The Hardy–Littlewood–Pólya inequality for analytic functions in Hardy–Sobolev spaces

For a function of a complex variable analytic in a strip the extremum of the $L_2(\mathbb R)$ norm of the $k$th derivative is found under a constraint on the $L_2(\mathbb R)$-norm of the function and the norm of its $n$th derivative in the metric of the Hardy–Sobolev space. The closely connected problem of the optimal recovery of the $k$th derivative of a function in the Hardy–Sobolev class from the inaccurately given trace of this function on the real axis is also studied. An optimal recovery method is found.
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