Roman Bilichenko

Oles Honchar Dnipro National University · Department of Mathematics

We obtained the value of the best approximation of unbounded functional $F_f(x) = (A^kx, f)$ on the class $\{ x\in D(A^r) \colon \| A^r x \| \leqslant 1 \}$ by linear bounded functionals ($A$ is a self-adjoint operator in the Hilbert space $H$, $f\in H$, $k < r$).

The well-known Taikov’s refined versions of the Hardy – Littlewood – Pólya inequality for the L 2 -norms of intermediate derivatives of a function defined on the real axis are generalized to the case of powers of self-adjoint operators in a Hilbert space.

We determine the best approximation of an arbitrary power A^k of an unbounded self-adjoint operator A in a Hilbert space H on the class {x ∈ D(A^r) : ∥A^rx∥ ≤ 1}, k r.