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Tauberian theorems for [J, f(x)] transformations

Published online by Cambridge University Press:  09 April 2009

A. Jakimovski  and
A. Livne
A. Jakimovski
Affiliation:
Tel Aviv University, Israel
A. Livne
Affiliation:
Tel Aviv University, Israel
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Let Σ∞n=0 an(sn=a0+…+an, n≧0) be a series of real or complex numbers. Denote by and two linear transforms T1 and T2 of {sn}. Estimates of the form for sequences {sn} satisfying where {dn} is a certain fixed linear transform of the sequence and depend on the transforms T1, T2 and {dn}, were considered for the first time by Hadwiger [2]. The smallest value of C satisfying (1.2) for all sequences {sn} satisfying (1.3) is known as the Tauberian constant associated with the pair of transforms T1, T2 and {dn}.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 15 , Issue 2 , March 1973 , pp. 202 - 221
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Agnew, R. P., ‘Abel transforms anad partial sums of Tauberian series’, Ann. of Math. 50 (1949), 110–117.CrossRefGoogle Scholar
[2]Hadwiger, H., ‘Über ein Destanztheorem bei der A-Limitierung’, Comment. Math. 16 (1944), 209–214.Google Scholar
[3]Hardy, G. H., Divergent series (Oxford University Press, 1949).Google Scholar
[4]Hobson, E. W., The theory of functions of a real variable, Vol. 1 (Dover, New York, 1957), 545.Google Scholar
[5]Jakimovski(Amir), A., ‘Some relations between the methods of summability of Abel, Borel Cesàro, Holder and Hausdorff,’ J. d'Analyse Math., 3 (1953/1954), 346–381.Google Scholar
[6]Jakimovski, A., ‘The sequence-to-function analogues to Hausdorff transformations’, Bull. Res. Council Israel 8F (1960), 135–154.Google Scholar
[7]Jakimovski, A., ‘Tauberian constants for the Abel and Cesàro transformations’, Proc. Amer. Math. Soc. 14 (1963), 228–238.Google Scholar
[8]Jakimovski, A. and Leviatan, D., ‘A property of approximation operators and applications to Tauberian constants’, Math. Z. 102 (1967), 177–204.CrossRefGoogle Scholar
[9]Jakimovski, A. and Livne, A., ‘Approximation operators and Tauberian constants’, Israel. J. Math. 7 (1969), 263–292.Google Scholar
[10]Knopp, K., Theory and applications of infinite series (Blackie and Son, London and Glasgow, 1944).Google Scholar
[11]Leviaten, D., ‘Tauberian constants for generalized Hausdorff transformations’, J. London Math. Soc. 43 (1968), 308–314.Google Scholar
[12]Leviatan, D., ‘Some Tauberian theorems for quasi-Hausdorff transforms’, Math. Z. 108 (1969), 213–222.CrossRefGoogle Scholar
[13]Lorentz, G.G., Bernstein polynomials (Toronto University Press, 1953).Google Scholar
[14]Meir, A., ‘Limit-distance of Hausdorff transforms of Tauberian series’, J. London Math. Soc. 40 (1965), 295–302.CrossRefGoogle Scholar
[15]Szasz, O., ‘Generalization of S. Bernstein's polynomials to the infinite interval’, Collected mathematical works (University of Cincinnati, 1955), 1401–1407.Google Scholar
[16]Widder, D. V., The Laplace transform (Princeton University Press, 1946).Google Scholar