The Riesz summability of logarithmic type
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- Proc. Amer. Math. Soc. 45 (1974), 365-370 Request permission
Abstract:
The series $\Sigma _{n = 1}^\infty {a_n}$ is said to be summable $(L)$ to $s$ if ${(\log (1 - x))^{ - 1}}\Sigma _{n = 1}^\infty {s_n}{x^{n + 1}}/n$, where ${s_n} = \Sigma _{v = 1}^n{a_v}$, converges for $0 \leq x < 1$ and tends to $s$ when $x \to 1 -$. The aim of this paper is to discuss the relation between summability $(L)$ and Riesz summability $(R,\log n,\kappa )$. It is proved that $(R,\log n,\kappa ) \subseteq (L)$ holds for $0 \leq \kappa \leq 1$ and is false for $\kappa > 1$. It is also proved that if $\Sigma _{n = 1}^\infty {a_n} = s(L)$ and bounded $(R,\log n,\kappa )$ for $\kappa \geq 0$ then $\Sigma _{n = 1}^\infty {a_n} = s(R,\log n,\kappa + \delta )$ for every $\delta > 0$.References
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Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 45 (1974), 365-370
- MSC: Primary 40D25
- DOI: https://doi.org/10.1090/S0002-9939-1974-0348326-9
- MathSciNet review: 0348326