Comparison Theorems of Kolmogorov Type for Classes Defined by Cyclic Variation Diminishing Operators and Their Application |
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Authors: | Fang Gensun Li Xuehua |
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Affiliation: | (1) School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China;(2) School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China |
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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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