Monotonicity of the Lebesgue constant for equally spaced knots
This resolves a theoretical question about the monotonicity of the Lebesgue constant for equally spaced knots, which is of interest to researchers in approximation theory and numerical analysis.
The paper proves that the L1-norm of the orthogonal projection onto the space of piecewise linear continuous functions with equally spaced knots is strictly increasing with the number of knots.
Let $t_{i}=\frac{i}{n}$ for $i=0,...,n$ be equally spaces knots in the unit interval $[0,1].$ Let $\mathcal{S}_{n}$ be the space of piecewise linear continuous functions on $[0,1]$ with knots $π_{n}=\{t_{i}:0\leq i\leq n\}.$ Then we have the orthogonal projection $P_{n}$ of $L^{2}([0,1])$ onto $\mathcal{S}_{n}.$ In Section 1 we collect a few preliminary facts about the solutions of the recurrence $f_{k-1}-4f_{k}+f_{k+1}=0$ that we need in Section 2 to show that the sequence $% a_{n}=\Vert P_{n}\Vert_{1}$ of $L^{1}-$norms of these projection operators is strictly increasing.