2017/03/11 by В. Ф. Бабенко, Babenko, Vladyslav, Yuliya Babenko +3
Mathematics · #Numerical methods in inverse problems #Differential Equations and Boundary Problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1703.04045
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 ‖ ψ(A)x‖ and ‖ x‖. We also present several applications of the obtained results. First, we find sharp constants in inequalities of the type used in H\rmormander 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 \frac1i \frac ddt in the spaces L2(\RR) and L2(\TT), as well as for integrals with respect to spectral measures, defined with the help of classical orthogonal polynomials.