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

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.