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 Mo:={z:|z|<1}, M1, M2, ⋯ is a collection of countably many Lebesgue measureable, disjoint sets. For k=1,2,⋯ let fk be a prescribed function, analytic on\(\overline {M_k } \). Then there exists a triangular matrix\(V = (c_{n^v } )\), such that the V-transform {σn(z)} of the geometric series has the following properties: {σn(z)} converges compactly to\(\frac{1}{{1 - z}}\) on Mo; for k=1,2,⋯ there are sets Bk, such that\(B_{k^\Delta } M_k \) has Lebesgue-measure zero and σn(z)→fk(z) for zεBk; 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 {σn(z)} diverges for zεB*.
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Luh, W., Trautner, R. Summierbarkeit der geometrischen Reihe auf vorgeschriebenen Mengen. Manuscripta Math 18, 317–326 (1976). https://doi.org/10.1007/BF01270492
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DOI: https://doi.org/10.1007/BF01270492