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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Best constants in norm inequalities for the difference operator
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by Hans G. Kaper and Beth E. Spellman PDF
Trans. Amer. Math. Soc. 299 (1987), 351-372 Request permission

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 \[ C(n, k) = {\operatorname {sup}}\{ {Q_{n,k}}(\xi ):\xi \in {l^\infty }({\mathbf {Z}}), {\Delta ^n}\xi \ne 0\} \] of the quotient \[ {Q_{n,k}}(\xi ) = \frac {{\left \| {{\Delta ^k}\xi } \right \|}} {{{{\left \| \xi \right \|}^{(n - k)/n}}{{\left \| {{\Delta ^n}\xi } \right \|}^{k/n}}}}\], 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
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 299 (1987), 351-372
  • MSC: Primary 39A70; Secondary 47B39
  • DOI: https://doi.org/10.1090/S0002-9947-1987-0869416-1
  • MathSciNet review: 869416