Analytical Representation of Infocommunication Data

The article discusses the question of possible extensions of families of approximating functions by superposition with some classes of transfer functions in order to improve the quality of approximation of continuous on a segment and tabular dependencies. Both in terms of accuracy and speed of computation. Particular attention is paid to the possibility of using the least squares method for such extensions, since this method provides the fastest solution compared to step-by-step approximation methods. The question of a possible change in the convergence criterion of the error function to expand the possibility of this method, which will allow more flexibility in solving this problem, is being investigated. Additionally, two practical problems are considered – the applicability of the Least Squares method to the training of artificial neural networks, as well as to extrapolation and prediction problems, to which polynomials are not very suitable. To justify the concept of transfer functions, theorems are formulated with proof of the linear transfer function and the completeness of the family of inverse polynomials. The applicability of the method of orthogonal polynomials, as well as rational functions for these purposes, is discussed. As a research method, it is proposed to supplement the approximating family with an example of polynomials with a strictly monotone transformation as a transfer function. Specific variants of such transfer functions are considered, it is indicated for what practical tasks they can be useful. The result is a demonstration of the visible advantage of a polynomial supported with a transfer function over a polynomial without it to approximate a specific type of dependencies often featured in info-communication data. The rest of the results have theoretical significance and can be used for forecasting and develop a new type of artificial neural networks architecture.