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A Class of Quadrature Formulas
| Content Provider | Scilit |
|---|---|
| Author | Kumar, Ravindra |
| Copyright Year | 1974 |
| Description | It is proved that there exists a set of polynomials orthogonal on $[ - 1,1]$ with respect to the weight function \begin{equation}\tag {$1$} w(t)/(t - x)\end{equation} 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{equation}\tag {$2$} \begin {array}{*{20}{c}} {w(t) = {{(1 - {t^2})}^{ - 1/2}},} \\ { = {{(1 - {t^2})}^{1/2}},} \\ { = {{((1 - t)/(1 + t))}^{1/2}},} \\ \end{array} \end{equation} and the generating functions and the recurrence relation are also given. Subsequently, a set of quadrature formulas given by \begin{equation}\tag {$3$} \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)} } \end{equation} 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. |
| Related Links | https://www.ams.org/mcom/1974-28-127/S0025-5718-1974-0373240-0/S0025-5718-1974-0373240-0.pdf |
| Ending Page | 778 |
| Page Count | 10 |
| Starting Page | 769 |
| ISSN | 00255718 |
| e-ISSN | 10886842 |
| DOI | 10.2307/2005698 |
| Journal | Mathematics of Computation |
| Issue Number | 127 |
| Volume Number | 28 |
| Language | English |
| Publisher | Duke University Press |
| Publisher Date | 1974-07-01 |
| Access Restriction | Open |
| Subject Keyword | Logic Functions Quadrature Formulas Respect Orthogonal Polynomials Simplified E_n Convergence |
| Content Type | Text |
| Resource Type | Article |
| Subject | Applied Mathematics Algebra and Number Theory Computational Mathematics |
