On sparse spectral Jacobi method for highly oscillatory integrals

One of the attractive research problems is the computation of highly oscillatory integrals in the field of applied sciences. For this purpose, many efforts are performed to get an accurate, stable, and efficient analysis of these integrals. The Levin method with Chebyshev polynomials and Chebyshev operators stands out as an attractive procedure in the literature. Despite the attractiveness of the numerical experiments in the literature, there remains a gap that requires further exploration. Therefore, the focus of this work is on connecting Jacobi operators and Jacobi polynomials with the Levin method. The proposed method performs better in accuracy and produces sparse and well-conditioned matrices, in contrast to the Levin method based on Chebyshev polynomials and operators in the literature [Li et al., A universal solution to one-dimensional oscillatory integrals, Sci. China Ser. F: Inf. Sci. 51(10) (2008), pp. 1614–1622; Ma and Liu, A well-conditioned Levin method for calculation of highly oscillatory integrals and its application, J. Comput. Appl. Math. 342 (2018), pp. 451–462]. We perform several numerical experiments to validate the claims of the proposed method.