The theory of spline interflatation of functions variables (r≥2)
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The theory of approximation of multidimensional functions of r variables using spline-interflatation operators is developed in this paper. A new method for constructing such operators, which is based on the approach of decomposing the multidimensional approximation problem into a sequence of one-dimensional problems, each solved using spline interpolation. is proposed in this paper. This makes it possible to investigate the interflatation properties of the constructed operators, as well as to analyze their effectiveness in approximating functions with several variables. The distinctive feature of the proposed method is the explicit representation of spline-interflatation operators in terms of one-dimensional spline interpolation operators, which are applied separately to each variable of the approximated function. This provides convenience in investigating the properties of operators and enables more in-depth analysis of their behavior. The expression for the approximation remainder of functions using these operators, in particular, in terms of the remainders that arise from applying one-dimensional spline interpolation operators is investigated in this paper. Special attention is paid to the analysis of approximation remainders of multidimensional functions and to proving that the approximation remainder calculated by means of the proposed interflatation operators is equal to the operator product of the approximation remaindes, defined separately for each variable. This means that total remainder can be considered as a combination of remainders obtained through one-dimensional operators, which significantly simplifies the analysis and makes it possible to investigate the approximation accuracy more thoroughly. Furthermore, a comparative analysis of the obtained results with classical multidimensional interpolation operators is carried out in this paper. This enables us to evaluate the advantages and disadvantages of the proposed method in the context of accuracy and efficiency of approximating functions with several variables. This opens up prospects for further development of the theory of multidimensional approximation and its application in various fields of science and engineerin , where efficient and accurate approximation of multidimensional functions is required.
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