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Convexity, moduli of smoothness and a Jackson-type inequality

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Abstract

For a Banach space B of functions which satisfies for some m>0

$$ \max ({\|F+G\|}_B,{\|F-G\|}_B)\geqq ({\|F\|}^s_B+m{\|G\|}^s_B)^{1/s},\quad \forall \,F,G\in B $$
(∗)

a significant improvement for lower estimates of the moduli of smoothness ω r(f,t) B is achieved. As a result of these estimates, sharp Jackson inequalities which are superior to the classical Jackson type inequality are derived. Our investigation covers Banach spaces of functions on ℝd or \(\mathbb{T}^{d}\) for which translations are isometries or on S d−1 for which rotations are isometries. Results for C 0 semigroups of contractions are derived. As applications of the technique used in this paper, many new theorems are deduced. An L p space with 1<p<∞ satisfies (∗) where s=max  (p,2), and many Orlicz spaces are shown to satisfy (∗) with appropriate s.

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Authors and Affiliations

  1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1

    Z. Ditzian

  2. Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada, R3T 2N2

    A. Prymak

Authors
  1. Z. Ditzian
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  2. A. Prymak
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Correspondence to Z. Ditzian.

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Ditzian, Z., Prymak, A. Convexity, moduli of smoothness and a Jackson-type inequality. Acta Math Hung 130, 254–285 (2011). https://doi.org/10.1007/s10474-010-0008-8

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  • DOI: https://doi.org/10.1007/s10474-010-0008-8

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2000 Mathematics Subject Classification