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ON APPROXIMATION OF A FUNCTION AND ITS DERIVATIVES BY A POLYNOMIAL AND ITS DERIVATIVES

Analysis in Theory and Applications 1994年第3期10卷 93-103页

作者： T.Kilgore j.szabados Aubunn University U.S.A Mathematical Institute of the Hungarian of Academy of Sciences. Hungaria

Let g∈C^q[-1, 1] be such that g^((k))(±1)=0 for k=0,…,q. Let P_n be an algebraic polynomial of degree at most n, such that P_n^((k))(±1)=0 for k=0,…,[_2~ (q+1)]. Then P_n and its derivatives P_n^((k)) for k≤q well approximate g and its respective derivatives, provided only that P_n well approxi- mates g itself in the weighted norm ‖g(x)-P_n(x) (1-x^2)^(1/2)~q‖ This result is easily extended to an arbitrary f∈C^q[-1, 1], by subtracting from f the polynomial of minnimal degree which interpolates f^((0))…,f^((q)) at±1. As well as providing easy criteria for judging the simultaneous approximation properties of a given Polynomial to a given function, our results further explain the similarities and differences between algebraic polynomial approximation in C^q[-1, 1] and trigonometric polynomial approximation in the space of q times differentiable 2π-periodic functions. Our proofs are elementary and basic in character, permitting the construction of actual error estimates for simultaneous approximation proedures for small values of q.

关键词： ON APPROXIMATION OF A FUNCTION AND ITS DERIVATIVES BY A POLYNOMIAL AND ITS DERIVATIVES ITS

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GLOBAL SMOOTHNESS PRESERVATION BY BIVARIATE INTERPOLATION OPERATORS

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Analysis in Theory and Applications 2003年第3期19卷 193-208页

作者： S.G. Gal j. szabados Deptartment of Mathematic University of Oradea Alfréd Rényi Institute of Mathematic Hungarian Academy of Sciences Str.Armatei Komane 53700 Oradea RomaniP.O.Box 127H-1364 Budapest Haungar

Extending the results of [4] in the univariate case, in this paper we prove that the bivariate interpolation polynomials of Hermite-Fejér based on the Chebyshev nodes of the first kind, those of Lagrange based on the Chebyshev nodes of second kind and ±1, and those of bivariate Shepard operators, have the property of partial preservation of global smoothness, with respect to various bivariate moduli of continuity.

关键词： bivariate interpolation polynomials and operators bivariate moduli of continuity global smoothness preservation

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Ultrasound-enhanced milling in the synthesis of phase-pure, highly crystalline ZnAl-layered double hydroxide of low Zn（II） content

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Particuology 2016年第4期14卷 29-33页

作者： M.szabados Cs. Bus M. Adok-Sipiczki Z. Konya A. Kukovecz P. Sipos I. Palinko Department Qf Organic Chemistry University of Szeged Domter 8 Szeged H-6720 Hungary Material and Solution Structure Research Group Institute of Chemistry University of Szeged Aradi Vertanflk tere 1 Szeged H-6720 Hungary Department of Applied and Environmental Chemistry University of Szeged Rerrich B. ter 1 Szeged H-6720. Hungary MTA-SZTE Reaction Kinetics and Surface Chemistry Research Croup Rerrich B ter 1 Szeged H-6720 Hungary MTA-SZTE "Lendulet" Porous Nanocomposites Research Group Rerrich B. ter 1 Szeged H-6720 Hungary Department of lnorganic and Analytical Chemistry University of Szeged Domter 7 Szeged H-6720 Hungary

Dry milling followed by ultrasonic irradiation in the presence of small amounts of aqueous NaOH, was used to synthesise ZnAl-layered double hydroxide （LDH）. The synthetic conditions were optimised and the Zn（OH）2, ZnO, and AI（OH）3 contaminants were removed by rinsing with aqueous NH3 solution from the sample prepared in the optimum conditions. Thus, phase-pure ZnAl-layered double hydroxide was obtained with high crystallinity. All materials produced during the syntheses were characterised by X- ray diffractometry, and a rose-like morphology was observed by scanning electron microscopy in the phase-pure sample. By changing the initial Zn/Al ratio, it was revealed that an LDH with low zinc content was always formed, probably through substitution of some of the Zn2. ions by AI3. ions in the gibbsite lattice.

关键词： ZnAl-layered double hydroxide Dry milling Ultrasonic irradiation

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ON BIRKHOFF INTERPOLATION (0;2) CASE

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Analysis in Theory and Applications 1995年第2期11卷 59-66页

作者： j. Burkett A. K. Varma University of Florida. USA University of Florida. USA

P. Turan and his associates considered in detail the problem of (0.2) interpolation based on the zeros of πn(x). Motivated by these results and an earlier result of szabados and Varma[9] here we consider the problem of existence, uniqueness and explicit representation of the interpolatory polynomial Rn (x) satisfying the function values at one set of nodes and the second derivative on the other set of nodes. It is important to note that this problem has a unique solution provided these two sets of nodes are chosen properly. We also promise to have an interesting convergence theorem in the second paper of this series, which will provide a solution to the related open problem of P. Turan.

关键词： CASE ON BIRKHOFF INTERPOLATION

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