等距节点上离散最小二乘方法的收敛性
数值分析
2025-10-20 v1 数值分析
摘要
我们考虑在区间 [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] N + 1 N+1 N + 1 n n n f ∈ C [ − 1 , 1 ] f\in\mathcal{C}\left[-1,1\right] f ∈ C [ − 1 , 1 ] N / n N/n N / n L S n N : C [ − 1 , 1 ] → P n {LS}_n^N:\mathcal{C}\left[-1,1\right]\rightarrow\mathcal{P}_n L S n N : C [ − 1 , 1 ] → P n Q k ( ⋅ ; α , β , N ) Q_k\left(\cdot;\alpha,\beta,N\right) Q k ( ⋅ ; α , β , N ) L S n N {LS}_n^N L S n N f ∈ C [ − 1 , 1 ] f\in\mathcal{C}\left[-1,1\right] f ∈ C [ − 1 , 1 ] α = β \alpha=\beta α = β f f f ( N n ) n ∈ N \left(N_n\right)_{n\in\mathbb{N}} ( N n ) n ∈ N α ≥ 0 \alpha\geq0 α ≥ 0 f ∈ { g ∈ C ∞ [ − 1 , 1 ] : lim n → ∞ sup x ∈ [ − 1 , 1 ] ∣ g ( n ) ( x ) ∣ n α + 1 / 2 2 n n ! = 0 } f\in\left\{g\in\mathcal{C}^\infty\left[-1,1\right]:\ \lim\limits_{n\to\infty}{\sup\limits_{x\in[-1,1]}{\left\lvert g^{(n)}(x)\right\rvert}\frac{n^{\alpha+1/2}}{2^nn!}}=0\right\} f ∈ { g ∈ C ∞ [ − 1 , 1 ] : n → ∞ lim x ∈ [ − 1 , 1 ] sup g ( n ) ( x ) 2 n n ! n α + 1/2 = 0 } ( N n ) n (N_n)_{n} ( N n ) n N n ≥ 2 n ( n + 1 ) N_n\geq2n(n+1) N n ≥ 2 n ( n + 1 ) L S n N n [ f ] {LS}_n^{N_n}[f] L S n N n [ f ] [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] K n + 1 : = { f ∈ C n + 1 [ − 1 , 1 ] : sup x ∈ [ − 1 , 1 ] ∣ f ( n + 1 ) ( x ) ∣ ≤ 1 } \mathcal{K}_{n+1}:=\left\{f\in\mathcal{C}^{n+1}\left[-1,1\right]:\ \sup\limits_{x\in[-1,1]}{\left\lvert f^{(n+1)}(x)\right\rvert\leq1}\right\} K n + 1 := { f ∈ C n + 1 [ − 1 , 1 ] : x ∈ [ − 1 , 1 ] sup f ( n + 1 ) ( x ) ≤ 1 }
引用
@article{arxiv.1905.00461,
title = {On convergence of discrete methods of least squares on equidistant nodes},
author = {René Goertz},
journal= {arXiv preprint arXiv:1905.00461},
year = {2025}
}
备注
17 pages