fast expansions, boundary function, fast trigonometric interpolation, relative error, differentiation.

The basics of fast universal trigonometric interpolation are given in the article. The error estimates are presented by the authors. The concept of various orders boundary function is introduced in this work. Analytical formulas used to solve engineering problems are written as well. The efficiency of fast universal trigonometric interpolation is shown by the example of solving a boundary value problem for a second-order nonlinear differential equation. To achieve the maximum relative error of the solution of less than one percent, it is sufficient to take into account only one first term with sines and one first term with cosines in the fast decomposition on a segment of unit length.

Chernyshov Alexander Danilovich, Dr. Phys. & Math. Sci., Professor Voronezh State University of Engineering Technology, Voronezh, Russia.

Nikiforova Olga Yurievna, Senior Lecturer, Voronezh State University of Engineering Technology, Voronezh, Russia.

Goryainov Vitaly Valerievich, Ph.D. in Phys. & Math., Assoc. Professor, Voronezh State Technical University, Voronezh, Russia.

Kuznetsov Sergey Fedorovich, Ph.D. in Phys. & Math., Assoc. Professor, Voronezh State University of Engineering Technology, Voronezh, Russia.

Rukin Ivan Gennadievich, post-graduate student, Voronezh State University of Engineering Technology, Voronezh, Russia.