Best approximation of a fractional-order differentiation operator in the uniform norm on the axis on the class of functions with integrable Fourier transform of the highest derivative
Abstract:
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 $L^2$ to the space $C$ on the class of functions $\mathcal{Q}^n$ whose Fourier transform of the $n$th-order fractional derivative, $0\le k<n,$ is integrable. The corresponding exact Kolmogorov inequality is given. A solution is obtained to the problem of optimal recovery of the differentiation operator of fractional order $k$ on functions of the class $\mathcal{Q}^n$ defined with a known error in the space $L^2.$
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