A class of quadrature formulas
Abstract: It is proved that there exists a set of polynomials orthogonal on with respect to the weight function
corresponding to the polynomials orthogonal on with respect to the weight function w. Simplified forms of such polynomials are obtained for the special cases
and the generating functions and the recurrence relation are also given. Subsequently, a set of quadrature formulas given by
for and (1, 1) is established; these formulas are valid for analytic functions. Convergence of the quadrature rules is discussed, using a technique based on the generating functions. This method appears to be simpler than the one suggested by Davis [2, pp. 311-312] and used by Chawla and Jain . Finally, bounds on the error are obtained.
-  G. Szegö, Orthogonal Polynomials, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1959.
-  Philip J. Davis, Interpolation and approximation, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR 0157156
-  M. M. Chawla and M. K. Jain, Error estimates for Gauss quadrature formulas for analytic functions, Math. Comp. 22 (1968), 82–90. MR 0223093, https://doi.org/10.1090/S0025-5718-1968-0223093-3
- G. Szegö, Orthogonal Polynomials, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1959.
- P. J. Davis, Interpolation and Approximation, Blaisdell, New York, 1963. MR 28 #5160. MR 0157156 (28:393)
- M. M. Chawla & M. K. Jain, "Error estimates for Gauss quadrature formulas for analytic functions," Math. Comp., v. 22, 1968, pp. 82-90. MR 36 #6142. MR 0223093 (36:6142)
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Keywords: Weight function, orthogonal polynomials, generating function, recurrence relation, quadrature formulas, convergence, bound of error
Article copyright: © Copyright 1974 American Mathematical Society