Approximation, regularization and smoothing of trigonometric splines

The methods of approximation, regularization and smoothing of trigonometric interpolation splines are considered in the paper. It is shown that trigonometric splines can be considered from two points of view - as a trigonometric Fourier series and as discrete trigonometric Fourier series according to certain systems of functions that are smoothness carriers. It is argued that with approximation and smoothing of trigonometric splines it is expedient to consider as discrete rows, since their differential properties are stored.