The Approximation of Functions by Partial Sums of the Fourier Series in Polynomials Orthogonal on Arbitrary Grids

Abstract

for arbitrary continuous function f(t) on the segment [−1,1] we construct discrete Fourier sums S_n,N(f,t) on a system of polynomials forming an orthonormal system on non-uniform grids ${T_N} = \left\{{{t_j}} \right\}\matrix{{N - 1} \cr {j = 0} \cr}$ of N points from segment [−1,1] with weight Δt_j=t_j+1−t_j. Approximation properties of the constructed partial sums S_n,N(f,t) of order n ≤ N−1 are investigated. A two-sided pointwise estimate is obtained for the Lebesgue function L_n,N(t) of considered discrete Fourier sums for $n = O\left({\delta \matrix{{- 1/5} \cr N \cr}} \right),{\delta _N} = {\max _0} \le j \le N - 1\Delta {t_j}$. The question of the convergence of S_n,N(f,t) to f(t) is also investigated. In particular, we obtain the deflection estimation of a partial sum S_n,N(f,t) from f(t) for $n = O\left({\delta \matrix{{- 1/5} \cr N \cr}} \right)$, which depends on n and the position of a point t ∊ [−1,1].