Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-16T17:33:30.060Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Nørlund Summability of a Class of Fourier Series

Published online by Cambridge University Press:  20 November 2018

Badri N. Sahney
Badri N. Sahney*
Affiliation:
The University of Calgary, Calgary, Alberta
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

1. Our aim in this paper is to determine a necessary and sufficient condition for N∅rlund summability of Fourier series and to include a wider class of classical results. A Fourier series, of a Lebesgue-integrable function, is said to be summable at a point by N∅rlund method (N, pn), as defined by Hardy [1], if pn → Σpn → ∞, and the point is in a certain subset of the Lebesgue set. The following main results are known.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 1 , 01 February 1970 , pp. 86 - 91
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Hardy, G. H., Divergent series (Oxford, at the Clarendon Press, 1949).Google Scholar
2. Hardy, G. H. and Littlewood, J. E., Notes on the theory of series XVIII: On the convergence of Fourier series, Proc. Cambridge Philos. Soc. 31 (1935), 317–323.Google Scholar
3. Hille, E. and Tamarkin, J. D., On the summability of Fourier series, Trans. Amer. Math. Soc. 84 (1932), 757–783.Google Scholar
4. Iyengar, K. S. K., A Tauberian theorem and its applications to convergence of Fourier series, Proc. Indian Acad. Sci. Sect. A 18 (1943), 81–87.Google Scholar
5. Iyengar, K. S. K., New convergence and summability test for Fourier series, Proc. Indian Acad. Sci. Sect. A 18 (1943), 113–120.Google Scholar
6. Iyengar, K. S. K., Notes on the summability. II. On the relation between summability by Nörlund means of a certain type and summability by Valiron means, Half-Yearly J. Mysore Univ. Sect. B (N.S.) 4 (1944), 161–166.Google Scholar
7. Jurkat, W., Zur Konvergenztheorie der Fourier-Reihen, Math. Z. 53 (1950-51), 309–339.Google Scholar
8. Rajagopal, C. T., Nörlund summability of Fourier series, Proc. Cambridge Philos. Soc. 59 (1963), 47–53.Google Scholar
9. Sahney, B. N., On the (H, p) summability of Fourier series, Boll. Un. Mat. Ital. 16 (1961), 156–163.Google Scholar
10. Sahney, B. N., On the Nörlund summability of Fourier series, Pacific J. Math. 13 (1963), 251–262.Google Scholar
11. Siddiqi, J. A., On the harmonic summability of Fourier series, Proc. Indian Acad. Sci. Sect. A 28 (1948), 527–531.Google Scholar
12. Varshney, O. P., On the relation between harmonic summability and summability by Riesz means of a certain type, Töhoku Math. J. (2) 11 (1959), 20–24.Google Scholar
13. Varshney, O. P., On the Nörlund summability of Fourier series, Acad. Roy. Belg. Bull. CI. Sci. (5) 52 (1966), 1552–1558.Google Scholar
14. Wang, F. T., On the Riesz summability of Fourier series, Proc. London Math. Soc. (2) 47 (1942), 308–325.Google Scholar
15. Zygmund, A., Trigonometrical series, 2nd ed. (Chelsea, New York, 1952).Google Scholar