An Estimate of Approximation of a Matrix-Valued Function by an Interpolation Polynomial - Kurbatov, Vitalii; Kurbatova, Irina (Eurasian Mathematical Journal, 2020)
Let A be a square complex matrix; z1, . . . , zn ∈ C be (possibly repetitive) points of interpolation; f be a function analytic in a neighborhood of the convex hull of the union of the spectrum of A and the points z1, . . . , zn; and p be the interpolation polynomial of f constructed by the points z1, . . . , zn. It is proved that under these assumptions‖f(A)− p(A)‖ ≤ 1n!maxt∈[0,1]µ∈co{z1,z2,...,zn}∥∥Ω(A)f (n)((1− t)µ1 + tA)∥∥,where Ω(z) =∏nk=1(z − zk) and the symbol co means the convex hull.