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 $π_{n}=\{t_{i}:0\leq i\leq n\}.$ 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.