Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-20T10:32:04.124Z Has data issue: false hasContentIssue false
  • English
  • Français

On non-uniqueness of the order of saturation

Published online by Cambridge University Press:  24 October 2008

B. Kuttner  and
B. Sahney
B. Kuttner
Affiliation:
University of Birmingham and University of Calgary
B. Sahney
Affiliation:
University of Birmingham and University of Calgary
Get access Rights & Permissions [Opens in a new window]

Extract

1. Let X be a normed linear space of periodic functions (of period 2π), which includes the class of trigonometric polynomials. We restrict ourselves throughout to functions f ∈ X. Let D) = (dn, k) be the matrix of a regular sequence-to-sequence transformation. (We could also consider sequence-to-function transformations, in which case n is replaced by a continuous variable.) We suppose that, for all n,

Let {Ln(x)} be the D transform of the Fourier series of f(x). If f(x) is a constant, it follows from (1) that, for all n, Ln(x) = f(x). However, it is often found that, roughly speaking, except in this trivial case, Ln(x) cannot tend to f(x) (in the topology given by the norm) with more than a certain degree of rapidity. This leads to the concept of ‘saturation’, which was first introduced by Favard (1) and Zamanski (8) and which has since been investigated by many authors (see (2), (3), (4), (5), (6), (7)). This is defined as follows.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 1 , July 1978 , pp. 113 - 116
Copyright
Copyright © Cambridge Philosophical Society 1978

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)Favard, J.Sur la saturation des procédés de sommation. J. de Math. 36 (1957), 359–372.Google Scholar
(2)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
(3)Ikeno, K. and Suzuki, Y.Some remarks of saturation problem in the local approximation. Tôhohu Math. J. 20 (1968), 214–233.Google Scholar
(4)Korovkin, P. P.Linear operators and approximation theory (Delhi, Hindustan Publ. Co. 1960).Google Scholar
(5)Sunouchi, G.On the class of saturation in the theory of approximation: I, II, III. Tôhohu Math. J. 12 (1960), 339–344; 13 (1961), 112–118, 320–328.Google Scholar
(6)Sunouchi, G. and Watari, C.On determination of the class of saturation in the theory of approximation of functions. Proc. Japan Acad. 34 (1958), 477–481.Google Scholar
(7)Suzuki, Y.Saturation of local approximation by linear positive operators. Tôhoku Math. J. 17 (1965), 210–221.Google Scholar
(8)Zamanski, M.Classes de saturation de certaines procédés d'approximation des séries de Fourier des functions continues. Ann. Sci. École Normale Sup. 66 (1949), 19–93.CrossRefGoogle Scholar
(9)Zygmund, A.Trigonometric series, vols. I and II (Cambridge University Press, 1968).Google Scholar