Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-12-05T04:25:17.598Z Has data issue: false hasContentIssue false
  • English
  • Français

Multivariate Landau–Kolmogorov-type inequality

Published online by Cambridge University Press:  24 October 2008

Z. Ditzian
Z. Ditzian
Affiliation:
Department of Mathematics, Faculty of Science, University of Alberta, Edmonton, CanadaT6G 2G1
Get access Rights & Permissions [Opens in a new window]

Abstract

Assuming that the nth iterate of the Laplacian Δnf belongs to L∞(ℝ), we show for 0 < k < 2n that

where ∂/∂ξi is the derivative in the ei direction. The result is also extended to other Banach spaces of functions on ℝd.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 2 , March 1989 , pp. 335 - 350
Copyright
Copyright © Cambridge Philosophical Society 1989

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]Boyadzhiev, K. N.. A many variable Landau–Kolmogorov inequality. Math. Proc. Camb. Philos. Soc. 101 (1987), 123–129.CrossRefGoogle Scholar
[2]Burzer, P. L. and Berens, H.. Semi-groups of Operators and Approximation (Springer-Verlag, 1967).Google Scholar
[3]Ditzian, Z.. Some remarks on inequalities of Landau and Kolmogorov. Aequationes Math. 12 (1975), 145–151.CrossRefGoogle Scholar
[4]Ditzian, Z.. Discrete and shift Kolmogorov type inequalities. Proc. Roy. Soc. Edinburgh Sect. A 93 (1983), 307–317.Google Scholar
[5]Johnen, H. and Scherer, K.. On the equivalence of the K-functional and moduli of continuity and some applications. In Constructive Theory of Functions of Several Variables, Lecture Notes in Math. vol. 571 (Springer-Verlag, 1976), pp. 119–140.CrossRefGoogle Scholar
[6]Kolmogorov, A. N.. On inequalities between the upper bounds of successive derivatives of an arbitrary function on an infinite interval. Amer. Math. Soc. Transl. Series 1, 2 (1962), 233–243.Google Scholar
[7]Timofeev, V. G.. Kolmogorov inequalities with Laplace operator. Theory of functions and approximation. Lecture Notes. Saratov State University (1983).Google Scholar
[8]Timofeev, V. G.. Landau inequality for function of several variables. Mat. Zametki 37 (1985), 676–689.Google Scholar