌ(ℱ,G), that is, ( , G ) = H / : ( f G ) ( h H ) Δ h f G , we try to characterize ℌ(ℱ,G) for some interesting classes of functions ℱ ⊃ G. We study classes of measurable functions on the circle group 𝕋 = / that are invariant for changes on null-sets (e.g. measurable...

Diffraction spectra of weighted Delone sets on beta-lattices with beta a quadratic unitary Pisot number

Jean-Pierre Gazeau, Jean-Louis Verger-Gaugry (2006)

Annales de l’institut Fourier

The Fourier transform of a weighted Dirac comb of beta-integers is characterized within the framework of the theory of Distributions, in particular its pure point part which corresponds to the Bragg part of the diffraction spectrum. The corresponding intensity function on this Bragg part is computed. We deduce the diffraction spectrum of weighted Delone sets on beta-lattices in the split case for the weight, when beta is the golden mean.

Disjointness results for some classes of stable processes

Michael Hernández, Christian Houdré (1993)

Studia Mathematica

We discuss the disjointness of two classes of stable stochastic processes: moving averages and Fourier transforms. Results on the incompatibility of these two representations date back to Urbanik. Here we extend various disjointness results to encompass larger classes of processes.

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