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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:
 4e+06 Tauberian theorems, general
Full Text:
##### References:
 [1] Bingham, N.H., Tauberian theorems and the central limit theorem, Ann. probab., 9, 221-231, (1981) · Zbl 0459.60021 [2] Bingham, N.H., Tauberian theorems for summability methods of random-walk type, J. London math. soc. (2), 30, 281-287, (1984) · Zbl 0518.40002 [3] Dunford, N.; Schwartz, J.T., Linear operators, (1958), Interscience New York [4] Fast, H., Sur la convergence statistique, Colloq. math., 2, 241-244, (1951) · Zbl 0044.33605 [5] Fridy, J.A., On statistical convergence, Analysis, 5, 301-310, (1985) · Zbl 0588.40001 [6] Fridy, J.A.; Miller, H.I., A matrix characterization of statistical convergence, Analysis, 11, 59-66, (1991) · Zbl 0727.40001 [7] Hardy, G.H., Divergent series, (1949), Oxford Univ. Press London · Zbl 0032.05801 [8] Hardy, G.H., Theorems relating to the summability and convergence of slowly oscillating series, Proc. London math. soc. (2), 8, 301-320, (1910) · JFM 41.0278.02 [9] Hardy, G.H.; Littlewood, J.E., Contributions to the arithmetic theory of series, Proc. London math. soc. (2), 11, 411-478, (1913) · JFM 43.0312.01 [10] Ibragimov, I.A.; Linnik, Yu.V., Independent and stationary sequences of random variables, (1971), Wolters-Noordhoff Groningen · Zbl 0219.60027 [11] Landau, E., Über die bedeutung einiger neuen grenzwertsätze der herren Hardy und axer, Prace mat-fiz., 21, 97-177, (1910) · JFM 41.0241.01 [12] Lorentz, G.G., Direct theorems on methods of summability, Canad. J. math., 1, 305-319, (1949) · Zbl 0034.03403 [13] Lorentz, G.G., Direct theorems on methods of summability, II, Canad. J. math., 3, 236-256, (1951) · Zbl 0042.29402 [14] Maddox, J., A Tauberian theorem for ordered spaces, Analysis, 9, 297-302, (1989) · Zbl 0677.40003 [15] Meyer-König, W.; Zeller, K., Abschnittskonvergenz und umkehrsätze beim Euler-verfahren, Math. Z., 161, 147-154, (1978) · Zbl 0365.40004 [16] Schieber, E., Über dasSα, Math. Z., 80, 19-43, (1962) · Zbl 0129.27303 [17] Schoenberg, I.J., The integrability of certain functions and related summability methods, Amer. math. monthly, 66, 361-375, (1959) · Zbl 0089.04002 [18] Schmaal, A.; Stam, A.J.; de Vries, T., Tauberian theorems for limitation methods admitting a central limit theorem, Math. Z., 150, 75-82, (1976) · Zbl 0318.40011 [19] Schmidt, R., Über divergenten folgen und lineare mittelbildungen, Math. Z., 22, 89-152, (1925) · JFM 51.0182.04 [20] Sitaraman, Y., On Tauberian theorems for theSα, Math. Z., 95, 34-49, (1967) · Zbl 0162.08103 [21] Sonnenschein, J., Sur LES series divergentes, Bul. acad. roy. belgique, 35, 594-601, (1949) · Zbl 0033.25701 [22] Tam, L., A Tauberian theorem for the general euler – borel summability method, Canad. J. math. (5), 44, 1100-1120, (1992) · Zbl 0756.40007 [23] Tauber, A., Ein satz aus der theorie der unendlichen reihen, Monatsh. math., 8, 273-277, (1897) · JFM 28.0221.02
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