Simultaneous approximation of Hilbert and Hadamard transforms on bounded intervals

In this paper, we propose a compound scheme of different product integration rules for the simultaneous approximation of both Hilbert and Hadamard transforms of a given function f . The advantages of such a scheme are multiple: a saving in the number of function evaluations and the avoidance of the derivatives of the density function f when approximating the Hadamard transform. Stability and convergence of the proposed method are proved in the space of locally continuous functions in (−1, 1) with possible algebraic singularities at the endpoints, equipped with weighted uniform norms. The theoretical estimates are confirmed by several numerical tests.