Hostname: page-component-7bb8b95d7b-lvwk9 Total loading time: 0 Render date: 2024-09-13T14:14:28.840Z Has data issue: false hasContentIssue false
  • English
  • Français

Saturation results for a class of linear operators

Published online by Cambridge University Press:  24 October 2008

B. Kuttner ,
R. N. Mohapatra  and
B. N. Sahney
B. Kuttner
Affiliation:
University of Birmingham
R. N. Mohapatra
Affiliation:
University of Birmingham
B. N. Sahney
Affiliation:
American University of Beirut
Get access Rights & Permissions [Opens in a new window]

Extract

Let B denote the space of bounded measurable functions with period 2π. We will suppose throughout that f(x) ∈ B. All norms considered are essential sup norms. Let the Fourier series of f(x) be given by

Let D = (dnk) (n, k = 0, 1, …) be an infinite matrix.

Let Ln(f; x) be the D transform of the Fourier series of f(t) at t = x, i.e.

where Sk(x) = A0(x) + A1(x) + … + Ak(x). Let us write

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 1 , July 1983 , pp. 133 - 148
Copyright
Copyright © Cambridge Philosophical Society 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Alexits, G.On the order of approximation of Fejér means. Hungarica Acta Math. 3 (1944), 20–25.Google Scholar
(2)Butzer, P. L.On the singular integral of de la Vallée Poussin. Archiv Math. 7 (1956), 295–309.CrossRefGoogle Scholar
(3)Butzer, P. L.Über den Grad der Approximation des Identitätsoperators durch Halbgruppen von linearen Operatoren und Anwendungen auf die Theorie der singularen Integrale. Math. Ann. 133 (1957), 97–110.CrossRefGoogle Scholar
(4)DeVore, R. The approximation of continuous functions by positive linear operators. Lecture Notes in Mathematics 293 (Springer-Verlag, Berlin, 1972).Google Scholar
(5)Favard, J.Sur la saturation des procédés de sommation. J. Math. 36 (1957), 359–372.Google Scholar
(6)Goel, D. S., Holland, A. S. B., Nasim, C. and Sahney, B. N.Best approximation by a saturation class of polynomial operators. Pacific J. Math. 55 (1974), 149–155.CrossRefGoogle Scholar
(7)Khan, H. H. and Rizvi, S. M.On the saturation of classes of functions by (N, pn, qn) method. Indian J. Pure Applied Math. 6 (1974), 1262–1269.Google Scholar
(8)Kuttner, B. and Sahney, B. N.On non-uniqueness of the degree of saturation. Math. Proc. Cambridge Philos. Soc. 84 (1978), 113–116.CrossRefGoogle Scholar
(9)Mamedov, R. G.Local saturation of a family of linear positive operators. Doklady Akad. Nauk 155 (1964), 499–502.Google Scholar
(10)Sunouchi, G.On the class of saturation in the theory of approximation II, III. Tôhoku Math. J. 13 (1961), 112–118; 320–328.Google Scholar
(11)Sunouchi, G.Saturation in the local approximation. Tôhoku Math. J. 17 (1965), 16–28.CrossRefGoogle Scholar
(12)Sunouchi, G. Local saturation in convolution operators. (To be published.)Google Scholar
(13)Sunouchi, G. and Watari, C.On determination of the class of saturation in the theory of approximation of functions II. Tôhoku Math. J. 11 (1959), 480–488.CrossRefGoogle Scholar
(14)Suzuki, Y.Saturation of local approximation by linear positive operators. Tôhoku Math. J. 17 (1965), 210–221.CrossRefGoogle Scholar
(15)Zamansky, M.Classes de saturation de certaines precédés d'approximation des séries de Fourier des fonctions continues. Ann. Sci. Ecole Normale Sup. 66 (1949), 19–93.CrossRefGoogle Scholar
(16)Zygmund, A.Trigonometric series, vols. 1 and 2 (Cambridge University Press, 1959).Google Scholar