Quadrature rules for \(L^1\)-weighted norms of orthogonal polynomials. (English) Zbl 1350.33012
Summary: In this paper, we obtain \(L^1\)-weighted norms of classical orthogonal polynomials (Hermite, Laguerre and Jacobi polynomials) in terms of the zeros of these orthogonal polynomials; these expressions are usually known as quadrature rules. In particular, these new formulae are useful to calculate directly some positive defined integrals as several examples show.
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 65D32 | Numerical quadrature and cubature formulas |
References:
| [1] | Abadias, L., Miana, P.J.: C0-semigroups and resolvent operators approximated by Laguerre expansions. pp. 1-26 (2013). ArXiv:1311.7542 · Zbl 1357.33012 |
| [2] | Abadias, L., Miana, P.J.: Hermite expansions of C0-groups and cosine functions. J. Math. Anal. Appl. 426, 288-311 (2015) · Zbl 1321.47032 |
| [3] | Beckermann B., Bustamante J., Martínez-Cruz R., Quesada J.M.: Gaussian, Lobatto and Radau positive quadrature rules with a prescribed abscissa. Calcolo 51, 319-328 (2014) · Zbl 1312.65032 · doi:10.1007/s10092-013-0087-3 |
| [4] | Bonis, M.C.de , della Vecchia, B., Mastroianni, G.: Approximation of the Hilbert transform on the real line using Hermite zeros. Math. Comput. 71, 1169-1188 (2001) · Zbl 0993.65145 |
| [5] | Criscuolo G., Scuderi L.: Error bound for product quadrature rules in L1-weighted norm. Calcolo 31, 73-93 (1994) · Zbl 0837.41020 · doi:10.1007/BF02575720 |
| [6] | Devore R.A., Ridgway L.: Scott: Error bounds for Gaussian quadrature and weighted-L1 polynomial approximation. Siam J. Numer. Anal. 21, 400-412 (1984) · Zbl 0565.41028 · doi:10.1137/0721030 |
| [7] | Gautschi W.: Orthogonal polynomials and quadrature. Electron. Trans. Numer. Anal. 9, 65-76 (1999) · Zbl 0963.33004 |
| [8] | Greenwood R.E., Miller J.J.: Zeros of the Hermite polynomials and weights for Gauss’ mechanical quadrature formula. Bull. Am. Math. Soc. 54(8), 765-769 (1948) · Zbl 0033.12402 · doi:10.1090/S0002-9904-1948-09075-9 |
| [9] | Hunter D.B.: Some Gauss-Type Formulae for the evaluation of Cauchy principal values of integrals. Numer. Math. 19, 419-424 (1972) · Zbl 0231.65028 · doi:10.1007/BF01404924 |
| [10] | Jacobi C.G.J.: Über Gaußs neue Methode, die Werthe der Integrale näherungsweise zu finde. J. Reine Angew. Math. 1, 301-308 (1826) · ERAM 001.0029cj · doi:10.1515/crll.1826.1.301 |
| [11] | Lebedev, N.N.: Special functions and their applications. Selected Russian Publications in the Mathematical Sciences, Prentice-Hall International, Inc., London (1965) · Zbl 0131.07002 |
| [12] | Santos-León J.C.: Szegö polynomials and Szegö quadrature for the Féjer kernel. J. Comput. Applied Math. 179, 327-341 (2005) · Zbl 1079.42018 · doi:10.1016/j.cam.2004.09.048 |
| [13] | Salzer H.E., Zucker R.: Table of the zeros and weight factors of the first fifteen Laguerre polynomials. Bull. Am. Math. Soc. 55, 1004-1012 (1949) · Zbl 0034.19102 · doi:10.1090/S0002-9904-1949-09327-8 |
| [14] | Szegö, G.: Orthogonal polynomials, American Mathematical Society Colloquium Publications Volume XXIII. American Mathematical Society (1967) · Zbl 0023.21505 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
