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On the Marchaud-type inequality

Author: Z. Ditzian
Journal: Proc. Amer. Math. Soc. 103 (1988), 198-202
MSC: Primary 26A15; Secondary 26D10, 41A25
MathSciNet review: 938668
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Abstract: Marchaud's inequality, which is valid in many function spaces, was strengthened by M. F. Timan [3] for $ {L_p},1 < p < \infty $, following a technique of A. Zygmund [5]. In both the above-mentioned articles the powerful Littlewood-Paley theorem is used. In the present paper a direct and, I believe much simpler, proof is given for that stronger Marchaud-type inquality. Moreover, the result will apply to a more general class of function spaces. It will be shown that it is sufficient that the "modulus of smoothness" of the norm is of "power-type" $ p$ and that translation is an isometry.

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  • [1] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
  • [2] Robert C. Sharpley, Cone conditions and the modulus of continuity, Second Edmonton conference on approximation theory (Edmonton, Alta., 1982), CMS Conf. Proc., vol. 3, Amer. Math. Soc., Providence, RI, 1983, pp. 341–351. MR 729339
  • [3] M. F. Timan, Inverse theorems of the constructive theory of functions in 𝐿_{𝑝} spaces (1≤𝑝≤∞), Mat. Sb. N.S. 46(88) (1958), 125–132 (Russian). MR 0100198
  • [4] A. F. Timan, Theory of approximation of functions of a real variable, Translated from the Russian by J. Berry. English translation edited and editorial preface by J. Cossar. International Series of Monographs in Pure and Applied Mathematics, Vol. 34, A Pergamon Press Book. The Macmillan Co., New York, 1963. MR 0192238
  • [5] A. Zygmund, A remark on the integral modulus of continuity, Univ. Nac. Tucumán. Revista A. 7 (1950), 259–269. MR 0042479

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Keywords: Marchaud type inequality, moduli of smoothness, norm modulus of smoothness of power type $ p$
Article copyright: © Copyright 1988 American Mathematical Society