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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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: It is proved that there exists a set of polynomials orthogonal on $ [ - 1,1]$ with respect to the weight function

$\displaystyle w(t)/(t - x)$ ($ 1$)

corresponding to the polynomials orthogonal on $ [ - 1,1]$ with respect to the weight function w. Simplified forms of such polynomials are obtained for the special cases

\begin{displaymath}\begin{array}{*{20}{c}} {w(t) = {{(1 - {t^2})}^{ - 1/2}},} \\... ...)}^{1/2}},} \\ { = {{((1 - t)/(1 + t))}^{1/2}},} \\ \end{array}\end{displaymath} ($ 2$)

and the generating functions and the recurrence relation are also given. Subsequently, a set of quadrature formulas given by

$\displaystyle \int_{ - 1}^1 {{{(1 + t)}^{p - 1/2}}{{(1 - t)}^{q - 1/2}}{{(1 + {a^2} + 2at)}^{ - 1}}f(t)dt = \sum\limits_{k = 1}^n {{H_k}f({t_k}) + {E_n}(f)} }$ ($ 3$)

for $ (p,q) = (0,0),(0,1)$ 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.

References [Enhancements On Off] (What's this?)

  • [1] G. Szegö, Orthogonal Polynomials, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1959.
  • [2] P. J. Davis, Interpolation and Approximation, Blaisdell, New York, 1963. MR 28 #5160. MR 0157156 (28:393)
  • [3] 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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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1974-0373240-0
PII: S 0025-5718(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