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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
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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

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