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].

中文翻译：

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任意网格上正交多项式的傅里叶级数部分和对函数的逼近

对于段[−1,1]上的任意连续函数f（t），我们在多项式系统上构造离散傅立叶和S _n，N（f，t），该多项式系统在非均匀网格\（{T_N} = \ left \ {{{{t_j}} \ right \} \ matrix {{N-1} \ cr {j = 0} \ cr} \）来自线段[−1,1]的N个点，权重为Δt _j = t _{j +1} - t _j。所构造的部分和的逼近性质小号_N，N-（F，T的顺序）ñ ≤ Ñ-1被调查。对于（\ n = O \ left（{\ delta \ matrix {{-1/5} \ cr N \ cr}的离散傅里叶总和的Lebesgue函数L _n，N（t）获得双向逐点估计}} \ right），{\ delta _N} = {\ max _0} \ le j \ le N -1 \ Delta {t_j} \）。还研究了S _n，N（f，t）到f（t）的收敛性问题。特别是，我们得到的部分和的偏转估计小号_N，N-（F，T从）˚F（吨）为1/5} \ CR N - \（N = O \左（{\增量\矩阵{{ \ cr}} \ right）\），具体取决于n和点t t [-1,1]的位置。