Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-25T05:15:07.324Z Has data issue: false hasContentIssue false
  • English
  • Français

Best constants in norm inequalities for derivatives on a half-line*

Published online by Cambridge University Press:  14 November 2011

Z. M. Franco ,
Hans G. Kaper ,
Man Kam Kwong  and
A. Zettl
Z. M. Franco
Affiliation:
Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL60439, U.S.A
Hans G. Kaper
Affiliation:
Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL60439, U.S.A
Man Kam Kwong
Affiliation:
Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL60439, U.S.A
A. Zettl
Affiliation:
Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL60439, U.S.A
Get access Rights & Permissions [Opens in a new window]

Synopsis

Let K be the class of all operators T in a Banach space × which have the property that, for any pair of integers (n, k) with n ≧2 and l≦ k ≦ n – l, there exists a constant Cnk such that

for all fϵdom Tn. If T ϵ K, then the best possible constant for the norm inequality (*) is the smallest non-negative value of the constant Cnk in (*). Any operator T which is the adjoint of a maximal symmetric operator in a Hilbert space belongs to the class K, as was shown by Ljubič [Izv. Akad. Nauk SSSR, Ser. Mat. 24 (1960), 825–864].

This article is concerned with the computation of the best possible constant for the differentiation operator Tf=if′ on the maximal domain in L2(0, ∞). Three algorithms, proposed by Ljubič [ibid.] and Kupcov [Trudy Mat. Inst. Steklov. 138 (1975)], are discussed and related to one another, asymptotic expressions (valid for large n) and numerical values of the best possible constant are presented, and the extremals (i.e. the elements / ∈ dom Tn for which equality holds in (*) with the best possible constant) are given.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 100 , Issue 1-2 , 1985 , pp. 67 - 84
Copyright
Copyright © Royal Society of Edinburgh 1985

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

1Landau, E.. Einige Ungleichungen für zweimal differentiierbare Funktionen. Proc. London. Math. Soc. 13 (1913), 43–49.Google Scholar
2Hardy, G. H. and Littlewood, J. E.. Some integral inequalities connected with the calculus of variations. Quart. J. Math. 3 (1932), 241–252.CrossRefGoogle Scholar
3Hardy, G. H., Littlewood, J. E. and Pólya, G.. Inequalities, 2nd edn (Edinburgh, 1952).Google Scholar
4Hadamard, J.. Sur le module maximum d'une fonction et de ses derivees. C.R. Acad. Sci. Paris 42 (1914), 68–72.Google Scholar
5Kolmogorov, A. N.. On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval. Moskov. Gos. Univ. Učen. Zap. Mat. 30 (1939), 3–13; English transl., Amer. Math. Soc. Tmnsl. no. 4, 1949; reprint, Amer. Math. Soc. Transl. (1) 2 (1962), 233–243.Google Scholar
6Ljubič, Ju. I.. On the belonging of the powers of an operator on a given vector to a certain linear class. Dokl. Akad. Nauk SSSR 102 (1955), 881–884 (Russian).Google Scholar
7Ljubič, Ju. I.. On inequalities between the powers of a linear operator. Izv. Akad. Nauk SSSR Ser. Mat. 24 (1960), 825–864; English transl., Amer. Math. Soc. Tmnsl. (2) 40 (1964), 39–84.Google Scholar
8Akhiezer, N. I. and Glazman, I. M.. Theory of Linear Operators in Hilbert Space, 2 Vols (New York: Ungar, 1961, 1963).Google Scholar
9Chernoff, P. R.. Optimal Landau-Kolmogorov Inequalities for Dissipative Operators in Hilbert and Banach Spaces. Adv. Math. 34 (1979), 137–144.CrossRefGoogle Scholar
10Kato, T. and Satake, I.. An algebraic theory of Landau-Kolmogorov inequalities. Tôhoku Math. J. 33 (1981), 421–428.CrossRefGoogle Scholar
11Kwong, M. K. and Zettl, A.. Norm Inequalities for Dissipative Operators onInner Product Spaces. Houston J. Math. 5 (1979), 543–557.Google Scholar
12Quôc Phong, Vu. On inequalities for powers of linear operators andfor quadratic forms. Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), 25–50.CrossRefGoogle Scholar
13Kupcov, N. P.. Kolmogorov estimates for derivatives in L2[0, ∞). Trudy Mat. Inst. Steklov 138 (1975); English transl., Proc. Steklov Inst. Math. 138 (1975), 101–125.Google Scholar
14Neta, B.. On Determination of Best-Possible Constants in Integral Inequalities Involving Derivatives. Math. Comp. 35 (1980), 1191–1193.CrossRefGoogle Scholar
15Moré, J. J., Garbow, B. S. and Hillstrom, K. E.. User Guide for Minpack-1, ANL-80-74 (Argonne, III.: Argonne National Laboratory, 1980).CrossRefGoogle Scholar