Approximation properties of some discrete Fourier sums for piecewise smooth discontinuous functions

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.

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Approximation Properties of Discrete Fourier Sums in Polynomials Orthogonal on Non-Uniform Grids

Владикавказский математический журнал ◽

10.46698/k4355-6603-4655-y ◽

2020 ◽

Author(s):

А.А. Нурмагомедов

Keyword(s):

Approximation Properties ◽

Uniform Grids ◽

Fourier Sums ◽

Alpha Beta

В данной работе для произвольной непрерывной на отрезке $[-1, 1]$ функции $f(x)$ в~случае целых положительных $\alpha$ и $\beta$ построены дискретные суммы Фурье $S_{n,N}^{\alpha,\beta}(f,x)$ по системе многочленов $\{\hat{p}_{k,N}^{\alpha,\beta}(x)\}_{k=0}^{N-1},$ образующих ортонормированную систему на неравномерных сетках $\Omega_N=\{x_j\}_{j=0}^{N-1},$ состоящих из конечного числа $N$ точек отрезка $[-1, 1]$ с весом типа Якоби. Исследуются аппроксимативные свойства построенных частных сумм $S_{n,N}^{\alpha,\beta}(f,x)$ порядка $n\leq{N-1}$ в~пространстве непрерывных функциий $C[-1, 1].$ А именно, получена двусторонняя поточечная оценка для функции Лебега $L_{n,N}^{\alpha,\beta}(x)$ рассматриваемых дискретных сумм Фурье при $n=O\big(\delta_N^{-1/(\lambda+3)}\big)$, $\lambda=\max\{\alpha, \beta\}$, $\delta_N=\max_{0\leq{j}\leq{N-1}}\Delta{t_j}$. Соответственно, исследован также вопрос сходимости $S_{n,N}^{\alpha,\beta}(f,x)$ к $f(x)$. В частности, получена оценка отклонения частичной суммы $S_{n,N}^{\alpha,\beta}(f,x)$ от $f(x)$ при $n=O\big(\delta_N^{-1/(\lambda+3)}\big),$ которая также зависит от~$n$ и положения точки $x\in[-1, 1].$

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The approximation of piecewise smooth functions by trigonometric Fourier sums

Daghestan Electronic Mathematical Reports ◽

10.31029/demr.12.3 ◽

2019 ◽

pp. 25-42

Author(s):

Magomedrasul Magomed-Kasumov

Keyword(s):

Fourier Series ◽

Piecewise Smooth ◽

Continuous Derivative ◽

Exact Order ◽

Smooth Functions ◽

Absolutely Continuous ◽

Piecewise Smooth Functions ◽

Order Of Magnitude ◽

Fourier Sums ◽

Magnitude Estimates

We obtain exact order-of-magnitude estimates of piecewise smooth functions approximation by trigonometric Fourier sums. It is shown that in continuity points Fourier series of piecewise Lipschitz function converges with rate $\ln n/n$. If function $f$ has a piecewise absolutely continuous derivative then it is proven that in continuity points decay order of Fourier series remainder $R_n(f,x)$ for such function is equal to $1/n$. We also obtain exact order-of-magnitude estimates for $q$-times differentiable functions with piecewise smooth $q$-th derivative. In particular, if $f^{(q)}(x)$ is piecewise Lipschitz then $|R_n(f,x)| \le c(x)\frac{\ln n}{n^{q+1}}$ in continuity points of $f^{(q)}(x)$ and $\sup_{x \in [0,2\pi]}|R_n(f,x)| \le \frac{c}{n^q}$. In case when $f^{(q)}(x)$ has piecewise absolutely continuous derivative it is shown that $|R_n(f,x)| \le \frac{c(x)}{n^{q+1}}$ in continuity points of $f^{(q)}(x)$. As a consequence of the last result convergence rate estimate of Fourier series to continuous piecewise linear functions is obtained.

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