Monotonicity of the Lebesgue constant for equally spaced knots

Abstract

Let t i = i n for i = 0, . . . , n be equally spaces knots in the unit interval [0, 1]. Let S n be the space of piecewise linear continuous functions on [0, 1] with knots π n = {t i : 0 ≤ i ≤ n}. Then we have the orthogonal projection P n of L 2 ([0, 1]) onto 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 = P n 1 of L 1 −norms of these projection operators is strictly increasing.