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Gibbs phenomenon for the Hausdorff means of double sequences1

Published online by Cambridge University Press:  09 April 2009

Fred Ustina
Fred Ustina
Affiliation:
University of AlbertaAlberta, Canada
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let g (u) be a regular Hausdorff weight function, and let hm (Ψ x m) denote the mthe corresponding Hausdorff transform, evaluated at x m, of the sequence of partial sums of the Fourier series of Ψ (x), where . In [3], Szász investigated the Gibbs phenomenon for Ψ(x) for these means. His main results are contained in the following two theorems: . (3) THEOREM 2. Taking the limit superior as. If this maximum is attained for τ = τ′ then.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 11 , Issue 2 , May 1970 , pp. 169 - 185
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Adams, C. R., ‘Hausdorff transformations for double sequences’, Bull. Amer. Math, Soc. 39 (1933), 303–312.CrossRefGoogle Scholar
[2]Hallenbach, F., Zur Theorie der Limitierungsverfahren von Doppelfolgen. (Inaugural-Dissertation, Rheinischen Friedrich-Wilhelms-Universität, Bonn 1933).Google Scholar
[3]Szász, O., ‘Gibbs phenomenon for Hausdorff means’, Trans. Amer. Math. Soc. 69 (1950), 440–456.Google Scholar
[4]Ustina, F., ‘Hausdorff means for double sequences’, Can. Math. Bull. 10 (1967), 347–352.CrossRefGoogle Scholar
[5]Ustina, F., ‘Gibbs phenomenon for functions of two variables’, Trans. Amer. Math. Soc. 129 (1967), 124–129.CrossRefGoogle Scholar
[6]Young, W. H., ‘On multiple integrals’. Proc. Royal Soc., London (A) 93 (1917), 28–41.Google Scholar