Best approximation of differentiation operators on the Sobolev class of functions analytic in a strip

A solution is obtained for interconnected extremal problems on the class of analytic functions in a strip with finite $L^2$-norms of limit values of functions on one boundary line and bounded $L^2$-norms of limit values of the derivative of order $n, n\ge 0,$ on the other boundary line: best approximation of the differentiation operators with respect to the uniform norm on an intermediate line by bounded operators; optimal recovery of the derivative of order k on an intermediate line from values of the function on the boundary line given with an error. An exact Kolmogorov-type inequality is obtained that estimates the uniform norm of the derivative of order $k$ on an intermediate line in terms of the $L^2$-norm of the limit boundary values of the function and the derivative of order $n.$