Some inclusion theorems
1964; Cambridge University Press; Volume: 6; Issue: 4 Linguagem: Inglês
10.1017/s204061850003495x
ISSN2051-2104
Autores Tópico(s)Advanced Optimization Algorithms Research
Resumo1. A number of inclusion theorems have been given in connection with methods of summation which include the Riesz method (R, λ, κ) . Lorentz [4, Theorem 10] gives necessary and sufficient conditions for a sequence to sequence regular matrix A = (a n, v ) to be such that A ⊃ (R, λ, 1)†. He imposes restrictions on the sequence { λ n } , so that A does not include all Riesz methods of order 1. In Theorem 1 below, we generalize the Lorentz theorem by giving a condition without restriction on λ n , If the matrix A is a series to sequence or series to function regular matrix, there do not appear to be any results concerning the general inclusion A ⊃ (R, λ, κ) . However, when A is the Riemann method (ℜ, λ, μ), Russell [7], generalizing earlier results, has given sufficient conditions for (ℜ, λ, μ) ⊃ (R, λ, κ) . Our Theorem 2 gives necessary and sufficient conditions for A ⊃ (R, λ, 1), where A satisfies the condition a n, v → 1 (n →co, ν fixed) . Thus Theorem 2 applies to any series to sequence regular matrix A. In Theorem 3 we give a further representation for matrices A which include ( R, λ, 1) , and finally make some remarks on the problem of characterizing matrices which include Riesz methods of any positive order κ.
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