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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.

The article presents a description of the history of high education development in the Arctic zone of the Russian Federation, provides an overview of the priority areas of popular professions for the region as a whole, and for its individual subjects; analyzes the main indicators that determine the opportunities for training in higher education institutions and identifies the distinctive features of the functioning of higher education in the Arctic region. A number of recommendations are proposed for certain areas of improving the system of training in higher education institutions of the studied region, taking into account its geographical location.

Using the fast expansion method, we obtained several exact solutions of the boundary value problem for the Poisson equation in a rectangular domain. We have given graphs of exact solutions corresponding to different boundary conditions and different types of the Poisson equation free term. We showed the influence of the type of boundary conditions and the right-hand side on the type of exact solution. We obtained a solution to the problem of membrane deflections under the action of variable load. We have given graphs of the stress components, from the analysis of which it follows that the greatest stress is in the middle of the rectangular membrane long sides.