Best Uniform Approximation 3 Theorem 1

We present a Chebyshev-type characterization for the best uniform approximations of periodic continuous functions by functions of the class M = ff(x) : f(x) = Z 2 0 K(x; y)h(y) dy; jh(y)j 1 a: e:; y 2 0; 2)g; where K(x; y) is a strictly cyclic variation diminishing kernel. x1 Introduction Let W r 1 a; b] be the Sobolev class of functions in C r?1 a; b] whose (r?1)st derivative is absolutely continuous and whose rth derivative is an element of the unit ball of L 1 a; b] and let W r 1 be the analogous class of 2-periodic functions. N. Korneichuk obtained in 1961 (see 4], p. 225) a characterization for the best approximation of a continuous 2-periodic function by functions of the class W 1 1. His characterization was evidently valid in the non-periodic case for the class W 1 1 a; b]. In 1980 U. Sattes 6] extended this result to all the classes W r 1 a; b] with r 2. It turned out that the best approximation coincides in some subinterval with a perfect spline of degree r satisfying some Chebyshev-type conditions. Shortly thereafter A. Pinkus 5], motivated by Sattes' result, considered the class M 0;1] = ff(x) : f(x) = Z 1 0 K(x; y)h(y) dy; l(y) h(y) u(y)g; where u; l 2 C0; 1], xed, u > l, and where K(x; y) is a continuous strictly totally positive kernel. For this class he obtained existence, uniqueness and