Numerical evaluation of finite Fourier integrals

A computationally efficient algorithm for evaluating Fourier integrals ∫¹_?1?(x)e^iωxdx using interpolatory quadrature formulas on any set of collocation points is presented. Examples are given to illustrate the performances of interpolatory formulas which are based on the applications of the Fejér, Clenshaw—Curtis, Basu and the Newton—Cotes points. Initially, the formulas for nonoscillatory integrals are generated and then generalizations to finite Fourier integrals are made. Extensions of this algorithm to some other weighted integrals are also considered.