Vol. 73, No. 1, 1977

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Mercerian theorems via spectral theory

Frank Peter Anthony Cass and Billy E. Rhoades

Vol. 73 (1977), No. 1, 63–71
Abstract

Given a regular matrix A, Mercerian theorems are concerned with determining the real or complex values of α for which αI + (1 α)A is equivalent to convergence. For α1, the problem is equivalent to determining the resolvent set for A, or, determining the spectrum σ(A) of A. where σ(A) = {λA λI is not invertible}. This paper treats the problem of determining the spectra of weighted mean methods; i.e., triangular matrices A = (ank) with an k = pk∕Pn, where p0 > 0, pn 0, k=0npk = Pn. It is shown that the spectrum of every weighted mean method is contained in the disc {λ|λ 12|12} (Theorem 1), and, if limpn∕Pn exists,

σ(A) = {λ|λ (2 𝜖)1|(1 𝜖)(2 𝜖)}
∪{pn∕Pnpn∕Pn < 𝜖∕(2 𝜖)},
where 𝜖 = limpn∕Pn.

Let γ = lim-pn∕Pn, δ = ---
limpn∕Pn, S = {pn∕Pnn 0}. When γ < δ, some examples are provided to indicate the difficulty of determining the spectrum explicitly. It is shown that {λ|λ (2 δ)1|(1 δ∕(2 δ))}∪ S σ(A) and

σ(A ) ⊆ {λ | |λ − (2 − γ)− 1| ≦ (1 − γ)∕(2− γ)} ∪S.

Theorem 1 is a generalization of the corresponding theorems of: S. Aljancic, L. N. Cakalov, K. Knopp, M. E. Landau, J. Mercer, Y. Okada, W. Sierpinski, and G. Sunouchi.

Mathematical Subject Classification 2000
Primary: 40D25
Milestones
Received: 29 March 1977
Published: 1 November 1977
Authors
Frank Peter Anthony Cass
Billy E. Rhoades