Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T08:25:40.194Z Has data issue: false hasContentIssue false
  • English
  • Français

Lipschitz classes and convolution approximation processes

Published online by Cambridge University Press:  24 October 2008

Z. Ditzian
Z. Ditzian
Affiliation:
University of Alberta, Edmonton
Get access Rights & Permissions [Opens in a new window]

Extract

For a continuous function f(x) on the reals or on the circle T (continuous and 2π periodic) we say that f(x) belongs to the generalized Lipschitz class, denoted by f ∈ Lip* α, if

where and Δhf(x) = f(x + ½h)−f(x−½h). For a convolution approximation process given by

where

we shall investigate equivalence relations between the asymptotic behaviour of (d/dx)rAn(f, x) and f ∈ Lip* α.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 90 , Issue 1 , July 1981 , pp. 51 - 61
Copyright
Copyright © Cambridge Philosophical Society 1981

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)Becker, M. and Nessel, R. S.An elementary approach to inverse approximation theory. J. of Approximation 23 (1978), 99–103.Google Scholar
(2)Berens, H. and Lorentz, G. G.Inverse theorems for Bernstein polynomials. Indiana Univ. Math. J., 21 (1972), 693–708.Google Scholar
(3)Ditzian, Z.Inverse theorems for functions in Lp and other spaces. Proc. Amer. Math. Soc. 54 (1976), 80–82.Google Scholar
(4)Ditzian, Z.Some remarks on inequalities of Landau and Kolmogorov. Aequationes Math. 12 (1975), 145–151.CrossRefGoogle Scholar
(5)Ditzian, Z. and May, C. P.Lp saturation and inverse theorems for modified Bernstein polynomials. Indiana Univ. Math. J. 25 (1976), 733–751.Google Scholar
(6)Stein, E.Singular integral and differentiability properties of functions. (Princeton Univ. Press, 1970).Google Scholar
(7)Timan, A. F.Theory of approximation of functions of real variables. (Macmillan, 1963).Google Scholar