Stechkin's problem for functions of a self-adjoint operator in a Hilbert space, Taikov-type inequalities and their applications

Abstract

In this paper we solve the problem of approximating functionals $(\phi (A)x,f)$ (where $\phi (A)$ is some function of self-adjoint operator $A$) on the class of elements of a Hilbert space that is defined with the help of another function $\psi (A)$ of the operator $A$. In addition, we obtain a series of sharp Taikov-type additive inequalities that estimate $|(\phi (A)x,f)|$ with the help of $\Vert \psi (A)x\Vert$ and $\Vert x\Vert$. We also present several applications of the obtained results. First, we find sharp constants in inequalities of the type used in H$\ddot{\mathrm{o}}$rmander theorem on comparison of operators in the case when operators are acting in a Hilbert space and are functions of a self-adjoint operator. As another application we obtain Taikov-type inequalities for functions of the operator $\frac{1}{i}\frac{d}{dt}$ in the spaces $L_2(\RR)$ and $L_2(\TT)$, as well as for integrals with respect to spectral measures, defined with the help of classical orthogonal polynomials.