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A remark on the centered n -dimensional Hardy-Littlewood maximal function

J. M. Aldaz (2000)

Czechoslovak Mathematical Journal

We study the behaviour of the n -dimensional centered Hardy-Littlewood maximal operator associated to the family of cubes with sides parallel to the axes, improving the previously known lower bounds for the best constants c n that appear in the weak type ( 1 , 1 ) inequalities.

A strengthening of a theorem of Marcinkiewicz

Konstantin E. Tikhomirov (2011)

Banach Center Publications

We consider a problem of intervals raised by I. Ya. Novikov in [Israel Math. Conf. Proc. 5 (1992), 290], which refines the well-known theorem of J. Marcinkiewicz concerning structure of closed sets [A. Zygmund, Trigonometric Series, Vol. I, Ch. IV, Theorem 2.1]. A positive solution to the problem for some specific cases is obtained. As a result, we strengthen the theorem of Marcinkiewicz for generalized Cantor sets.

Approach regions for the square root of the Poisson kernel and boundary functions in certain Orlicz spaces

M. Brundin (2007)

Czechoslovak Mathematical Journal

If the Poisson integral of the unit disc is replaced by its square root, it is known that normalized Poisson integrals of L p and weak L p boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning and the author, respectively. In this paper we characterize the approach regions for boundary functions in two general classes of Orlicz spaces. The first of these classes contains spaces L Φ having the property L ∞ ⊂ L Φ ⊂ L p , 1 ≤ p < ∞ . The second contains spaces L Φ that...

Convergence of series of dilated functions and spectral norms of GCD matrices

Christoph Aistleitner, István Berkes, Kristian Seip, Michel Weber (2015)

Acta Arithmetica

We establish a connection between the L² norm of sums of dilated functions whose jth Fourier coefficients are ( j - α ) for some α ∈ (1/2,1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L² and for the almost everywhere convergence of series of dilated functions.

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.

Efficient computation of delay differential equations with highly oscillatory terms

Marissa Condon, Alfredo Deaño, Arieh Iserles, Karolina Kropielnicka (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This paper is concerned with the asymptotic expansion and numerical solution of systems of linear delay differential equations with highly oscillatory forcing terms. The computation of such problems using standard numerical methods is exceedingly slow and inefficient, indeed standard software is practically useless for this purpose. We propose an alternative, consisting of an asymptotic expansion of the solution, where each term can be derived either by recursion or by solving a non-oscillatory...

Efficient computation of delay differential equations with highly oscillatory terms

Marissa Condon, Alfredo Deaño, Arieh Iserles, Karolina Kropielnicka (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is concerned with the asymptotic expansion and numerical solution of systems of linear delay differential equations with highly oscillatory forcing terms. The computation of such problems using standard numerical methods is exceedingly slow and inefficient, indeed standard software is practically useless for this purpose. We propose an alternative, consisting of an asymptotic expansion of the solution, where each term can be derived either by recursion or by solving a non-oscillatory...

Efficient computation of delay differential equations with highly oscillatory terms

Marissa Condon, Alfredo Deaño, Arieh Iserles, Karolina Kropielnicka (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is concerned with the asymptotic expansion and numerical solution of systems of linear delay differential equations with highly oscillatory forcing terms. The computation of such problems using standard numerical methods is exceedingly slow and inefficient, indeed standard software is practically useless for this purpose. We propose an alternative, consisting of an asymptotic expansion of the solution, where each term can be derived either by recursion or by solving a non-oscillatory...

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