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Displaying similar documents to “Some properties of Holmgren-Riesz transform in two dimensions”

«Approximate approximations» and the cubature of potentials

Gunther Schmidt (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The paper discusses new cubature formulas for classical integral operators of mathematical physics based on the «approximate approximation» of the density with Gaussian and related functions. We derive formulas for the cubature of harmonic, elastic and diffraction potentials approximating with high order in some range relevant for numerical computations. We prove error estimates and provide numerical results for the Newton potential.

Riesz means of Fourier transforms and Fourier series on Hardy spaces

Ferenc Weisz (1998)

Studia Mathematica

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Elementary estimates for the Riesz kernel and for its derivative are given. Using these we show that the maximal operator of the Riesz means of a tempered distribution is bounded from H p ( ℝ ) to L p ( ℝ ) (1/(α+1) < p < ∞) and is of weak type (1,1), where H p ( ℝ ) is the classical Hardy space. As a consequence we deduce that the Riesz means of a function ⨍ ∈ L 1 ( ℝ ) converge a.e. to ⨍. Moreover, we prove that the Riesz means are uniformly bounded on H p ( ℝ ) whenever 1/(α+1) < p < ∞. Thus, in case ⨍ ∈ H p ( ℝ ) , the Riesz...