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Zur Lototsky — Transformation über kompakten Räumen von Wahrscheinlichkeitsmassen

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Abstract

As is well known, the theory of the classical Bernstein polynomials is connected with the theory of probability on the one hand and with the theory of matrix transformations and summability on the other hand. It is the purpose of the present paper to define and to investigate the Lototsky method of summability on the space of Radon probability measures on a compact topological space T. By the aid of an extended version of the Bohman-Korovkin approximation theorem we shall prove a convergence theorem for the sequence (Ln,ρ,P)n≧1 of so-called Lototsky-Schnabl operators, having ρ as its sequence of “ray functions”. By specializing in an appropriate manner the underlying space T as well as the matrix P of weights, we shall deduce from this general theorem a result concerning the approximation properties of the sequence (Ln,ρ)n≧1 of Lototsky-Bernstein operators acting on the space of real-valued functions which are continuous on a compact N-simplex.

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Literatur

  1. BOURBAKI, N.: Intégration, Chap. I–IV, 2e éd. Paris: Hermann 1965.

    Google Scholar 

  2. BOURBAKI, N.: Intégration, Chap. V, 2e éd. Paris: Hermann 1967.

    Google Scholar 

  3. BOURBAKI, N.: Intégration, Chap. VII, VIII. Paris: Hermann 1963.

    Google Scholar 

  4. FELBECKER, G., and W. SCHEMPP: A generalization of Bohman-Korovkin's theorem. Math. Z.122, 63–70 (1971).

    Google Scholar 

  5. JAKIMOVSKI, A.: A generalization of the Lototsky method of summability. Michigan Math. J.6, 277–290 (1959).

    Google Scholar 

  6. KING, J. P.: The Lototsky transform and Bernstein polynomials. Canad. J. Math.18, 89–91 (1966).

    Google Scholar 

  7. LORENTZ, G. G.: Bernstein polynomials. Toronto: University of Toronto Press 1953.

    Google Scholar 

  8. MEYER-KÖNIG, W., und K. ZELLER: Bernsteinsche Potenzreihen. Studia Math.19, 89–94 (1960).

    Google Scholar 

  9. SCHEMPP, W.: A note on Korovkin test families. (Erscheint demnächst).

  10. SCHNABL, R.: Eine Verallgemeinerung der Bernsteinpolynome. Math. Ann.179, 74–82 (1968).

    Google Scholar 

  11. SCHNABL, R.: Zur Approximation durch Bernsteinpolynome auf gewissen Räumen von Wahrscheinlichkeitsmaßen. Math. Ann.180, 326–330 (1969).

    Google Scholar 

  12. STANCU, D. D.: De l'approximation, par des polynômes du type Bernstein, des fonctions de deux variables. Com. Akad. R. P. Romîne9, 773–777 (1959).

    Google Scholar 

  13. VOGEL, W.: Wahrscheinlichkeitstheorie. Göttingen: Vandenhoeck & Ruprecht 1970.

    Google Scholar 

  14. WOOD, B.: Convergence and almost convergence of certain sequences of positive linear operators. Studia Math.34, 113–119 (1970).

    Google Scholar 

  15. ZELLER, K., und W. BEEKMANN: Theorie der Limitierungsverfahren, 2. Aufl. Berlin-Heidelberg-New York: Springer 1970.

    Google Scholar 

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  1. Institut für Mathematik der Ruhr-Universität Bochum, Buscheystraße NA, D-4630, Bochum

    Walter Schempp

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  1. Walter Schempp
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Schempp, W. Zur Lototsky — Transformation über kompakten Räumen von Wahrscheinlichkeitsmassen. Manuscripta Math 5, 199–211 (1971). https://doi.org/10.1007/BF01443253

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  • DOI: https://doi.org/10.1007/BF01443253

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