The Lebesgue Constant for the Periodic Franklin System

Abstract

We identify the torus with the unit interval $[0,1)$ and let $n,ν\in\mathbb{N}$, $1\leq ν\leq n-1$ and $N:=n+ν$. Then we define the (partially equally spaced) knots \[ t_{j}=\{[c]{ll}% \frac{j}{2n}, & \text{for}j=0,...,2ν, \frac{j-ν}{n}, & \text{for}j=2ν+1,...,N-1.] Furthermore, given $n,ν$ we let $V_{n,ν}$ be the space of piecewise linear continuous functions on the torus with knots $\{t_j:0\leq j\leq N-1\}$. Finally, let $P_{n,ν}$ be the orthogonal projection operator of $L^{2}([0,1))$ onto $V_{n,ν}.$ The main result is \[\lim_{n\rightarrow\infty,ν=1}\|P_{n,ν}:L^\infty\rightarrow L^\infty\|=\sup_{n\in\mathbb{N},0\leq ν\leq n}\|P_{n,ν}:L^\infty\rightarrow L^\infty\|=2+\frac{33-18\sqrt{3}}{13}.\] This shows in particular that the Lebesgue constant of the classical Franklin orthonormal system on the torus is $2+\frac{33-18\sqrt{3}}{13}$.