manuscripta mathematica

, Volume 36, Issue 3, pp 309–321

Universelle Approximation durch Riesz-Transformierte der geometrischen Reihe

  • Karin Faulstich
  • Wolfgang Luh
  • Ludwig Tomm
Article

DOI: 10.1007/BF01322495

Cite this article as:
Faulstich, K., Luh, W. & Tomm, L. Manuscripta Math (1981) 36: 309. doi:10.1007/BF01322495

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.

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