On sequences that do not increase the number of real roots of polynomials

Abstract

A complete description is given for the sequences {λ_k} ^∞_k=0 such that, for an arbitrary real polynomial$f(t) = \sum\nolimits_{k = 0}^n {a_k t^k }$, an arbitraryA ε (0,+∞), and a fixedC ε (0,+∞), the number of roots of the polynomial$(Tf)(t) = \sum\nolimits_{k = 0}^n {a_k \lambda _k t^k }$ on [0,C] does not exceed the number of roots off(t) on [0,A].