Dolomites Research Notes on Approximation

The paper deals with the approximation of integrals of the type where f is a sufficiently smooth function and the kernel k collects criticisms of many different types (highly oscillating, weakly singular, "nearly" singular, etc.). We propose an extended product rule based on the approximation of f by an extended Lagrange process at Laguerre zeros. We prove that the rule is stable and convergent with order of the best polynomial approximation in suitable function spaces. Furthermore, by combining the stated rule with a related product formula, we define a pattern that allows a significant saving in number of function evaluations. We give details on the construction of the coefficients of the rule for some selected kernels. Finally, some numerical tests are proposed to show the efficiency of the compounded quadrature scheme.