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On orthogonal polynomials for certain nondefinite linear functionals. (English) Zbl 0942.42014
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.

MSC:
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
26C10 Real polynomials: location of zeros
65D07 Numerical computation using splines
41A15 Spline approximation
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