Mean convergence of Lagrange interpolation for Freud’s weights with application to product integration rules
نویسنده
چکیده
The connection between convergence of product integration rules and mean convergence of Lagrange interpolation in L, (1 <p < 00) has been thoroughly analysed by Sloan and Smith [37]. Motivated by this connection, we investigate mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials associated with Freud weights on R. Our results apply to the weights exp(-x”/2), m = 2,4,6. . . , and for the Hermite weight (m = 2) extend results of Nevai [28] and Bonan [2] in at least one direction. The results are sharp in L,, , 1 < p G 2. As a consequence, we can improve results of Smith, Sloan and Opie [38] on convergence of product integration rules based on the zeros of the orthogonal polynomials associated with the Hermite weight. In the process, we prove a new Markov-Stieltjes inequality for Gauss quadrature sums, and solve a problem of Nevai on how to estimate certain quadrature sums.
منابع مشابه
Convergence of Product Integration Rules for Weights on the Whole Real Line II
We continue our investigation of product integration rules associated with weights on the whole real line, such as exp jxj ; > 1. In an earlier paper, we considered interpolatory integration rules whose abscissas are the zeros of an orthogonal polynomial associated with the weight. In this paper, we show the advantage of adding two extra points to the zeros, following an idea of J. Szabados. Th...
متن کاملQuadrature Sums and Lagrange Interpolation for General Exponential Weights
where > 0. Once the theory had been developed in its entirety, it became clear that one could simultaneously treat not only weights like those above, but also not necessarily even weights on a general real interval. See [3], [12], [16] for various perspectives on this type of potential theory and its applications. One important application is to Lagrange interpolation. Mean convergence of Lagra...
متن کاملHermite and Hermite-fejér Interpolation of Higher Order and Associated Product Integration for Erdős Weights
Using the results on the coefficients of Hermite-Fejér interpolations in [5], we investigate convergence of Hermite and Hermite-Fejér interpolation of order m, m = 1, 2, . . . in Lp(0 < p < ∞) and associated product quadrature rules for a class of fast decaying even Erdős weights on the real line.
متن کاملPointwise convergence of derivatives of Lagrange interpolation polynomials for exponential weights
For a general class of exponential weights on the line and on (−1, 1), we study pointwise convergence of the derivatives of Lagrange interpolation. Our weights include even weights of smooth polynomial decay near ±∞ (Freud weights), even weights of faster than smooth polynomial decay near ±∞ (Erdős weights) and even weights which vanish strongly near ±1, for example Pollaczek type weights. 1991...
متن کاملIteration-Free Computation of Gauss-Legendre Quadrature Nodes and Weights
Gauss–Legendre quadrature rules are of considerable theoretical and practical interest because of their role in numerical integration and interpolation. In this paper, a series expansion for the zeros of the Legendre polynomials is constructed. In addition, a series expansion useful for the computation of the Gauss–Legendre weights is derived. Together, these two expansions provide a practical ...
متن کامل