# 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 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## Best constants in norm inequalities for the difference operatorHTML articles powered by AMS MathViewer

by Hans G. Kaper and Beth E. Spellman
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 $n = 2, \ldots ,5$.
References
J. Hadamard, Sur le module maximum d’une fonction et de ses dérivées, C. R. Acad. Sci. Paris Ser. A 42 (1914), 68-72. E. Landau, Einige Ungleichungen für zweimal differentiierbare Funktionen, Proc. London Math. Soc. 13 (1913), 43-49. G. E. Šilov, On inequalities between derivatives, Moskov. Gos. Univ. Sb. Rabot Stud. Nauchn. Kruzhkov (1937), 17-27. (Russian) A. N. Kolmogorov, Dokl. Akad. Nauk SSSR (N.S.) 15 (1937), 107-112.
• A. Kolmogoroff, On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval, Amer. Math. Soc. Translation 1949 (1949), no. 4, 19. MR 0031009
• Yu. I. Lyubič, On the belonging of the powers of an operator on a given vector to a certain linear class, Dokl. Akad. Nauk SSSR (N.S.) 102 (1955), 881–884 (Russian). MR 0071731
• Ju. I. Ljubič, On inequalities between powers of a linear operator, Izv. Akad. Nauk SSSR Ser. Mat. 24 (1960), 825–864 (Russian). MR 0130576
• Herbert A. Gindler and Jerome A. Goldstein, Dissipative operator versions of some classical inequalities, J. Analyse Math. 28 (1975), 213–238. MR 482361, DOI 10.1007/BF02786813
• Herbert A. Gindler and Jerome A. Goldstein, Dissipative operators and series inequalities, Bull. Austral. Math. Soc. 23 (1981), no. 3, 429–442. MR 625184, DOI 10.1017/S0004972700007309
• Z. Ditzian, Discrete and shift Kolmogorov type inequalities, Proc. Roy. Soc. Edinburgh Sect. A 93 (1982/83), no. 3-4, 307–317. MR 688793, DOI 10.1017/S0308210500015997
• Z. Ditzian and D. J. Newman, Discrete Kolmogorov-type inequalities, Preprint (1984).
• I. J. Schoenberg, Cardinal spline interpolation, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0420078
• Carl de Boor, A practical guide to splines, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. MR 507062
• Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
• Larry L. Schumaker, Spline functions: basic theory, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. MR 606200
Similar Articles
• Retrieve articles in Transactions of the American Mathematical Society with MSC: 39A70, 47B39
• Retrieve articles in all journals with MSC: 39A70, 47B39