Nonlinear trigonometric approximation and the Dirac delta function

The nonlinear approximations based on two types of trigonometric generating functions are developed. It is shown that such nonlinear approximations to the Dirac delta function on [-1,1] are the corresponding Gaussian quadratures applied to some Stieltjes integrals, whose integrands contain weights and the two types of generating functions. In addition, the convergence is proved and the error terms are obtained. Some numerical tests are also shown.