Indefinite integration of oscillatory functions by the Chebyshev series expansion (Q1819569)

The integral \(I(\omega,x)=\int^{x}_{0}f(t)e^{i\omega t}dt\), \(0\leq x\leq 1\), is considered. There exists much literature for the case \(x=1\) for the case \(\omega =0\). For I(\(\omega\),x) the function f(t) is approximated by a finite sum of shifted Chebyshev polynomials of the first kind. An error estimate of the approximated integral is given. The method obtained, combined with the extrapolation method due to \textit{A. Sidi} [J. Inst. Math. Appl. 26, 1-20 (1980; Zbl 0464.65002), Math. Comput. 38, 517-529 (1982; Zbl 0508.65011)] provides a quadrature scheme for the integrals \(\int^{\infty}_{a}f(x)\cos \omega x dx\), \(\int^{\infty}_{a}f(x)\sin \omega x dx\).