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

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

2006-02-12

arxiv.org/abs/math/0602251v1

Mathematics

Numerical Analysis

22 pages

Scientific paper

Using present a unified approach, we establish a Kolmogorov type comparison theorem for the classes of $2\pi$-periodic functions defined by a special class of operators having certain oscillation properties, which includes the classical Sobolev class of functions with 2$\pi$-periodic, the Achieser class, and the Hardy-Sobolev class as its special examples. Then, using these results, we prove a Taikov type inequality, and calculate the exact values of Kolmogorov, Gel$'$fand, linear and information $n$--widths of this class of functions in some space $L_{q}$, which is the classical Lebesgue integral space of 2$\pi$--periodic with the usual norm.

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