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A Best Possible Tauberian Theorem for the Collective Continuous Hausdorff Summability Method

Published online by Cambridge University Press:  20 November 2018

G. E. Peterson
G. E. Peterson*
Affiliation:
University of Missouri, St. Louis, Missouri
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The purpose of this paper is to prove that o(l/x) is the best possible Tauberian condition for the collective continuous Hausdorff method of summation. The analogue of this result for the collective (discrete) Hausdorff method is known [1, pp. 229, ff.; 7, p. 318; 8, p. 254]. Our method involves generalizing a well-known Abelian theorem of Agnew [2] to locally compact spaces and then applying the analogue for integrals of a result Lorentz obtained for series [6, Theorem 1].

Let T and X denote locally compact, non compact, σ-compact Hausdorff spaces. Let T′ = T ∪ (∞) and X′ = X ∪ (∞) denote the onepoint compactifications of T and X, respectively. Let B(T) denote the set of locally bounded, complex valued Borel functions on T and let B∞(T) denote the bounded functions in B(T).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 544 - 549
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Agnew, R. P., Analytic extension by Hausdorff methods, Trans. Amer. Math. Soc. 52 (1942), 217–237.Google Scholar
2. Agnew, R. P., Equivalence of methods for evaluation of sequences, Proc. Amer. Math. Soc. 3 (1952), 550–556.Google Scholar
3. Bourbaki, N., Elements of mathematics, General topology, Part 1 (Addison-Wesley, Reading, Massachusetts, 1966).Google Scholar
4. Eberlein, W. F., Banach-Hausdorff limits, Proc. Amer. Math. Soc. 1 (1950), 662–664.Google Scholar
5. Hardy, G. H., Divergent series (Oxford, at the Clarendon Press, 1949).Google Scholar
6. Lorentz, G. G., Tauberian theorems and Tauberian conditions, Trans. Amer. Math. Soc. 63 (1948), 226–234.Google Scholar
7. Lorentz, G. G., Direct theorems on methods of summability, Can. J. Math. 1 (1949), 305–319.Google Scholar
8. Lorentz, G. G., Direct theorems on methods of summability. II, Can. J. Math. 3 (1951), 236–256.Google Scholar
9. Persson, A., Summation methods on locally compact spaces, Dissertation, University of Lund, 1965.Google Scholar
10. Peterson, G. E., Tauberian theorems for integrals. I (submitted for publication).Google Scholar
11. Peterson, G. E., Tauberian theorems for integrals. II (submitted for publication).Google Scholar
12. Rudin, W., Real and complex analysis (McGraw-Hill, New York, 1966).Google Scholar
13. Rogosinski, W. W., On Hausdorff1 s methods of summability. II, Proc. Cambridge Philos. Soc. 38 (1942), 344–363.Google Scholar