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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
DOI: https://doi.org/10.1090/S0002-9947-1987-0869416-1
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$.

References [Enhancements On Off] (What's this?)

  • [1] J. Hadamard, Sur le module maximum d'une fonction et de ses dérivées, C. R. Acad. Sci. Paris Ser. A 42 (1914), 68-72.
  • [2] E. Landau, Einige Ungleichungen für zweimal differentiierbare Funktionen, Proc. London Math. Soc. 13 (1913), 43-49.
  • [3] G. E. Šilov, On inequalities between derivatives, Moskov. Gos. Univ. Sb. Rabot Stud. Nauchn. Kruzhkov (1937), 17-27. (Russian)
  • [4] A. N. Kolmogorov, Dokl. Akad. Nauk SSSR (N.S.) 15 (1937), 107-112.
  • [5] -, On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval, Moscov. Gos. Univ. Uchen. Zap. Mat. 30 (1939), 3-13; English transl., Amer. Math. Soc. Transl. 4 (1949); reprint, Amer. Math. Soc. Transl. (1) 2 (1962), 233-243. MR 0031009 (11:86d)
  • [6] Ju. I. Ljubič, On the belonging of the powers of an operator of a given vector to a certain linear class, Dokl. Akad. Nauk SSSR 102 (1955), 881-884. (Russian) MR 0071731 (17:176c)
  • [7] -, On inequalities between the powers of a linear operator, Izv. Akad. Nauk SSSR Ser. Mat. 24 (1960), 825-864; English transl., Amer. Math. Soc. Transl. (2) 40 (1964), 39-84. MR 0130576 (24:A436)
  • [8] H. A. Gindler and J. A. Goldstein, Dissipative operator versions of some classical inequalities, J. Analyse Math. 28 (1975), 213-238. MR 0482361 (58:2434)
  • [9] -, Dissipative operators and series inequalities, Bull. Austral. Math. Soc. 23 (1981), 429-442. MR 625184 (83d:47040)
  • [10] Z. Ditzian, Discrete and shift Kolmogorov type inequalities, Proc. Roy. Soc. Edinburgh A 93 (1983), 307-313. MR 688793 (84m:47038)
  • [11] Z. Ditzian and D. J. Newman, Discrete Kolmogorov-type inequalities, Preprint (1984).
  • [12] I. J. Schoenberg, Cardinal spline interpolation, CBMS 12, SIAM, Philadelphia, Pa., 1973. MR 0420078 (54:8095)
  • [13] C. de Boor, A practical guide to splines, Applied Mathematical Sciences, no. 27, Springer-Verlag, New York, 1978. MR 507062 (80a:65027)
  • [14] M. Abramowitz and I. A. Stegun (eds.), Handbook of mathematical functions, NBS Applied Mathematics Series, no. 55, Washington, D. C., 1964. MR 0167642 (29:4914)
  • [15] L. L. Schumaker, Spline functions: Basic theory. Wiley, New York, 1981. MR 606200 (82j:41001)

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Additional Information

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

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