Solution of an extremal problem for classes of discontinuous functions of two variables in permutations

1979 ◽  
Vol 31 (1) ◽  
pp. 75-80 ◽  
Author(s):  
A. I. Stepanets
Keyword(s):  
Extremal Problem ◽  
Discontinuous Functions ◽  
Functions Of Two Variables

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 DISCONTINUOUS FUNCTIONS OF TWO VARIABLES BY DISCONTINUOUS INTERLINATION SPLINES USING TRIANGULAR ELEMENTS

2020 ◽  
Vol 14 (1) ◽  
pp. 75-89
Author(s):  
V. Mezhuyev ◽  
O. M. Lytvyn ◽  
I. Pershyna ◽  
O. Nechuiviter
Keyword(s):  
Experimental Data ◽  
Gibbs Phenomenon ◽  
Discontinuous Functions ◽  
Functions Of Two Variables ◽  
Improve Approximation ◽  
Triangular Elements

The paper develops a method for approximation of the discontinuous functions of two variables by discontinuous interlination splines using arbitrary triangular elements. Experimental data are one-sided traces of a function given along a system of lines (such data are commonly used in remote methods, in particular in tomography). The paper also proposes a method for approximating the discontinuous functions of two variables taking into account triangular elements having one curved side. The proposed methods improve approximation of the discontinuous functions, allowing an application to complex domains of definition and avoiding the Gibbs phenomenon.

An extremal problem for operators

1986 ◽  
Vol 84 ◽  
pp. 213-223 ◽  
Author(s):  
Vlastimil Pták
Keyword(s):  
Extremal Problem

scholarly journals On Generalized Step-Functions and Superposition Operators

2004 ◽  
Vol 11 (4) ◽  
pp. 753-758
Author(s):  
A. Kharazishvili
Keyword(s):  
Step Functions ◽  
Functions Of Two Variables ◽  
Superposition Operators

Abstract For a given σ-ideal of sets, the notion of a generalized stepfunction is introduced and investigated in connection with the problem of sup-measurability of certain functions of two variables, regarded as superposition operators.

On students’ understanding of the differential calculus of functions of two variables

2015 ◽  
Vol 38 ◽  
pp. 57-86 ◽  
Author(s):  
Rafael Martínez-Planell ◽  
Maria Trigueros Gaisman ◽  
Daniel McGee
Keyword(s):  
Differential Calculus ◽  
Functions Of Two Variables
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