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A note on summability factors

Published online by Cambridge University Press:  26 February 2010

L. S. Bosanquet  and
J. B. Tatchell
L. S. Bosanquet
Affiliation:
University College, London.
J. B. Tatchell
Affiliation:
University College, London.
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Extract

Our main object in this note is to establish (Theorem 1) a necessary and sufficient condition to be satisfied by a sequence {εn} so that a series Σ an εnmay be summable | A |whenever the series Σanis summable (C, — 1). We suppose that an and εn are complex numbers. The condition is unchanged if the an are restricted to be real, but our proof is adapted to the case where they may be complex. Theorem 1 has been quoted by Bosanquet and Chow [12] in order to fill a gap in the theory of summability factors. We also obtain some related results, which are discussed in the Appendix.

Type
Research Article
Information
Mathematika , Volume 4 , Issue 1 , June 1957 , pp. 25 - 40
Copyright
Copyright © University College London 1957

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References

1.Anderaen, A. F., “Comparison theorems in the theory of Cesàro summability”, Proc. London Math. Soc. (2), 27 (1928), 39–71.Google Scholar
2.Anderaen, A. F., “Über die Anwendung von Differenzen nicht ganzer Ordnung in der Reihentheorie”, Report of the 8th Scandinavian Congress of Mathematicians in Stockholm (1934), 326–348.Google Scholar
3.Anderaen, A. F., “Summation af ikke hel Orden”, Mat. Tidsskrift (1946), 33–52.Google Scholar
4.Anderaen, A. F., “On the extensions within the theory of Cesaro summability of a classical theorem of Dedekind”, Proc. London Math. Soc. (in the press).Google Scholar
5.Banach, S., “Théoréme sur les ensembles de première catégorie”, Fundamenta Math., 16 (1930), 395–398.CrossRefGoogle Scholar
6.Banach, S., Théorie des opérations linéaires (Warsaw, 1932).Google Scholar
7.Bosanquet, L. S., “A mean value theorem”, Journal London Math. Soc., 16 (1941), 146–148.CrossRefGoogle Scholar
8-10.Bosanquet, L. S., “Note on convergence and summability factors (I-III)”, Journal London Math. Soc., 20 (1945), 39–48;CrossRefGoogle Scholar
Proc. London Math. Soc. (2), 50 (1948–1949), 295–304 and 482–496.Google Scholar
11.Bosanquet, L. S. and Chow, H. C., “Some analogues of a theorem of Andersen”, Journal London Math. Soc., 16 (1941), 42–48.CrossRefGoogle Scholar
12.Bosanquet, L. S. and Chow, H. C., “Some remarks on convergence and summability factors”, Journal London Math. Soc., 32 (1957), 73–82.CrossRefGoogle Scholar
13.Fekete, M., “Summabilitási factor-sorozatok”, Math, és Termés Ért., 35 (1917), 309–324.Google Scholar
13a.Fekete, M., “On the absolute summability (A) of infinite series”, Proc Edinburgh Math. Soc. (2), 3 (1933), 132–134.Google Scholar
14.Hardy, G. H., Divergent series (Oxford, 1949).Google Scholar
15.Jacob, M., “Über die Verallgemeinerung einiger Theorems von Hardy in der Theorie der Fourier' schen Beihen”, Proc London Math. Soc. (2), 26 (1927), 470–492.CrossRefGoogle Scholar
15a.Jurkat, W., Über Rieszsche Mittel und verwandte Klassen von Matrixtransformationen”, Math. Zeitschrift, 57 (1953), 353–394.CrossRefGoogle Scholar
16.Jurkat, W., and Peyerimhoff, A., “Mittelwertsätze bei Matrix- und Integraltransformationen”, Math. Zeitschrift, 55 (1951), 92–108.CrossRefGoogle Scholar
16a.Jurkat, W., and Peyerimhoff, A., “Summierbarkeitsfaktoren”, Math. Zeitschrift, 58 (1953), 186–203.CrossRefGoogle Scholar
17.Orlica, W., “Beiträge zur Theorie der Orthogonalentwicklungen II”, Studia Math., 1 (1929), 241–255.CrossRefGoogle Scholar
18.Peyerimhoff, A., “Konvergenz- und Summierbarkeitsfaktoren”, Math. Zeitschrift, 55 (1951), 23–54.CrossRefGoogle Scholar
19.Tatchell, J. B., “On some integral transformations”, Proc. London Math. Soc. (3), 3 (1953), 257–266.CrossRefGoogle Scholar
20.Tatchell, J. B. “A note on a theorem by Bosanquet”, Journal London Math. Soc., 29 (1954), 207–211.CrossRefGoogle Scholar
21.Whittaker, J. M., “The absolute summability of Fourier series”, Proc. Edinburgh Math. Soc., (2), 2 (1930-1931), 1–5.CrossRefGoogle Scholar
22.Young, W. H., “On the convergence of the derived series of a Fourier series”, Proc. London Math. Soc. (2), 17 (1918), 195–236.CrossRefGoogle Scholar
23.Zaanen, A. C., Linear analysis (Amsterdam, 1953).Google Scholar