Vol. 67, No. 1, 1976

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ISSN: 0030-8730
Inclusion relations between power methods of limitation

Abraham Ziv

Vol. 67 (1976), No. 1, 251–275
Abstract

Let p(x) = pkxk be a power series with pk (k = 0,1,) complex numbers and 0 < ρp its radius of convergence, and assume that P(x)0 for 0 αp x < ρp. The power method of limitation, P, is defined by

            ∞∑      k
limp s = xl→im0p−   pkskx ∕P (x) (x real)
k=0

(provided the series converges in [αpp) and the limit exists and is finite). Abel and Borel methods are the best known power methods. In this article inclusion relations between two power methods are investigated. Several theorems are proved, which lead to necessary and sufficient conditions, for inclusion, that are correct under some fairly moderate restrictions.

Mathematical Subject Classification 2000
Primary: 40D25
Milestones
Received: 5 December 1974
Revised: 9 November 1975
Published: 1 November 1976
Authors
Abraham Ziv