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Comparison Theorems of Kolmogorov Type for Classes Defined by Cyclic Variation Diminishing Operators and Their Application

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Abstract

Presenting a unified approach, we establish a Kolmogorov-type comparison theorem for classes of 2π-periodic functions defined by a special class of operators having certain oscillation properties, which include the classical Sobolev class of 2π-periodic functions, the Achieser class, and the Hardy-Sobolev class as examples. Then, using these results, we prove a Taikov-type inequality, and calculate the exact values of the Kolmogorov, Gel'fand, linear, and information n-widths of these classes of functions in the space Lq, which is the classical Lebesgue integral space of 2π-periodic functions with the usual norm.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

    Fang Gensun

  2. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

    Li Xuehua

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  1. Fang Gensun
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  2. Li Xuehua
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Correspondence to Fang Gensun.

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Gensun, F., Xuehua, L. Comparison Theorems of Kolmogorov Type for Classes Defined by Cyclic Variation Diminishing Operators and Their Application. Constr Approx 27, 99–120 (2008). https://doi.org/10.1007/s00365-006-0647-2

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  • DOI: https://doi.org/10.1007/s00365-006-0647-2

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