Vol. 26, No. 2, 1968

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Bounded series and Hausdorff matrices for absolutely convergent sequences

Philip C. Tonne

Vol. 26 (1968), No. 2, 415–420
Abstract

If f is a function from [0,1] to the complex plane and c is a complex sequence, then the Hausdorff matrix H(c) for c and a sequence L(f,c) are defined:

H(c)np = (n )
p q=0np(1)q(n − p)
qcp+q
L(f,c)n = p=0nH(c) npf(p∕n).
This paper consists of the following theorem and two converses to it.

Theorem 1. If A is a complex sequence and p=0Ap is bounded (there is a number B such that if n is a nonnegative integer then | p=0nAp| < B), f is a function from [0,1] to the complex plane such that if 0 x < 1 then f(x) = p=0Apxp, and c is an absolutely convergent sequence ( p=0|cp+1 cp| converges), then L(f,c) converges. Furthermore, if c has limit d, L(f,c) has limit p=0Ap(cp d) + f(1)d.

Mathematical Subject Classification
Primary: 30.30
Milestones
Received: 9 January 1968
Published: 1 August 1968
Authors
Philip C. Tonne