Abstract
The problem of approximating of a surface given by the values of a function of two arguments in a finite number of points of a certain region in the classical formulation is reduced to solving a system of algebraic equations with tightly filled matrixes or with band matrixes. In the case of complex surfaces, such a problem requires a significant number of arithmetic operations and significant computer time spent on such calculations. The curvilinear boundary of the domain of general type does not allow using classical orthogonal polynomials or trigonometric functions to solve this problem. This paper is devoted to an application of orthogonal splines for creation of approximations of functions in form of finite Fourier series. The orthogonal functions with compact supports give possibilities for creation of such approximations of functions in regions with arbitrary geometry of a boundary in multidimensional cases. A comparison of the fields of application of classical orthogonal polynomials, trigonometric functions and orthogonal splines in approximation problems is carried out. The advantages of orthogonal splines in multidimensional problems are shown. The formulation of function approximation problem in variational form is given, a system of equations for coefficients of linear approximation with a diagonal matrix is formed, expressions for Fourier coefficients and approximations in the form of a finite Fourier series are written. Examples of approximations are considered. The efficiency of orthogonal splines is shown. The development of this direction associated with the use of other orthogonal splines is discussed.
Highlights
Solution of approximation problem, how rule, leads to2
It is shown that the use of orthogonal splines allows to obtain approximations of functions of many variables in the general case of multidimensional regions having a curvilinear boundary
There are no restrictions on the use of orthogonal splines associated with the shape of the region and the geometry of the boundary
Summary
The first in the world orthogonal splines were proposed in [6,7,8] without the Gram-Sch midt procedure of. Where h − the step of an grid x1 < x2 < ⋯ < xn These functions are orthogonal on every a grid. A grid can be nonuniform and steps of grid depend on i These splines are continuous, consist of linear parts, are the generalization of B-splines (x − xi−1)/h, x ∈ [xi−1, xi]; ψi(x) = { (xi+1 − x)/h, x ∈ [xi, xi+1]; 0. The generalization φi(x) was carried [8] out by adding to the B-spline ψi(x) two similar B-splines, having a compact supports, the sizes h of which in two times less in comparison with the size 2h of the compact support [xi−1, xi+1] of ψi (x).
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