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

M. S. Sultanakhmedov

Dagestan Scientific Center of RAS, 45, M.Gadzhieva st., Makhachkala, 367025, Russia

Abstract: 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.

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

UDC: 517.521

MSC: 42C10, 41A10, 33F05

Received: 03.06.2019
Revised: 22.10.2019
Accepted: 18.10.2019

Language: English

DOI: 10.15393/j3.art.2019.6410

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