A Note on Finite Quadrature Rules with a Kind of Freud Weight Function

We introduce a finite class of weighted quadrature rules with the weight function |x|−2aexp (−1/x2) on (−∞,∞) as ∫−∞∞|x|−2aexp (−1/x2)f(x)dx=∑i=1nwif(xi)+Rn[f], where xi are the zeros of polynomials orthogonal with respect to the introduced weight function, wi are the corresponding coefficients, and...

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Published in:Mathematical Problems in Engineering 2009-01, Vol.2009 (1), p.1280-1287-76
Main Authors: Aghigh, Kamal, Masjed-Jamei, M.
Format: Article
Language:English
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Summary:We introduce a finite class of weighted quadrature rules with the weight function |x|−2aexp (−1/x2) on (−∞,∞) as ∫−∞∞|x|−2aexp (−1/x2)f(x)dx=∑i=1nwif(xi)+Rn[f], where xi are the zeros of polynomials orthogonal with respect to the introduced weight function, wi are the corresponding coefficients, and Rn[f] is the error value. We show that the above formula is valid only for the finite values of n. In other words, the condition a≥{max n}+1/2 must always be satisfied in order that one can apply the above quadrature rule. In this sense, some numerical and analytic examples are also given and compared.
ISSN:1024-123X
1563-5147
DOI:10.1155/2009/421546