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

I. I. Sharapudinov ; T. I. Sharapudinov

Daghestan Electronic Mathematical Reports, Tome 4 (2015), pp. 74-117

Résumé

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 $$ {|\frac1{h^{\nu}}\Delta_h^\nu\left[ f(x_{j-\nu})-\mathcal{ X}_{n+2r,N}(f,x_{j-\nu})\right]|\over\left(\sqrt{1-x_{j}^2}+{1\over m}\right)^{r-\nu-\frac12}}. $$

Détail

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