Abstract:
We will discuss Stechkin's problem on the best approximation of the differentiation operator of order $k$ on the class of $n$ times differentiable functions $(0\le k < n)$ by bounded linear operators in the spaces $L_p$ on the real line and on the half-line as well as the related problem on the optimal recovery of smooth functions given with a known error. A solution of Stechkin's problem will be given in the space $L_2(0,\infty)$ for $k = 1$ and $n = 2;$ this result was obtained jointly with M.A.Filatova.
References
S. B. Stechkin, “Nailuchshee priblizhenie lineinykh operatorov”, Mat. zametki, 1:2 (1967), 137–148
V. V. Arestov, “Priblizhenie neogranichennykh operatorov ogranichennymi i rodstvennye ekstremalnye zadachi”, Uspekhi mat. nauk, 51:6 (1996), 89–124
V. I. Berdyshev, “Nailuchshee priblizhenie v $L[0,\infty)$ operatora differentsirovaniya”, Mat. zametki, 9:5 (1971), 477–481
V. V. Arestov, M. A. Filatova, “Best approximation of the differentiation operator in the space $L_2$ on the semiaxis”, JAT, 187 (2014), 65–81
