Θ-summability of Fourier series

A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ we show that if the maximal Fejér operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means of these Fourier series is bounded from H _p to L _p (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function f∈L ₁ converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallée-Poussin, Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.     

The maximal Fejér operator on real Hardy spaces

We prove that the maximal Fej'er operator is not bounded on the real Hardy spaces H ¹, which may be considered over and . We also draw corollaries for the corresponding Hardy spaces over ² and ². This revised version was published online in August 2006 with corrections to the Cover Date.     

The length of harmonic forms on a compact Riemannian manifold

We study -dimensional Riemannian manifolds with harmonic forms of constant length and first Betti number equal to showing that they are -step nilmanifolds with some special metrics. We also characterize, in terms of properties on the product of harmonic forms, the left-invariant metrics among them. This allows us to clarify the case of equality in the stable isosytolic inequalities in that setting. We also discuss other values of the Betti number.

     

Production of biologically inert Teflon thin layers on the surface of allergenic metal objects by pulsed laser deposition technology

Allergic-type diseases are current nowadays, and they are frequently caused by certain metals. We demonstrated that the metal objects can be covered by Teflon protective thin layers using a pulsed laser deposition procedure. An ArF excimer laser beam was focused onto the surface of pressed PTFE powder pellets; the applied fluences were 7.5–7.7 J/cm². Teflon films were deposited on fourteen-carat gold, silver and titanium plates. The number of ablating pulses was 10000. Post-annealing of the films was carried out in atmospheric air at oven temperatures between 320 and 500 °C. The thickness of the thin layers was around 5 μm. The prepared films were granular without heat treatment or after annealing at a temperature below 340 °C. At 360 °C a crystalline, contiguous, smooth, very compact and pinhole-free thin layer was produced; a melted and re-solidified morphology was observed above 420 °C. The adhesion strength between the Teflon films and the metal substrates was determined. This could exceed 1–4 MPa depending on the treatment temperature. It was proved that the prepared Teflon layers can be suitable for prevention of contact between the human body and allergen metals and so for avoidance of metal allergy. Received: 12 June 2002 / Accepted: 13 June 2002 / Published online: 4 November 2002 RID="*" ID="*"Corresponding author. E-mail: bhopp@physx.u-szeged.hu     

Unwinding of DNA polymerases by the antitumor drug,cis-diamminedichloroplatinum(II)

The helix-turn-helix motifs of the DNA binding domains of human polymerase-alpha and polymerase-kappa are dramatically perturbed upon binding to cisplatin with concomitant release of zinc.     

Development of chemically modified glass surfaces for nucleic acid,protein and small molecule microarrays

Microarrays have become a widely used tool to investigate the living cell at different levels. DNA microarrays enable the expression analysis of thousand of genes simultaneously, while protein arrays investigate the properties and interactions of proteins with other proteins and with non-proteinaceous molecules. One crucial step in producing such microarrays is the permanent immobilization of samples on a solid surface. Our goal was to develop diverse linker systems capable of anchoring different biological samples, especially DNA and drug-like small molecules. We developed 6 different chemical surfaces having a 3-D-like linker system for biomolecule immobilization, and compared them to previously described immobilization strategies. The attachment chemistry utilizes the amino reactive properties of acrylic and epoxy functions. The capacity of the support was increased by creating a branching structure holding the reactive functions. The method of anchoring was investigated through a model reaction. From HPLC and mass spectrometry measurements we concluded that the covalent binding of DNA occurs through nucleobases. The tested systems offer the capability to permanently immobilize several biomolecular species in an array format.     

(C,α) Summability of Walsh—Fourier Series

The one-dimensional dyadic martingale Hardy spaces H _p are introduced and it is proved that the maximal operator of the (C,) means of a Walsh—Fourier series is bounded from H _p to L _p (1/( + 1) < p < ) and is of weak type (L ₁,L ₁). As a consequence, we obtain the summability result due to Fine; more exactly, the (C,) means of the Walsh—Fourier series of a function f L ₁ converge a.e. to the function in question. Moreover, we prove that the (C,) means are uniformly bounded on H _p whenever 1/( + 1) < p < . We define the two-dimensional dyadic hybrid Hardy space H ₁ and verify that the maximal operator of the (C,,) means of a two-dimensional function is of weak type H ₁ ,L ₁). Consequence, the Walsh—Fourier series of every function f H ₁ is (C,,) summable to the function f.     

Extension of a theorem of Fejér to double Fourier-Stieltjes series

A theorem of Fejér states that if a periodic function F is of bounded variation on the closed interval [0, 2π], then the nth partial sum of its formally differentiated Fourier series divided by n converges to π^-1[F(x+0)-F(x-0)] at each point x. The generalization of this theorem for Fourier-Stieltjes series of (nonperiodic) functions of bounded variation is also well known. The aim of the present article is to extend these results to the (m, n)th rectangular partial sum of double Fourier or Fourier-Stieltjes series of a function F(x, y) of bounded variation over the closed square [0, 2π]×[0, 2π] in the sense of Hardy and Krause. As corollaries, we also obtain the following results:

Commensurate water monolayer at the RuO2(110)/water interface

We demonstrate the presence of two types of commensurate, registered water monolayers with different densities at the RuO2(110)/bulk-water (0.1 M NaOH solution) interface with off-specular, oxygen crystal truncation rods. At anodic potentials (close to oxygen evolution), the extraneous water layer and the surface hydroxide layer form a bilayer with O-H-O bond distances similar to that of ice X. At cathodic potentials, the water molecules converted from the bridging OH molecules form a low-density water layer.