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Tauberian theorems via statistical convergence. (English) Zbl 0919.40006
The authors define a type of regular summability method over an abstract metric space as follows: Let \(\{f(n)\mid n= 0,1,2,\dots\}\) be a sequence taking values in a metric space \((X,\rho)\), and let \(A= [a_{nk}]\) be a nonnegative summability method. Then \(L\) is the \((A)\) st-lim of \(f\) (or \(f\) is \((A)\)-statistically convergent to \(L\)) if, for any \(\varepsilon> 0\), \[ \lim_{n\to \infty} \sum_{k:\rho(f(k),L)\geq \varepsilon} a_{nk}= 0. \] By using probabilistic tools some Tauberian theorems are proved which have best possible order Tauberian conditions. Further the authors use their methods to unify and improve (i) the classical Tauberian theorems of summability, (ii) the theory for the random walk type method as proved by Bringham, and (iii) the Hausdorff methods as proved by Lorentz.
Reviewer: I.L.Sukla (Orissa)

MSC:
40E05 Tauberian theorems, general
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