Computation of highly oscillatory Bessel transforms with algebraic singularities

In this paper, we consider the Clenshaw-Curtis-Filon method for the highly oscillatory Bessel transform $\int_0^1x^α(1-x)^βf(x) J_ν(ωx)dx$, where $f$ is a smooth function on $[0, 1]$, and $ν\geq0.$ The method is based on Fast Fourier Transform (FFT) and fast computation of the modified moments. We give a recurrence relation for the modified moments and present an efficient method for the evaluation of modified moments by using recurrence relation. Moreover, the corresponding error bound in inverse powers of $N$ for this method for the integral is presented. Numerical examples are provided to support our analysis and show the efficiency and accuracy of the method.