On orthogonal polynomials for certain nondefinite linear functionals
Nondefinite linear functionals \(L_n[f]= \int_{\mathbb{R}} w(x) f^{(n)}(x) dx\), i.e., polynomials \(P_m\) of degree \(\leq m\) satisfying the relation \(\int_{\mathbb{R}} w(x)(P_m(x) x^k)^{(n)} dx= 0\) are considered. The problem is studied whether there exist polynomials \(P_m\) which satisfy the foregoing relation and all of whose zeros are real. Nonexistence of orthogonal polynomials with all zeros real in several cases are proved. The cases studied include different relations between \(w\), \(n\), and \(r\). The connection with moment preserving spline approximation is used for the proofs.
scientific article; zbMATH DE number 1326361
scientific article