Bases derived from trigonometry and their advantages

A specific use of trigonometric functions with respect to any interval possesses a high approximate quality. In this case, a solution of integral equations with kernels of the form $K(x-t)$ by the Galerkin method allows one to reduce the double integral to a very simple single integration. Also, a specific base of functions for solving problems with an elliptic operator with disconnected coefficients is proposed. A distinctive feature of this base is automatic realization of conjugate conditions in locations of discontinuities of coefficients of equations. Another essential property is a high-precise approximation of piecewise-smooth solutions of the above problems by means of a small number of base functions. All the proofs of the results obtained follow from the two theorems presented.