Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Landau's inequality for the difference operator


Authors: Man Kam Kwong and A. Zettl
Journal: Proc. Amer. Math. Soc. 104 (1988), 201-206
MSC: Primary 39A70
DOI: https://doi.org/10.1090/S0002-9939-1988-0958067-2
MathSciNet review: 958067
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The best constants for Landau's inequality with the classical $ p$-norms are known explicitly only when $ p = 1,2{\text{ and }}\infty $. This is true for both the discrete and the continuous versions of the inequality and for both the "whole line" and "half line" cases. In each of the six known cases the best constant in the discrete version is the same as the best constant for the continuous version. Here we show that for many other values of $ p$ the discrete constants are strictly greater than the corresponding continuous ones. In addition, we show that the "three norm version" of the inequality, established by Nirenberg and Gabushin in the continuous case is also valid in the discrete case.


References [Enhancements On Off] (What's this?)

  • [Z] Ditzian [1983], Discrete and shift Kolmogorov type inequalities, Proc. Roy. Soc. Edinburgh 93A, 307-313. MR 688793 (84m:47038)
  • [Z] Franco, H. G. Kaper, M. K. Kwong and A. Zettl [1983], Bounds for the best constants in Landau's inequality on the line, Proc. Roy. Soc. Edinburgh 95A, 257-262. MR 726877 (85m:26018)
  • [V] N. Gabushin [1967], Inequalities for norms of a function and its derivatives in $ {L_p}$ metrics, Mat. Zametki 1, 291-298. MR 0206700 (34:6518)
  • [H] G. Kaper and B. E. Spellman [1987], Best constants in norm inequalities for the difference operator, Trans. Amer. Math. Soc. 299, 351-372. MR 869416 (88d:39012)
  • [M] K. Kwong and A. Zettl [1980], Ramifications of Landau's inequality, Proc. Roy. Soc. Edinburgh 86, 175-212. MR 592549 (82e:26015)
  • 1. -[1988], Best constants for discrete Kolmogorov inequalities, Houston J. Math. (to appear). MR 1002084 (91c:26021)
  • [L] Nirenberg [1955], Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math. 8, 648-674. MR 0075415 (17:742d)
  • [C] de Boor [1978], A practical guide to splines, Springer-Verlag, Berlin and New York. MR 507062 (80a:65027)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 39A70

Retrieve articles in all journals with MSC: 39A70


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0958067-2
Keywords: Landau's, Kolmogorov, Hardy-Littlewood inequality, best constants, $ p$-norm inequalities
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society