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 linearcontinuous 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\[\lim_{n\rightarrow\infty,\nu=1}\|P_{n,\nu}:L^\infty\rightarrowL^\infty\|=\sup_{n\in\mathbb{N},0\leq \nu\leq n}\|P_{n,\nu}:L^\infty\rightarrowL^\infty\|=2+\frac{33-18\sqrt{3}}{13}.\] This shows in particular that theLebesgue constant of the classical Franklin orthonormal system on the torus is$2+\frac{33-18\sqrt{3}}{13}$.
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