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Some Theorems on Strong Nörlund Summability

Published online by Cambridge University Press:  20 November 2018

Frank P. Cass
Frank P. Cass*
Affiliation:
University of Western Ontario, London, Ontario
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Throughout this paper H, H1 etc. will denote positive constants which will not necessarily be the same at different occurrences. If is a series, we shall use the notation For a real, define

1

Let {pn} be a sequence with p0>0 and pn≥0 for n > 0. Define

2

The following identities are immediate:

3

4

where

5

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 1 , March 1972 , pp. 63 - 69
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Borwein, D., On strong and absolute summability, Proc. Glasgo. Math. Assoc. 4 (1960), 122-139.Google Scholar
2. Borwein, D. and Cass, F. P., Strong Norlund summability, Math. Z. 103 (1968), 94-111.Google Scholar
3. Borwein, D. and Cass, F. P., Multiplication theorems for strong Nörlund summability, Math. Z. 107 (1968), 33-42.Google Scholar
4. Cass, F. P., Convexity theorems for Nörlund and strong Nörlund summability, Math. Z. 112 (1969), 357-363.Google Scholar
5. Flett, T. M., Some remarks on strong summability, Quart. J. Math. Oxford Ser. 10 (1959), 115-139.Google Scholar
6. Garabedian, H. L. and Randels, W. C., Theorems on Reisz means,Duke Math. J. 4 (1938), 529-533.Google Scholar
7. Hardy, G. H., Divergent series, Oxford, 1949.Google Scholar
8. Hardy, G. H. and Littlewood, J. E., Some properties of fractional integrals I, Math. Z. 27 (1928), 565-606.Google Scholar
9. Hardy, G. H., Littlewood, J. E., and Pölya, G., The maximum of a certain bilinear form, Proc. London Math. Soc. (2) 25 (1926), 265-282.Google Scholar
10. Hardy, G. H., Littlewood, J. E., and Pölya, G., Inequalities, Cambridge, 1934.Google Scholar
11. Kuttner, B. and Thorpe, B., On the strong Nörlund summability of a Cauchy product series, Math. Z. 111 (1969), 69-86.Google Scholar
12. Wilansky, A., Functional analysis, Blaisdell, Waltham, Mass., 1964.Google Scholar