Monotonicity of the Lebesgue constant for equally spaced knots

Abstract

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 ${\pi }_n=\{t_i:0\le i\le n\}.$ 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 of $L^1-$norms of these projection operators is strictly increasing.