Open questions around the spline orthoprojector

An open problem about the max-norm of the spline orthoprojector is formulated. It gives rise to several conjectures about the orthoprojectors onto some spaces of polynomials. We discuss in this note several open questions concerning the orthogonal projector onto spline spaces. The main advance on the subject has of course been Shadrin’s proof [9] of de Boor’s conjecture [1] that the max-norm of the orthoprojector is bounded independently of the breakpoint sequence underlying the spline space. To give a more precise statement, we shall consider a breakpoint sequence ∆ = (−1 = t0 < t1 < · · · < tN < tN+1 = 1), and introduce the space Sk,m(∆) := { s ∈ Cm−1[−1, 1] : s|(ti−1,ti) is a polynomial of degree < k, i = 1, . . . , N + 1 } of splines of order k satisfying m smoothness conditions at each interior breakpoint of ∆. Notice that we have to require 0 ≤ m ≤ k − 1. Following the practice of writing PV for the orthogonal projector onto a space V , the orthoprojector from L2[−1, 1] onto Sk,m(∆) is denoted by PSk,m(∆). We shall regard this projector as an operator on L∞[−1, 1], then consider its associated norm, and finally examine the supremum over all breakpoint sequences. In short, we are interested in the quantity Λk,m := sup ∆ ∥∥PSk,m(∆)∥∥∞ = sup ∆ sup ‖f‖∞≤1 ∥∥PSk,m(∆)(f)∥∥∞. Shadrin’s theorem ensures that Λk,m ≤ Λk,k−1 <∞. But estimating the quantity Λk,m, or merely its order, remains a challenge — only for m = 1, 2 is it known [2] that the ratio Λk,m/ √ k is bounded from above and from below. We are going to