Mercerian theorems via spectral theory

Frank Peter Anthony Cass and Billy E. Rhoades

Vol. 73 (1977), No. 1, 63–71

DOI: 10.2140/pjm.1977.73.63

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 α≠1, the problem is equivalent to determining the resolvent set for A, or, determining the spectrum σ(A) of A. where σ(A) = {λ∣A − λI 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_{n k} = p_k∕P_n, where p₀ > 0, p_n ≧ 0, ∑ _k=0ⁿp_k = P_n. It is shown that the spectrum of every weighted mean method is contained in the disc {λ∣|λ − 1∕2|≦ 1∕2} (Theorem 1), and, if limp_n∕P_n exists,

σ(A)	= {λ∣|λ − (2 − 𝜖)−1|≦ (1 − 𝜖)∕(2 − 𝜖)}
	∪{pn∕Pn∣pn∕Pn < 𝜖∕(2 − 𝜖)},

where 𝜖 = limp_n∕P_n.

Let γ = p_n∕P_n, δ = p_n∕P_n, S = {p_n∕P_n∣n ≧ 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

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