Nonexistence of universally accelerating linear summability methods

AbstractLet A = [amn], B = [bmn] be two regular Toeplitz matrices. Suppose that for every convergent sequence {{sn} its B-transform σ(B)m, where σ(B)m = ∑∞n = 1bmnsn, converges not slower than σ(A)m, in the sens. 0 e that ¦ σ(B)m − s ¦ ⩽ K ¦ σ(A)m − s ¦, where s = limn → ∞sn, and K does not depend on m (only on {sn}). Then A and B are equivalent (i.e., they sum the same sequences, even unbounded), and, moreover, A and B essentially coincide: there exists a number p such that amn = bmn if m ≥/ p