Abstract
The article deals with some interpolation representations of stochastic processes with non-equidistance interpolation knots. Research is based on observations of the process and its derivatives of the first and second orders at some types of knots and observations of the process and its derivatives of the first orders at other types of knots. The necessary results from the theory of entire functions of complex variable are formulated. The function bounded on any bounded region of the complex plane is considered. The estimate of the residual of the interpolation series is obtained. The interpolation formula that uses the value of the process and its derivatives at the knots of interpolation is proved. Considering the separability of the process and the convergence of a row that the interpolation row converges to the stochastic process uniformly over in any bounded area of changing of parameter is obtained. The main purpose of this article is the obtained convergence with probability 1 of the corresponding interpolation series to a stochastic process in any bounded domain of changes of parameter. Obtained results may be applied in the modern theory of information transmission.
Highlights
The one of the fundamental results in the Theory of Information Transmission is a theorem of expression of the function with a bounded specter of values in the periodic sequence of initial moments
The results in this work are principally new and they are related to the interpolation representations of stochastic processes with non-equidistance interpolation knots
The work is devoted to investigation of interpolation representations of a class of stochastic processes
Summary
The one of the fundamental results in the Theory of Information Transmission is a theorem of expression of the function with a bounded specter of values in the periodic sequence of initial moments. The significance of that fact was first introduced in [1]. Further these questions were studied in [2, 3]. Kotelnikov-Shannon theorem is generally well-known [4]. The investigations related to the construction of interpolation polynoms are attracting significant interest. Many of the questions concerning the construction of a spline approximation are known as representation of a motion in 3D modeling implemented with help of interpolation and approximation [5]. Many questions in modern physics [6] as well as physics of materials [7] and the modern theory of signal transmission [8] are based on the Kotelnikov-Shannon theorem. The present work is concerned on the questions stated above
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