On sequences that do not increase the number of real roots of polynomials - Bakan, A. G.; Holub, A. P. (Ukrains’kyi Matematychnyi Zhurnal, 1993)
A complete description is given for the sequences $\{λ_k}_{k = 0}^{ ∞}$ such that, for an arbitrary real polynomial $f(t) = \sum olimits_{k = 0}^n {a_k t^k }$, an arbitrary $A \in (0, +∞)$, and a fixed $C \in (0,+∞)$, the number of roots of the polynomial $(Tf)(t) = \sum olimits_{k = 0}^n {a_k λ _k t^k }$ on $[0,C]$ does not exceed the number of roots off $(t)$ on $[0, A]$.