Summierbarkeit der geometrischen Reihe auf vorgeschriebenen Mengen

Abstract

In this paper we are concerned with the summability of the geometric series$\sum\limits_{v = 0}^\infty {z^v }$ by matrix methods. We prove the following theorem: Suppose M_o:={z:|z|<1}, M₁, M₂, ⋯ is a collection of countably many Lebesgue measureable, disjoint sets. For k=1,2,⋯ let f_k be a prescribed function, analytic on$\overline {M_k }$. Then there exists a triangular matrix$V = (c_{n^v } )$, such that the V-transform {σ_n(z)} of the geometric series has the following properties: {σ_n(z)} converges compactly to$\frac{1}{{1 - z}}$ on M_o; for k=1,2,⋯ there are sets B_k, such that$B_{k^\Delta } M_k$ has Lebesgue-measure zero and σ_n(z)→f_k(z) for zεB_k; if$M^ * = [\mathop u\limits_{k \geqq 0} M_k ]^c$ there is a set B^*, such that B^*ΔM^* has Lebesgue-measure zero and {σ_n(z)} diverges for zεB^*.