Stechkin's problem for functions of a self-adjoint operator in a Hilbert space, Taikov-type inequalities and their applications
In this paper we solve the problem of approximating functionals (ϕ(A)x, f) (where ϕ(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 ψ(A) of the operator A. In addition, we obtain a series of sharp Taikov-type additive inequalities that estimate |(ϕ(A)x, f)| with the help of kψ(A)xk and kxk. We also present several applications of the obtained results. First, we find sharp constants in inequalities of the type used in H¨ormander 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 1 i d dt in the spaces L2(R) and L2(T), as well as for integrals with respect to spectral measures, defined with the help of classical orthogonal polynomials.