The Lebesgue constant for the periodic Franklin system
Volume 205 / 2011
Abstract
We identify the torus with the unit interval $[0,1)$ and let $n,\nu\in\mathbb{N}$ with $0\leq \nu\leq n-1$ and $N:=n+\nu$. Then we define the (partially equally spaced) knots \[ t_{j}=\begin{cases} {j}/({2n})&\text{for $j=0,\ldots,2\nu$,}\\ ({j-\nu})/{n}&\text{for $j=2\nu+1,\ldots,N-1$.} \end{cases} \] 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 from $L^{2}([0,1))$ onto $V_{n,\nu}.$ The main result is \begin{align*} \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}. \end{align*} This shows in particular that the Lebesgue constant of the classical Franklin orthonormal system on the torus is $2+\frac{33-18\sqrt{3}}{13}$.
