Let us suppose that
formula here
with
formula here
converges for s>0. If the series converges to sum A for
s = 0, then f(s)→A as s→0+,
but the converse will not hold without an additional, Tauberian, hypothesis. The
usual form of such a hypothesis is a condition on {an},
but in the ‘high-indices’ theorem of Hardy and Littlewood [4],
the hypothesis is a condition on {λn}. They
showed that if
formula here
then the existence and finiteness of lims→0+f(s)
imply the convergence of [sum ]an.
The condition (1.3) is usually referred to as the ‘Hadamard gap condition’ and a
series (1.1) such that {λn} satisfies (1.3) is called
a lacunary series, although this
terminology is occasionally used for series satisfying other gap conditions.
Zygmund has shown [15] that if the function f(s),
defined as in (1.1) by a lacunary
series, is of bounded variation in (0, ∞), then [sum ]an
converges absolutely. Clearly, the converse implication holds.
The result of Zygmund has the form
formula here
an uppercase letter with subscripts here denoting, as it will throughout this paper, a
constant determined by the indicated subscripts.
The result of Zygmund and that of Hardy and Littlewood are the extreme cases,
corresponding to m = 1 and m = ∞ of the inequality
formula here
which holds for m[ges ]1 and which, with more general results, is to be found in the first
part of this paper [8].
In all of these results, the interval of integration can be reduced to a finite interval
(0, Bq,λ1).