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In this paper we solve the problem of approximating functionals$(\varphi(A)x, f)$ (where $\varphi(A)$ is some function of self-adjointoperator $A$) on the class of elements of a Hilbert space that is defined withthe help of another function $\psi (A)$ of the operator $A$. In addition, weobtain a series of sharp Taikov-type additive inequalities that estimate$|(\varphi(A)x, f)|$ with the help of $\| \psi (A)x\|$ and $\| x\|$. We alsopresent several applications of the obtained results. First, we find sharpconstants in inequalities of the type used in H${\rm{\ddot{o}}}$rmander theoremon comparison of operators in the case when operators are acting in a Hilbertspace and are functions of a self-adjoint operator. As another application weobtain Taikov-type inequalities for functions of the operator $\frac1i \frac{d}{dt}$ in the spaces $L_2(\RR)$ and $L_2(\TT)$, as well as for integrals withrespect to spectral measures, defined with the help of classical orthogonalpolynomials.

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