Vol. 21, No. 1, 1967

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ISSN: 0030-8730
Power-series and Hausdorff matrices

Philip C. Tonne

Vol. 21 (1967), No. 1, 189–198
Abstract

The purpose of this paper is to pair classes of continuous functions from [0,1] to the complex numbers with classes of complex sequences. If f is a function from [0,1] to the complex numbers and c is a complex sequence, a sequence L(f,c) is defined:

         ∑n       (n) n∑−p     (n − p)
L(f,c)n =   f(p∕n) p     (− 1)q  q   cp+q.
p=0          q=0

A class A of continuous functions is paired with a class B of sequences provided that

(1) if f is in A and c is in B then L(f,c) converges,

(2) if f is a continuous function and L(f,c) converges for each c in B then f is in A, and

(3) if c is a sequence and L(f,c) converges for each f in A then c is in B.

We establish the following pairings:

CONTINUOUS

SEQUENCES



all continuous functions

Hausdorff moment sequences



power-series absolutely convergent at 1

bounded sequences



power-series absolutely convergent at r(r < 1)

sequences dominated by geometric sequences having ratio r



entire functions

all sequences dominated by geometric sequences



polynomials

all sequences



Mathematical Subject Classification
Primary: 40.20
Milestones
Received: 15 June 1965
Published: 1 April 1967
Authors
Philip C. Tonne