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Extended quadrature rules for oscillatory integrands. (English) Zbl 1025.41016
This paper is concerned with the approximation of the integral \(I[y]=\int _{-1}^1 y(x)dx\) by a quadrature rule of the form \[ Q_N[y]=\sum _{k =1}^N\left ((w _k^{(0)}y(x_k)+w _k^{(1)}y^{(1)}(x_k)+\dots +w _k^{(p)}y^{(p)}(x_k)\right), \] i.e., by a rule which uses the values of both \(y\) and its derivatives up to \(p\)-th order at \(N\) given nodes. The case \(p=1\) has been considered by the same authors [J. Comput. Appl. Math. 140, 479-497(2002; Zbl 1002.41015)]. As in this previous paper the function \(y\) is assumed to be a polynomial or an \(\omega \) dependent function of the form \(y(x)=f_1(x)\cos(\omega x)+f_2(x)\sin(\omega x) \) with smoothly varying \(f_1\) and \(f_2\). Formalisms for deriving polynomial-fitting quadrature rules or versions tuned on oscillatory integrands are generated. A numerical illustration is presented.

41A55 Approximate quadratures
EXPFIT4; Maple
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