We give a natural extension of
the classical definition of Césaro convergence of a divergent sequence/function. This
involves understanding the spectrum of eigenvalues and eigenvectors of a certain
Césaro operator on a suitable space of functions or sequences. The essential
idea is applicable in identical fashion to other summation methods such as
Borel’s. As an example we show how to obtain the analytic continuation of the
Riemann zeta function ζ(z) for ℜ(z) ≤ 1 directly from generalised Césaro
summation of its divergent defining series. We discuss a variety of analytic and
symmetry properties of these generalised methods and some possible further
applications.
