Nonlinear approximations to the derivative of delta function

The nonlinear approximations based on trigonometric generating functions are studied. It is shown that, by properly choosing generating functions, such nonlinear approximations to the derivative of the Dirac delta function on [−1, 1] are the corresponding Gaussian quadratures applied to its Stieltjes integral representations by the generating functions. Moreover, the approximations are proved to be convergent and the error terms are obtained.