A new product integration rule for the finite Hilbert transform

For the finite weighted Hilbert transform we consider two different product integration rules, the VP rule and the L-rule, based on the same nodes and obtained by approximating the density function with filtered de la Vallée Poussin and classical Lagrange interpolation polynomials, respectively. The L-rule is well known and widely studied. The VP rule is here introduced and we will prove the convergence in suitable weighted uniform spaces. Hence we will examine the performance of both the product rules, showing that in case of density functions that have some pathologies (peaks, cusps, etc.) localized in isolated points, VP rules inherit the good properties of the filtered de la Vallée Poussin type approximation, providing better performances than L-rules.