an:01326361
Zbl 0942.42014
Ehrich, Sven
On orthogonal polynomials for certain nondefinite linear functionals
EN
J. Comput. Appl. Math. 99, No. 1-2, 119-128 (1998).
00059139
1998
j
42C05 33C45 26C10 65D07 41A15
nondefinite linear functionals; orthogonal polynomials; zeros; spline approximation
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.
V??clav Burjan (Praha)