Best approximation of the differentiation operator on the class of functions analytic in a strip (Q2516589)

Let \(Y\) be the horizontal strip in the complex plane which consists of the numbers with imaginary values between \(0\) and \(1\). Let \(1 \leq p \leq \infty\). Let \(H\) be the set of real-valued functions which are analytic on \(Y\) and which have bounded \(L_p\) norm on each horizontal line \(h\) within \(Y\). An operator \(D\) -- which carries a function, differentiable on \(h\), to its derivative -- is an unbounded operator. This work addresses the important problem of approximating \(D\) restricted to \(H\), by a bounded linear operator. The questions investigated include the error of approximation, the modulus of continuity of the differentiation operator on \(H\), and an optimal recovery problem for the derivative of a functions.