The object is scientific bases of telecommunication system updating. The aim is the theory forming and constructive methods of spline approximation allowing more effectively to solve different problems in telecommunications for raising the quality characteristics of telecommunication systems. Research methods: usage of communication theory, system theory, mathematic analysis, functional analysis, theory of models and simulation modelling, tensor analysis, combinatorial topology, spline approximation, interpolation, linear algebra, differential geometry, synthesis of signal impulses, optimization of selectiv e signals parameters under its synthesis. Theoretical and practical results: 1.On the base of cubic splines the engineering technics of analysis and synthesis of new kinds of multiparameter selective signals with finite spectrum, differing with that while its usage the beat interferences (intersymbol and interchannel) have been minimized is formed. The set of necessary parameters of the synthesized signals and admitted region of these parameters allowing to provide effective characteristics and physical realization of these signals on practice have been defined. 2. Bank occurence of multiparameter selective signals with finite spectrum free of intersymbol interference (ISI) allows researcher to form more rational kinds of signals depending on conditions in communication channels and requirements, specifying to the telecommunication system itself. 3. The recommendations on signals synthesis effective on two traditional criteria (dependence of eye-pattern aperture value from coefficient of signal spectrum rounding approximated in the frequency domain by cubic spline and cubic B-spline, and dependence of selective signals energy concentration from coefficient of rounding) as for the usage of new signal functions, the extremal properties of which allow to improve technical characteristics of telecommunications systems have been got. 4. It has been proved that the wide used Kotelnikov series on practice for approximation of signals and processes on finite interval is not the best method of approximation. The more effective apparatus of approximation are spline-functions that allow to improve accuracy of approximation. Apart from it under the spline approximation it is practically absent the Gibbs effect. 5. Under the receiving the interval evaluations of the random processes the evaluation sequence approximation by cubic splines is more effective then linear received by the method of Kalman-Byusi. The possible measures of inaccuracy are examined, and its limit values have been got. 6. While the solving of nonlinear problems particularly the problems of optimal control, received the measures of inaccuracy in calculations for linear , cubic, nonlocal splines, discrete cubic splines, hermite cubic and B-splines, uniform and nonuniform mesh of interval fragmentation on which the function of control is given. It is shown that the further scale-up the smoothness of control function does not give the rise of approximation order-saturation of interpolation of random process is occured. Hence the limit of approachable accuracy under the realization of telecommunications systems management is defined. 7. It is shown that in the telecommunications systems models presented by hub network the application of calculating tensor methods allows to analyze structural and functional properties of these systems, to provide the more general results supplying the minimal time of multipath routing that allows also find simpler optimal solutions. Scientific novelty: 1 the method of synthesis of multiparameter selective signals with finite spectrum free of ISIbased on the usage of cubic and cubic B-splines for approximation of frequency properties has been developed firstly. The analytical method of these signal analysis and synthesis in frequency and time domains has been developed. 2. The full energy analysis of selective signals which spectral density approximated by cubic and B-cubic splines has been firstly conducted. The bases for multiparameter selective signals bank with finite spectrum free from ISI have been created. 3. The possibility of random signal recovery quantity increasing by spline-approximation, in comparison with the used on practice the methods of Kotelnikov series has been prooved. 4. The possibility to get more precise interval evaluation of random processes and signals by nonlinear approximation, the results of linear evaluation of network elements state and networks in general has been proved firstly, so, under the evaluation sequence interpolation got by method of Kalman-Byusi, the approximation by cubic splines is more effective then linear. 5. The method of linear not uniform differential equation by spline functions (linear, cubic, B-splines) that allows to get stable solutions of problems for optimization control in telecommunications networks firstly developed. 6. It is firstly suggested under the recovery of discrete vector processes to generalyze the concept of spline functions by the concept of tensor, the components of which are spline functions that allow to get invariant to dimensions and to the system of the coordinates solutions and to make wider the group of these solutions, combining structural and functional properties of telecommunication system, the concept of tensor splines has been firstly introduced and it has been shown the correctness of algebra operations under the tensor splines: addition, multiplication, contraction of tensor splines that allow to make proper mathematical procedures under the discrete processes and fields, characterizing the multivariable state of telecommunication systems, 7. Firstly for the solution of nonlinear problems generalized the method of linearization by transition to Riemannian space, using the covariant differentiation with the help of tensor splines, got the method of nonlinear problem solution on the variety of tensor splines with the help of the tensor linearization of discrete nonlinear neighborhood systems by which the research of telecommunication systems functional properties is simplified. The results of work are realized on such organizations as: Kharkov Scientific Research Institute of Expert Testimony in Court named after M.S.Bokarius, Ukraine Scientific research Institute of Communication, State University of Information Telecommunication Technologies, Odessa National Academy of Telecommunications named after A.S. Popov. The results have been used in six scientific research works of ONAT named after A.S. Popov, and suggested to use in the educational process of HEI and for SRW of lecturers, aspirants.
