The Lebesgue Constant for the Periodic Franklin System

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