Application of atomic functions for construction of new classes of quadrature formulas

Solving of many equations includes a phase of numerical integration using methods of Gaussian type quadrature. It was spread out using an approximation of the function with compact support (e.g. wavelet or atomic function). These functions are used in solving a different kind of problem e.g. for antenna synthesis, for solving a boundary value problem, for filter synthesis, for signal processing, and for image compression . Naturally it computed Gaussian quadrature with weight including the corresponding atomic function. Using these quadratures allows one to reduce computational costs, because one does not need to compute the value of atomic functions, and perhaps to increase the accuracy as compared with traditional Gaussian quadrature.