A class of quadrature formulas

Author:
Ravindra Kumar

Journal:
Math. Comp. **28** (1974), 769-778

MSC:
Primary 65D30

MathSciNet review:
0373240

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Abstract | References | Similar Articles | Additional Information

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 [3]. Finally, bounds on the error are obtained.

**[1]**G. Szegö,*Orthogonal Polynomials*, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1959.**[2]**Philip J. Davis,*Interpolation and approximation*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR**0157156****[3]**M. M. Chawla and M. K. Jain,*Error estimates for Gauss quadrature formulas for analytic functions*, Math. Comp.**22**(1968), 82–90. MR**0223093**, 10.1090/S0025-5718-1968-0223093-3

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DOI:
https://doi.org/10.1090/S0025-5718-1974-0373240-0

Keywords:
Weight function,
orthogonal polynomials,
generating function,
recurrence relation,
quadrature formulas,
convergence,
bound of error

Article copyright:
© Copyright 1974
American Mathematical Society