周期Franklin系统的Lebesgue常数
数值分析
2015-03-04 v3
摘要
我们将环面等同于单位区间[ 0 , 1 ) [0,1) [ 0 , 1 ) n , ν ∈ N n,\nu\in\mathbb{N} n , ν ∈ N 1 ≤ ν ≤ n − 1 1\leq \nu\leq n-1 1 ≤ ν ≤ n − 1 N : = n + ν N:=n+\nu N := n + ν t j = { [ c ] l l t_{j}=\{[c]{ll}% \frac{j}{2n}, & \text{对于}j=0,...,2\nu, \frac{j-\nu}{n}, & \text{对于}j=2\nu+1,...,N-1.] t j = {[ c ] l l n , ν n,\nu n , ν V n , ν V_{n,\nu} V n , ν { t j : 0 ≤ j ≤ N − 1 } \{t_j:0\leq j\leq N-1\} { t j : 0 ≤ j ≤ N − 1 } P n , ν P_{n,\nu} P n , ν L 2 ( [ 0 , 1 ) ) L^{2}([0,1)) L 2 ([ 0 , 1 )) V n , ν V_{n,\nu} V n , ν lim n → ∞ , ν = 1 ∥ P n , ν : L ∞ → L ∞ ∥ = sup n ∈ N , 0 ≤ ν ≤ n ∥ P n , ν : L ∞ → L ∞ ∥ = 2 + 33 − 18 3 13 . \lim_{n\rightarrow\infty,\nu=1}\|P_{n,\nu}:L^\infty\rightarrow L^\infty\|=\sup_{n\in\mathbb{N},0\leq \nu\leq n}\|P_{n,\nu}:L^\infty\rightarrow L^\infty\|=2+\frac{33-18\sqrt{3}}{13}. n → ∞ , ν = 1 lim ∥ P n , ν : L ∞ → L ∞ ∥ = n ∈ N , 0 ≤ ν ≤ n sup ∥ P n , ν : L ∞ → L ∞ ∥ = 2 + 13 33 − 18 3 . 2 + 33 − 18 3 13 2+\frac{33-18\sqrt{3}}{13} 2 + 13 33 − 18 3
引用
@article{arxiv.1103.1950,
title = {The Lebesgue Constant for the Periodic Franklin System},
author = {Markus Passenbrunner},
journal= {arXiv preprint arXiv:1103.1950},
year = {2015}
}
备注
Mathematica Notebook for creating Table 1 on page 21 is attached