Efficient Filon-type methods for \(\int_a^b f(x)\,e^{i\omega g(x)}\, dx\) (Q868673)

The paper represents a modification of the Filon-type method for computing oscillatory integrals \(\int^h_a\exp(i\overline\omega g(x))f(x)\,dx\) with the phase \(g\) and amplitude \(f\). The main idea is as follows. Outside the neighborhood of the stationary point \(\eta\) coordinate \(y=g(x)\) is used; inside the neighborhood of the stationary point \(\eta\) coordinate \(t:t^p=g(x)-g(\eta)\), \(p\geq 2\) is used. Furthermore, the amplitude \(f\) in the oscillatory integral is replaced by the polynomial \(q\) and can be computed explicitly based on the first few generalized moments [\textit{M. Abramowitz} and \textit{I. A. Stegun}, Handbook of Mathematical Functions with formulas, graphs and mathematical tables. US. Department of Commerce, Washington (1964; Zbl 0171.38503)]. There is upper estimate of oscillatory integrals with amplitude \(f-q\).