On quadrature of highly oscillatory Bessel function via asymptotic analysis of simplex integrals

In this article, two methods for evaluating highly oscillatory Bessel integrals are explored. Firstly, a polynomial is analyzed as an effective approximation of the simplex integral of a highly oscillatory Bessel function based on Laplace transform, and its error rapidly decreases as the frequency increases. Furthermore, the inner product of $f$ and highly oscillatory Bessel function can be approximated by two other forms of inner product by which one depends on a polynomial and the higher derivatives of $f$, another depends on Bessel function and the interpolation polynomial of $f$. In addition, three issues related to highly oscillatory Bessel integrals have also been discussed: inequalities for the convergence rate of Filon-type methods, evaluation of Cauchy principal values, and simplified evaluation on infinite intervals. Through some preliminary numerical experiments, our theoretical analysis has been preliminarily confirmed, and the proposed numerical method is accurate and effective.