O. Davydov, Characterization of the best uniform approximation of periodic functions by convex classes defined by strictly CVD kernels, in "Approximation Theory VIII" (C.K.Chui and L.L.Schumaker, Eds.), Vol. 1: "Approximation and Interpolation," pp. 177-184, World Scientific Publishing Co., 1995.

Abstract: We present a Chebyshev-type characterization for the best uniform approximations of periodic continuous functions by functions of the class $${\cal M} = \{ f(x): \enskip f(x) = \int_0^{2\pi}K(x,y) h(y)\,dy , \enskip |h(y)| \leq 1 {\rm \enskip a.\ e.} , \enskip y \in [0, 2\pi ) \} ,$$ where $K(x,y)$ is a strictly cyclic variation diminishing kernel.

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