The Lebesgue Constant for the Periodic Franklin System

TL;DR

This paper determines the exact Lebesgue constant for the Franklin system on the torus, showing it converges to a specific value, which is significant for understanding the system's approximation properties.

Contribution

The paper precisely calculates the Lebesgue constant for the Franklin system on the torus, providing new insights into its approximation behavior.

Findings

Lebesgue constant converges to a specific value as n→∞ with ν=1

Exact Lebesgue constant for the Franklin system is 2 + (33 - 18√3)/13

Results improve understanding of the Franklin system's approximation properties

Abstract

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 $2+\frac{33-18\sqrt{3}}{13}$.