Monotonicity of the Lebesgue constant for equally spaced knots

TL;DR

This paper proves that the Lebesgue constant, represented by the L1-norm of orthogonal projections onto piecewise linear functions with equally spaced knots, increases monotonically as the number of knots grows.

Contribution

It establishes the monotonicity of the Lebesgue constant for equally spaced knots, providing new insights into the behavior of projection operators in approximation theory.

Findings

The sequence of L1-norms of the projections is strictly increasing.

Preliminary recurrence solutions are used to analyze the projection norms.

Monotonicity holds for all numbers of 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 $\pi_{n}=\{t_{i}:0\leq i\leq n\}.$ 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.