On biorthogonal systems associated to orthogonal polynomials

Abstract

We consider biorthogonal systems of functions associated to derivatives of orthogonal polynomials in the case of general weights. For Freud polynomials, it is proved that the derivatives of any orders of them constitute Hilbertian bases in the space of weighted square integrable functions. §1. IntroductionIt is known a characteristic property of the classical orthogonal polynomials, which states that their derivatives are also orthogonal polynomials. Such a property has many applications in the theory of orthogonal expansions and numerical computations of derivatives. Our purpose in the present paper is to deal with derivatives of non classical orthogonal polynomials from the aspect of expansions and approximations.In Section 2 we consider biorthogonal systems associated to derivatives of orthogonal polynomials in the cases of general weights. Section 3 deals with Freud weights. Among others it is proved that the derivatives of any orders of Freud orthonormal polynomials constitute Hilbertian bases in weighted spaces. Some further results concerning the derivatives of Freud-type polynomials can be found in [1], [2]. Some notations will be used. L p (a, b) (−∞ a < b ∞, 1 p ∞) denotes the space of all functions f defined on (a, b) with the norms