Simultaneous generic approximation by the iterates of the Cesaro operator

TL;DR

This paper demonstrates that for a generic sequence in a subset of a locally convex space, the iterates of the Cesaro operator can simultaneously approximate any given sequence, showing topological and algebraic genericity.

Contribution

It establishes the topological and algebraic genericity of simultaneous approximation by Cesaro iterates in locally convex spaces.

Findings

Sequences of Cesaro iterates are dense in the convex hull of A.

Simultaneous approximation of arbitrary sequences by Cesaro iterates is topologically generic.

In vector spaces, this approximation property is algebraically generic.

Abstract

We show that for the generic sequence (a) of elements in a subset A of a separable locally convex metrisable space V, the sequences [T^k(a)]_n, n=1,2,... are dense in the convex hull convA of A for all k=1,2,...; where T is the Cesaro operator. Further, if convA is dense in V, then for every sequence x_k of elements of V, k=1,2,... there exists a sequence of indices l_n, n=1,2,..., such that, we have the simultaneous approximation that [T^k(a)]_(l_n) converges to x_k as n tends to infinity for all k=1,2,... These phenomena are topologically generic and in the case where A is a vector space they are algebraically generic.