Mercerian theorems via spectral theory
Frank P. Cass, Billy E. Rhoades · Pacific Journal of Mathematics · 1977
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 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, ^ (1 -β)/(2 -β)} U {pJPn I pJPn < β/(2-ε)} , where e -lim pJP n .Let r = limpJP n , δ = 1STpJP n , S = {pJP 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 1 ^ (1 -3/(2 -8)}\J8Q σ(A) and σ(A) Q{λ\\λ-(2-r)" 1 1 ^ (1 " r)/(2 ~ ΐ)} U S .