On the core of weighted means of sequences

Abstract

Let(p(n)) be a sequence of nonnegative numbers such that p(0)>0 and P-n:= Sigma(n)(k=0) pk. The sequence (t(n)) of n-th weighted means of a equence(u(n)) is defined by t(n):=1/P-n (k=0)Sigma(n)p(k)u(k) (n=0,1,2, ...). It is well-known from the Knopp's core theorem that K - core(t) subset of K-core(u) for every real sequence(u(n)). But the converse of this inclusion is not true in general. In this paper, we obtain sufficient conditions under which the converse inclusion holds.