文档介绍：MATHEMATICS
ON THE STRUCTURE OF WELL DISTRIBUTED SEQUENCES (III)
BY
G. M. PETERSEN AND M. T. McGREGOR
(Communicated by Prof. J. PoPKEN at the meeting of November 27, 1965)
l. Firstly, we shall discuss the relation of Tauberian theorems to the
distribution of sequences. We begin with some definitions and notation.
Let (sn) be a sequence of real numbers satisfying 0 < Sn < 1 for every n,
n= 1, 2, 3, .... We take O.;;;a<b< 1 and let l[a,bJ(x) denote the charac•
teristic function of the interval [a, b ], so that
1ifxE[a,b]
{
I [ab ' J( X ) = 0 othel'Wlse..
The sequence (sn) is said to be well distributed 1) if
1 n+p
lim- · ! l[a,bJ(Sk) = b-a
~coP k=n+l
holds uniformly inn for every interval [a, b]. In other words, (sn) is well
distributed if the sequence (I[a,bJ(Sn)) is almost convergent 2) to b-a for
every subinterval [a, b] of the interval [0, 1].
00
Consider the series ! Wk with partial sums
k=l
and let A= (amn) be a regular summability matrix (or the method of
almost convergence d). If A (or d)-summability of the sequence (sn),
together with the condition
(1)