Abstract
In this paper we will discuss how to construct and compute a new set of orthogonal polynomials from an existing one. For a given pair of positive integers (n, r) and a given positive measure dσ(t), we will construct a set of orthogonal polynomials corresponding to the modified measure \( d\hat \sigma \left( t \right) = {\left( {{\pi _n}\left( t \right)} \right)^{2r}}d\sigma \left( t \right)\). For r = 2 and the first-kind Chebyshev measure we are able to find explicit formulas for the recurrence coefficients of the new set of polynomials. A conjecture is made on those coefficients for any positive integer r and the first-kind Chebyshev measure. For r = 2 and arbitrary measures, a computational method is proposed. Other results are also stated.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Bellen, Alcuni problemi aperti sulla convergenza in media dell’interpolazione Lagrangiana estesa, Rend. Ist. Mat. Univ. Trieste, 20 (1988), Fasc. suppl., 1–9.
W. Gautschi and S. Li, A set of orthogonal polynomials induced by a given orthogonal polynomial, Technical Report CSD-TR-92–075, Purdue University, October 1992.
W. Gautschi and S. Li, A set of orthogonal polynomials induced by a given orthogonal polynomial, Aequationes Math., 46 (1993), 174–198.
W. Gautschi and S. Li, On quadrature convergence of extended Lagrange interpolation, Math. Comp., 65 (1996), 1249–1256.
W. Gautschi and S.E. Notaris, Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegö type, J. Comput. Appl. Math., 25 (1989), 199–224.
J. Kautsky and G.H. Golub, On the calculation of Jacobi matrices,Linear Algebra Appl., 52/53 (1983), 439–455.
S. Li, Kronrod extension of Turán formula, Studia Sci. Math. Hungar., 29 (1994), 71–83.
P. Nevai, A new class of orthogonal polynomials,Proc. Amer. Math. Soc., 91 (1984), 409–415.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Basel AG
About this paper
Cite this paper
Li, S. (1999). Construction and Computation of a New Set of Orthogonal Polynomials. In: Gautschi, W., Opfer, G., Golub, G.H. (eds) Applications and Computation of Orthogonal Polynomials. International Series of Numerical Mathematics, vol 131. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8685-7_10
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8685-7_10
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9728-0
Online ISBN: 978-3-0348-8685-7
eBook Packages: Springer Book Archive