On biorthogonal systems associated to orthogonal polynomials (Q532061)

A characteristic property of the classical orthogonal polynomials states that their dervatives are also orthogonal polynomials. Such a property has many applications in the theory of orthogonal expansions and numerical computations of derivatives. The purpose of the paper is to deal with derivatives of non classical orthogonal polynomials from the aspect of expansion and approximation. The paper is organized in 3 sections. Section 1, ``Introduction'', presents shortly the content of the paper. In Section 2, biorthogonal systems associated to derivatives of orthogonal polynomials in the case of general weights are considered. The main result of this section is Theorem 1, where the properties of coordinate functionals are established. Section 3 deals with Freud weights: it is proved that the derivatives of Freud orthonormal polynomials constitute Hilbertian basis in weighted spaces. Results relating to derivatives of Freud-type polynomials (see \textit{S. Bonan} and \textit{P. Nevai} [J. Approximation Theory 40, 134--147 (1984; Zbl 0533.42015)] and \textit{S. Bonan, D. S. Lubinsky} and \textit{P. Nevai} [SIAM J. Math. Anal. 18, 1163--1176 (1987; Zbl 0638.42023)]) are generalized.