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JOURNALS // Daghestan Electronic Mathematical Reports // Archive

Daghestan Electronic Mathematical Reports, 2015 Issue 4, Pages 74–117 (Mi demr19)

On the simultaneous approximation of functions and their derivatives by Chebyshev polynomials orthogonal on uniform grid

I. I. Sharapudinovab, T. I. Sharapudinovab

a Daghestan Scientific Centre of Russian Academy of Sciences
b Vladikavkaz Scientific Centre of the RAS

Abstract: The article is dedicated to investigation of approximative properties of polynomial operator $\mathcal{ X}_{m,N}(f)=\mathcal{ X}_{m,N}(f,x)$, which is defined in the space $C[-1,1]$ and based on the use of only discrete values of the function $f(x)$, given in the nodes of uniform grid $\{x_j=-1+jh\}_{j=0}^{N+2r-1}\subset [-1,1]$. This operator can be used for solving the problem of simultaneous approximation of a differentiable function $f(x)$ and its multiple derivatives $f'(x), \ldots, f^{(p)}(x)$. Construction of operators $\mathcal{ X}_{m,N}(f)$ is based on Chebyshev polynomials $T_n^{\alpha,\beta}(x,N)$ $(0\le n\le N-1)$, which form an orthogonal system on the set $\Omega_N=\{0,1,\ldots,N-1\}$ with weight
$$ \mu(x)=\mu(x;\alpha,\beta,N)=c{\Gamma(x+\beta+1) \Gamma(N-x+\alpha)\over \Gamma(x+1)\Gamma(N-x)}, $$
i.e.
$$ \sum_{x\in\Omega_N}\mu(x)T_n^{\alpha,\beta}(x,N)T_m^{\alpha,\beta}(x,N) =h_{n,N}^{\alpha,\beta}\delta_{nm}. $$
There were obtained upper bounds for the Lebesgue functions of an operator $\mathcal{ X}_{m,N}(f)=\mathcal{ X}_{m,N}(f,x)$ and weighted approximations of the following form
$$ {|\frac1{h^{\nu}}\Delta_h^\nu\left[ f(x_{j-\nu})-\mathcal{ X}_{n+2r,N}(f,x_{j-\nu})\right]|\over\left(\sqrt{1-x_{j}^2}+{1\over m}\right)^{r-\nu-\frac12}}. $$


Keywords: Chebyshev polynomials orthogonal on the grid; Chebyshev polynomials of the first kind; approximation of functions and derivatives.

UDC: 517.587

Received: 27.10.2015
Revised: 22.12.2015
Accepted: 23.12.2015

DOI: 10.31029/demr.4.5



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