The Lebesgue constants for the Franklin orthogonal system
Volume 164 / 2004
Studia Mathematica 164 (2004), 55-73
MSC: 41A44, 41A15.
DOI: 10.4064/sm164-1-4
Abstract
To each set of knots $t_i = {i/ 2n}$ for $i=0,\dots ,2\nu$ and $t_i= {(i-\nu) / n}$ for $i=2\nu +1,\dots, n+\nu $, with $1\leq \nu\leq n$, there corresponds the space ${\cal S}_{\nu ,n}$ of all piecewise linear and continuous functions on $I=[0,1]$ with knots $t_i$ and the orthogonal projection $P_{\nu ,n}$ of $L^2(I)$ onto ${\cal S}_{\nu ,n}$. The main result is $$ \lim_{(n-\nu)\wedge\nu\to\infty}\|P_{\nu ,n}\|_1 = \sup_{\nu ,n\,:\,1\leq \nu\leq n}\|P_{\nu ,n}\|_1 = 2+(2-\sqrt{3})^2. $$ This shows that the Lebesgue constant for the Franklin orthogonal system is $2+(2-\sqrt{3})^2$.
