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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.