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