Mercerian theorems via spectral theory

Abstract

Given a regular matrix A, Mercerian theorems are concerned with determining the real or complex values of a for which al + (1a)A is equivalent to convergence. For al, the problem is equivalent to determining the resolvent set for A, or, determining the spectrum (A) of A, where (A) = { IA -l is not invertible}. This paper treats the problem of determining the spectra of weighted mean methods; i.e., triangular matrices A = (a nk ) with a nkp k /P n , where p 0 > 0, Pn ^ 0, &=o Pk ~ P n It is shown that the spectrum of every weighted mean method is contained in the disc {\\ 1/2} (Theorem 1), and, if lim pJP n exists,