On the convergence of the least square method in case of non-uniform grids

M. S. Sultanakhmedov

Problemy analiza, Tome 8 (2019) no. 3, pp. 166-186

Résumé

Let $f(t)$ be a continuous on $[-1, 1]$ function, which values are given at the points of arbitrary non-uniform grid $\Omega_N= \{ t_j \}_{j=0}^{N-1}$, where nodes $t_j$ satisfy the only condition $\eta_{j}\!\leq \!t_{j}\!\leq\!\eta_{j+1},$ $0\leq j \leq N-1,$ and nodes $\eta_{j}$ are such that $-1=\eta_{0}\eta_{1}\eta_{2}\cdots\eta_{N-1}\eta_{N}=1$. We investigate approximative properties of the finite Fourier series for $f(t)$ by algebraic polynomials $\hat{P}_{n,\,N}(t)$, that are orthogonal on $\Omega_N = \{ t_j \}_{j=0}^{N-1}$. Lebesgue-type inequalities for the partial Fourier sums by $\hat{P}_{n,\,N}(t)$ are obtained.

Détail

Keywords: random net, non-uniform grid, least square method, Fourier series, function approximation.
Mots-clés : orthogonal polynomials, Legendre polynomials