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