Approximation by triangular fourier sums on classes of continuous periodic functions of two variables

1983 ◽  
Vol 35 (2) ◽  
pp. 215-220
Author(s):  
A. I. Stepanets ◽  
V. I. Rukasov
Keyword(s):  
Periodic Functions ◽  
Functions Of Two Variables ◽  
Fourier Sums

scholarly journals Analysis of the results of a computational experiment to restore the discontinuous functions of two variables using projections

2021 ◽  
pp. 12-17
Author(s):  
Oleg Lytvyn ◽  
Oleg Lytvyn ◽  
Oleksandra Lytvyn
Keyword(s):  
Differentiable Function ◽  
Fourier Coefficients ◽  
Gibbs Phenomenon ◽  
Discontinuous Function ◽  
Projection Data ◽  
Approximation Accuracy ◽  
Discontinuous Functions ◽  
Functions Of Two Variables ◽  
Fourier Sums ◽  
The Difference

This article presents the main statements of the method of approximation of discontinuous functions of two variables, describing an image of the surface of a 2D body or an image of the internal structure of a 3D body in a certain plane, using projections that come from a computer tomograph. The method is based on the use of discontinuous splines of two variables and finite Fourier sums, in which the Fourier coefficients are found using projection data. The method is based on the following idea: an approximated discontinuous function is replaced by the sum of two functions – a discontinuous spline and a continuous or differentiable function. A method is proposed for constructing a spline function, which has on the indicated lines the same discontinuities of the first kind as the approximated discontinuous function, and a method for finding the Fourier coefficients of the indicated continuous or differentiable function. That is, the difference between the function being approximated and the specified discontinuous spline is a function that can be approximated by finite Fourier sums without the Gibbs phenomenon. In the numerical experiment, it was assumed that the approximated function has discontinuities of the first kind on a given system of circles and ellipses nested into each other. The analysis of the calculation results showed their correspondence to the theoretical statements of the work. The proposed method makes it possible to obtain a given approximation accuracy with a smaller number of projections, that is, with less irradiation.

scholarly journals Approximation of the periodical functions of high smoothness by the rightangled Fourier sums

2013 ◽  
Vol 5 (1) ◽  
pp. 102-109
Author(s):  
O.O. Novikov ◽  
O.G. Rovenska
Keyword(s):  
Upper Bounds ◽  
Functions Of Two Variables ◽  
High Smoothness ◽  
Fourier Sums ◽  
The Right

We obtain asymptotic equalities for upper bounds of the deviations of the right-angled Fourier sums taken over classes of periodical functions of two variables of high smoothness. These equalities in corresponding cases guarantee the solvability of the Kolmogorov–Nikol’skii problem for the right-angled Fourier sums on the specified classes of functions.

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