Best Approximation of a Fractional-Order Differentiation Operator in the Uniform Norm on the Axis on the Class of Functions with Summable Fourier Transform of the Highest Derivative

A solution is given to Stechkin’s problem on the best approximation, in the uniform norm on the real axis, of differentiation operators of fractional (more precisely, real) order k by bounded linear operators from the space L₂ to the space C on the class Qⁿ of functions whose n th-order fractional derivative, 0≤ k<n, has a summable Fourier transform. The corresponding sharp Kolmogorov inequality is presented. A solution is obtained to the problem of optimal recovery of the differentiation operator of fractional order k on functions of the class Qⁿ defined with a known error in the space L₂ .