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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Best constants in norm inequalities for the difference operator


Authors: Hans G. Kaper and Beth E. Spellman
Journal: Trans. Amer. Math. Soc. 299 (1987), 351-372
MSC: Primary 39A70; Secondary 47B39
MathSciNet review: 869416
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Abstract: Let $ \xi = {({\xi _m})_{m \in {\mathbf{Z}}}}$ be an arbitrary element of the sequence space $ {l^\infty }({\mathbf{Z}})$, and let $ \Delta $ be the difference operator on $ {l^\infty }({\mathbf{Z}}):\Delta \xi = {({\xi _{m + 1}} - {\xi _m})_{m \in {\mathbf{Z}}}}$. The object of this investigation is the best possible value

$\displaystyle C(n,\,k) = {\operatorname{sup}}\{ {Q_{n,k}}(\xi ):\xi \in {l^\infty }({\mathbf{Z}}),\,{\Delta ^n}\xi \ne 0\} $

of the quotient

$\displaystyle {Q_{n,k}}(\xi ) = \frac{{\left\Vert {{\Delta ^k}\xi } \right\Vert... ...i \right\Vert}^{(n - k)/n}}{{\left\Vert {{\Delta ^n}\xi } \right\Vert}^{k/n}}}}$

, where $ n = 2,\:3, \ldots $; $ k = 1, \ldots ,\,n - 1$. It is shown that $ C(n,\,k)$ is at least equal to the corresponding constant $ K(n,\,k)$, determined by Kolmogorov [Moscov. Gos. Univ. Uchen. Zap. Mat. 30 (1939), 3-13; Amer. Math. Soc. Transl. (1) 2 (1962), 233-243] for the differential operator $ D$ on $ {L^\infty }({\mathbf{R}})$, and exactly equal to $ K(n,\,k)$ if $ k = n - 1$. Lower bounds for $ C(n,\,k)$ are derived that show that $ C(n,\,k)$ is generally greater than $ K(n,\,k)$. The values of $ C(n,\,k)$, $ k = 1, \ldots ,\,n - 1$, are computed for $ n = 2, \ldots ,5$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0869416-1
PII: S 0002-9947(1987)0869416-1
Keywords: Best constants, norm inequalities, difference operator, powers of operators, sequence spaces, Kolmogorov constants, spline functions
Article copyright: © Copyright 1987 American Mathematical Society