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Extremals in Landau's inequality for the difference operator*

Published online by Cambridge University Press:  14 November 2011

Man Kam Kwong  and
A. Zettl
Man Kam Kwong
Affiliation:
Department of Mathematical Sciences, Northern Illinois University and Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, U.S.A.
A. Zettl
Affiliation:
Department of Mathematical Sciences, Northern Illinois University and Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, U.S.A.
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Synopsis

The best constants in Landau's inequality for the difference operator in the classical sequence spaces lp are known explicitly only for p = 1, 2, ∞. This is true in both the infinite N = (0, 1, 2, …) and biinfinite Z= (… − 1, 0, 1, …) cases. It is known that there are no extremals when p = 2 in both the infinite and biinfinite cases. Also, it is known that there are extremals when p = ∞ in the biinfinite case. Here we prove that there are no extremals in the other three cases where the best constants are known explicitly. The proofs for these three cases are quite different from each other.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 107 , Issue 3-4 , 1987 , pp. 299 - 311
Copyright
Copyright © Royal Society of Edinburgh 1987

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