High accurate sine and cosine interpolation with repeated differentiation ability obtained by fast expansions technique

Fast trigonometric interpolations on a finite interval with different basis functions and increased accuracy are provided. Sine interpolation exactly coincides with this function at the interpolation points, and derivatives of cosine interpolation of the second example exactly coincide with derivatives of the said function at the interpolation points. In all cases interpolations allow differentiation for a given number of times being equal to the order of a certain boundary function. The applied Fourier series converge quickly and hence they can be limited to a small number of terms, thus greatly saving computing time. These special features allow their use in solving complex applied multi-dimensional problems of integro-differential type. The compact formulas for trigonometric interpolations in an explicit form are given which facilitate their application. An error estimate is provided and, algorithms for applying the described interpolations are setup. (paper)