APPROXIMATION OF ONE CLASS OF SMOOTH FUNCTIONS BY ANOTHER CLASS OF SMOOTHER FUNCTIONS ON THE AXIS

This paper investigates the problem of best and best linear approximation in the space of functions on the real axis with bounded Fourier transform. The study focuses on approximating the class \(B_1^{n-k}(1)\) of functions whose derivatives of order \(n-k-1\) have variation bounded by 1 by the class \(B_{2}^n(N)\) of functions whose \(n\)th-order derivative (\(0 \le k < n\)) belongs to the space \(L_2(-\infty,\infty)\) with norm bounded by \(N > 0\). This problem is related to Stechkin's problem and the corresponding sharp Kolmogorov inequality, both previously studied by the author. Stechkin's problem concerns the best approximation in the uniform norm on the real axis of \(k\)th-order differentiation operators by bounded linear operators from \(L_2\) to \(C\), considered on the class of functions whose Fourier transform of the \(n\)th-order derivative (\(0 \le k < n\)) is summable.