Universelle Approximation durch Riesz-Transformierte der geometrischen Reihe

Abstract

Let p={p_v} 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={m_k} of nonnegative integers. The matrix A=(α_kv) 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⊂¯G^c 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⊂G^c 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.