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

Abstract EN

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≥{maxn}+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.