On some problems of optimal recovery of analytic and harmonic functions from inaccurate data

Some general theorems on optimal recovery are proved which are then applied to solving recovery problems in the Hardy $H_p$ and Bergman $A_p$ spaces as well as in the analogous spaces of harmonic functions. In particular, we obtain a generalization of the Schwartz lemma for these spaces. For the space $H_p$, we solve the problem of finding an optimal formula for numeric differentiation which uses the values, known to within an error $\delta$, of functions at the points $-h$ and $h$. For any fixed $\delta$ and $p=\infty$, we find the optimal value of $h$.