Mercerian theorems via spectral theory

Frank Cass, Billy Rhoades
1977 Pacific Journal of Mathematics  
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 aΦl, the problem is equivalent to determining the resolvent set for A, or, determining the spectrum σ(A) of A, where σ(A) = {λ IA -λl 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, Σ&=o Pk ~ P n It
more » ... Σ&=o Pk ~ P n It is shown that the spectrum of every weighted mean method is contained in the disc {λ\\λ 1/2} (Theorem 1), and, if lim pJP n exists, where e -lim pJP n . Let r = limpJP n , δ = 1STpJP n , S = {pJP n \n^0}. When γ < δ, some examples are provided to indicate the difficulty of determining the spectrum explicitly. It is shown that {λ I U -(2δ)-1
doi:10.2140/pjm.1977.73.63 fatcat:nqnky7ponbdehfqtow2xcdr4hy