Nonexistence of universally accelerating linear summability methods (Q1344342)

The equivalence of two Toeplitz matrices is deduced from their transformation properties with regard to convergent sequences. Let \(a(m,n)\) and \(b(m,n)\) \((m,n \geq 1)\) be the elements of two regular Toeplitz matrices \(A\) and \(B\). \({\mathcal S}\) being a representative convergent sequence, denote its members by \(S(n)\) \((n \geq 1)\) and its limit by \(L({\mathcal S})\). Set \(\sigma (A \| S,m) = \sum a(m,n) S(n)\) \((1 \leq n < \infty)\) for \(m \geq 1\) and define \(\sigma (B \| {\mathcal S}, m)\) for \(m \geq 1\) similarly. If, for every convergent sequence \({\mathcal S}\), \(| \sigma (B \| {\mathcal S},m) - L({\mathcal S}) | \leq K ({\mathcal S}) | \sigma (A \| {\mathcal S}, m) - L({\mathcal S}) |\) \((m \geq 1)\), where \(K({\mathcal S})\) depends upon \({\mathcal S}\) but not upon \(m\), then (a) for some \(p \geq 1\), \(a(m,n) = b(m,n)\) \((n \geq 1)\) for all \(m \geq p\) and (b) \(A\) and \(B\) sum the same sequences.