manuscripta mathematica

, Volume 36, Issue 3, pp 309–321

Universelle Approximation durch Riesz-Transformierte der geometrischen Reihe

  • Karin Faulstich
  • Wolfgang Luh
  • Ludwig Tomm
Article

Abstract

Let p={pv} be a fixed sequence of complex numbers. Define\(p_n : = \mathop \Sigma \limits_{\nu = o}^n p_\nu \) and suppose that\(p_{m_k } \ne o\) for a subsequence M={mk} of nonnegative integers. The matrix A=(αkv) with the elements
$$\alpha _{k\nu } = p_\nu /p_{m_k } if o \leqslant \nu \leqslant m_k ,\alpha _{k\nu } = oif \nu > m_k $$
generates a summability method (R,p,M) which is a refinement of the well known Riesz methods. The (R,p,M) methods have been introduced in [4].

In the present paper we are concerned with the summability of the geometric series\(\mathop \Sigma \limits_{\nu = o}^n z^\nu \) by (R,p,M) methods. We prove the following theorem. Suppose G is a simply connected domain with\(\{ z:|z|< 1\} \subset G,1 \varepsilon | G \). Then there exists a universal, regular (R,p,M) method having the following properties: (1)\(\mathop \Sigma \limits_{\nu = o}^\infty z^\nu \) is compactly summable (R,p,M) to\(\tfrac{1}{{1 - z}}\) on G. (2) For every compact set B⊂¯Gc which has a connected complement and for every function f which is continuous on B and analytic in its interior there exists a subsequence M(B,f) of M such that\(\mathop \Sigma \limits_{\nu = o}^\infty z^\nu \) is uniformly summable (R,p,M(B,f)) to f(z) on B. (3) For every open set U⊂Gc which has simply connected components in ℂ and for every function f which is analytic on U there exists a subsequence M(U,f) of M such that\(\mathop \Sigma \limits_{\nu = o}^\infty z^\nu \) is compactly summable (R,p,M(U,f)) to f(z) on U.

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Literaturverzeichnis

  1. [1]
    ALEXITS, G.: Konvergenzprobleme der Orthogonalreihen. VEB Deutscher Verlag der Wissenschaften, Berlin 1960MATHGoogle Scholar
  2. [2]
    BIRKHOFF, G.D.: Démonstration d'un théorème élémentaire sur les fonctions entières. C.R. Acad. Sci. Paris189 (1929), 473–475MATHGoogle Scholar
  3. [3]
    CHUI, C.K. und PARNES, M.N.: Approximation by over-convergence of a power series. J. Math. Anal. Appl.36 (1971), 693–696MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    FAULSTICH, K.: Summierbarkeit von Potenzreihen durch Riesz-Verfahren mit komplexen Erzeugendenfolgen. Mitt. Math. Sem. Gießen, Heft139, (1979)Google Scholar
  5. [5]
    HEINS, M.: A universal Blaschke product. Arch. Math.6 (1955), 41–44MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    LORENTZ, G.G.: Bernstein polynomials. In: Mathematical Expositions, no. 8, University of Toronto Press, Toronto, 1953Google Scholar
  7. [7]
    LUH, W.: Approximation analytischer Funktionen durch überkonvergente Potenzreihen und deren Matrix-Transformierten. Mitt. Math. Sem. Gießen, Heft88 (1970)Google Scholar
  8. [8]
    LUH, W.: Über den Satz von Mergelyan. J. Approximation Theory16, No. 2 (1976), 194–198MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    LUH, W.: Über die Summierbarkeit der geometrischen Reihe. Mitt. Math. Sem. Gießen, Heft113 (1974)Google Scholar
  10. [10]
    LUH, W.: Über cluster sets analytischer Funktionen. Acta Math. Acad. Sei. Hung.33 (1–2), (1978), 137–142MathSciNetMATHGoogle Scholar
  11. [11]
    LUH, W.: und TRAUTNER, R.: Summiertaarkeit der geometrischen Reihe auf vorgeschriebenen Mengen. Manuscripta Math.18 (1976), 317–326MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    MARCINKIEWICZ, J.: Quelques théorèmes sur les séries orthogonales. Ann. Soc. Polon. Math.16 (1938), 84–96MATHGoogle Scholar
  13. [13]
    MENCHOFF, D.: Über die Partialsummen der trigonometrischen Reihen (Russian). Mat. Sb.20 (62), (1947), 197–238MathSciNetGoogle Scholar
  14. [14]
    MERGELYAN, S.N.: Uniform approximations of functions of a complex variable (Russian). Uspehi Mat. Nauk. (N.S.)7, no. 2 (48), (1952), 31–122MathSciNetMATHGoogle Scholar
  15. [15]
    PAL, J.: Zwei kleine Bemerkungen. Tôhoku Math. J.6 (1914/15), 42–43Google Scholar
  16. [16]
    RUDIN, W.: Real und complex analysis. McGraw-Hill. New York-Toronto-London, 1966MATHGoogle Scholar
  17. [17]
    SEIDEL, W. und WALSH, J.L.: On approximation by euclidean and noneuclidean translations of analytic functions. Bull. Amer. Math. Soc.47 (1941), 916–920MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    SMIRNOV, V.I. und LEBEDEV, N.A.: Functions of a complex variable: Constructive theory. In: The M.I.T. Press. Cambridge, Mass. 1968MATHGoogle Scholar
  19. [19]
    TALALYAN, A.A.: Über die Konvergenz fast überall von Teilfolgen der Partialsummen allgemeiner Orthogonalreihen (Russian). Akad. Nauk. Armjan. SSR Dokl.10 (1957), 17–34MathSciNetMATHGoogle Scholar
  20. [20]
    TOMM, L.: Über die Summierbarkeit der geometrischen Reihe mit regulären Verfahren. Dissertation, Ulm 1979Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Karin Faulstich
    • 1
  • Wolfgang Luh
    • 1
  • Ludwig Tomm
    • 2
  1. 1.Fachbereich IV/MathematikUniversität TrierTrier
  2. 2.Abteilung für Mathematik IVUniversität UlmUlm

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