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# On linear summation methods for Jacobi series in the case of half integer α

Analysis Mathematica
Author: с. кАльНЕИ

## Abstract

An upper estimate is proved for the Lebesgue function with respect to Jacobi polynomials

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$P_m^{(\alpha ,\beta )} (x)$$ \end{document}
in the case of half integerα and it is expressed in terms of the matrix coefficients determining the linear summation method. The author also proves the analogue of the well-known theorem by S. M. Nikol'skii on the necessary and sufficient condition for the summability of trigonometric Fourier series.

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# On the divergence of linear summation methods of Fourier series

Analysis Mathematica
Author: M. A. Skopina

Пусть функция ϕ задан а на [0, 1],f∈L(0,2π),

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sigma _{\varphi , m} (f, x) = \sum\limits_{k = - n}^n {\varphi \left( {\frac{{\left| k \right|}}{n}} \right) } \hat f_k e^{ikx} ,$$ \end{document}
гдеf k — коэффициенты Ф урье функцииf. Получе ны условия на ϕ, при котор ых существует такая функцияf∈L(0, 2π), чт о последовательност ь {σϕ, n(f, x)} расходится для почти всехx∈(0,2π).

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# On necessary and sufficient conditions of the regularity of summation methods for Laguerre-Fourier series

Analysis Mathematica
Author: M. D. Burmistrova

## Summary

The problem of convergence of linear means is considered for the Laguerre-Fourier  series of continuous functions. An upper estimate is obtained for the Laguerre-Lebesgue function  in terms  of the entries of the matrix which determines the linear summability method  in question.  This allows us to prove for such series an analogue of the well-known theorem by S. M. Nikol'skii which provides necessary and sufficient conditions for the summability  of trigonometric Fourier series. A theorem on the regularity of the summability methods  is also  established.

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# On the universal A.S. central limit theorem

Acta Mathematica Hungarica
Author: S. Hörmann

## Abstract

Let (X k) be a sequence of independent r.v.’s such that for some measurable functions gk : R kR a weak limit theorem of the form

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$g_k (X_1 , \ldots ,X_k )\xrightarrow{\mathcal{L}}G$$ \end{document}
holds with some distribution function G. By a general result of Berkes and Csáki (“universal ASCLT”), under mild technical conditions the strong analogue
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac{1} {{D_N }}\sum\limits_{k = 1}^N {d_k I\left\{ {g_k (X_1 , \ldots ,X_k ) \leqq x} \right\} \to G(x)} a.s.$$ \end{document}
is also valid, where (d k) is a logarithmic weight sequence and D N = ∑k=1 N d k. In this paper we extend the last result for a very large class of weight sequences (d k), leading to considerably sharper results. We show that logarithmic weights, used traditionally in a.s. central limit theory, are far from optimal and the theory remains valid with averaging procedures much closer to, in some cases even identical with, ordinary averages.

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# On summability of weighted Lagrange interpolation. I

Acta Mathematica Hungarica
Authors: László Szili and Péter Vértesi
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# Possibility of deriving kinetic parameters from integrated DTA peaks

Journal of Thermal Analysis and Calorimetry

Three DTA peaks (two endo and one exothermic) were replotted in the form of ∑ peak area.Δa, or ∑ peak magnitude,ΔT, as a function of temperature. The integral sigmoid curves were plotted in the form of logg(α) vs. 1/T or log logg(α)/T 2 vs. 1/T. Both of the proposed summation methods gave satisfactory straight lines (F 1 function), characterized by the same activation energies, correlation coefficients and standard deviations. Integration of the peak areas by Simpson's rule resulted in the same values as obtained by the summation procedure. Analysis by the suggested integral method resulted in activation energies that show a logarithmic divergence relative to the magnitude ofE a estimated directly from the DTA peaks.

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# On summability of weighted Lagrange interpolation. II

Acta Mathematica Hungarica
Authors: László Szili and Péter Vértesi

## Abstract

The aim of this paper is to continue our investigations started in [15], where we studied the summability of weighted Lagrange interpolation on the roots of orthogonal polynomials with respect to a weight function w. Starting from the Lagrange interpolation polynomials we constructed a wide class of discrete processes which are uniformly convergent in a suitable Banach space (C ρ, ‖‖ρ) of continuous functions (ρ denotes (another) weight). In [15] we formulated several conditions with respect to w, ρ, (C ρ, ‖‖ρ) and to summation methods for which the uniform convergence holds. The goal of this part is to study the special case when w and ρ are Freud-type weights. We shall show that the conditions of results of [15] hold in this case. The order of convergence will also be considered.

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# Contribution to the use of data processing to establish instrumental multi-element activation analysis as a routine method

Journal of Radioanalytical and Nuclear Chemistry
Authors: A. Gierl and F. Lux

## Abstract

A computer program has been worked out for the following requirements: (1) The program is to render a listing of the elements detected, together with their respective concentrations, not just an evaluation of the gamma-spectrum (peak energies and areas). (2) There should be no necessity to intermediate decisions, i.e. execution of the program should be possible by auxiliary personnel. (3) Gamma-ray spectra recorded under widely different conditions should be amenable to evaluation. This implies a large range of variation of the number of channels per peak. (4) In order to have the most extensive capability of executing multi-element analyses instrumentally, it must be possible to evaluate complex spectra with many superpositions. The program involves the fitting of Gaussians. It is shown that this evaluation method gives more precise peak area determinations than the summation method and also yields reasonable results in the case of strong superposition.

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# Seismic ground motion and site effect modelling along two profiles in the city of Debrecen, Hungary

Acta Geodaetica et Geophysica Hungarica
Authors: K. Gribovszki and F. Vaccari

The aim of our study is to determine the design ground acceleration values at different parts of Debrecen along two profiles crossing the city. Synthetic seismograms are computed by the so called “hybrid technique”. This technique consists of the modal summation method, followed by finite difference modelling. Two independent computations have been performed using two different seismic sources and profiles. In both computations the seismic sources have been located in the so called “Mobile Zone”. The Mobile Zone is a seismically active fault system between the villages of Hosszúpályi and Gálospetri. The focal mechanism and the homogeneous and heterogeneous parts of the profile are known from geophysical and geological data of the investigated area. The maximum response spectra ratio values of the horizontal component are found below 1 Hz all along the profile and the frequencies below 1 Hz are in good agreement with the natural frequencies of the multi-storeyed buildings. Computed effective peak acceleration (EPA) values are found to be in good agreement also with the higher than 6° MSK macroseismic intensity values estimated from the assessment records of damages wrought by the 1834 Érmellék earthquake.

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# Seismic microzonation with the use of GIS - Case study for Debrecen, Hungary

Acta Geodaetica et Geophysica Hungarica
Authors: K. Gribovszki and G. F. Panza

The aim of our study is to determine the design ground acceleration values at the whole territory of Debrecen and to accomplish the seismic risk map of Debrecen using synthetic seismograms. Synthetic seismograms are computed by the so called hybrid technique" along 11 different profiles crossing the city. The hybrid technique consists of the modal summation method, followed by finite difference modelling. 11 independent computations have been performed using the same seismic source but different profiles. The seismic source has been located in the so called “Mobile Zone”, which is a seismically active fault system in Érmellék region. The focal mechanism and the homogeneous and heterogeneous parts of the profiles are  known from geophysical and geological data of the investigated area. As the results of the computations PGA grid maps of Debrecen for the 3 different components and the spectral  acceleration (response spectra, SA) charts of the synthetic seismograms for the transversal components came  into existence. The seismic risk map of the city has been completed from the SA charts created from the synthetic seismograms and from the grid map of the buildings in Debrecen with different number of floors by applying GIS tools.

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