A dilation quadrature formula for hypersingular and highly oscillatory integrals on the positive half-line

The aim of this paper is to introduce a new quadrature rule for approximating integrals with highly oscillatory and hypersingular integrands defined on the positive half-line. After the integration interval is split into the subintervals [0, M] and [M,+1), so that the part on [M,+1) is negligible, the interval [0,M] is suitably dilated and decomposed into a sum of integrals, where each of them is approximated by a Gaussian quadrature rule. We prove that the formula is convergent when the function f is bounded on R+ together with a certain number of its derivatives. Numerical tests compare the performance of the proposed rule with other formulas available in the literature.