Polynomial approximation using projections whose kernels contain the Chebyshev polynomials
The authors study projections of \(C^{(n+1)}[-1,1]\) onto the space \(P_ n\) of polynomials of degree \(\leq n\) with the property that the error in the Chebyshev norm can be estimated by \[ (*)\quad \| f-Pf\| \leq \frac{1}{2^ n(n+1)!}\| f^{(n+1)}\|. \] They find two necessary conditions one of which states that \(P(T_{n+1})=0\) for the Chebyshev polynomial \(T_{n+1}\) of degree \(n+1\). These conditions are not sufficient for (*). A complete characterization of projections satisfying (*) is given only in the case \(n=0\). It is well known that the projector \(P^*\) which assigns to f the polynomial \(P^*f\) interpolating f at the zeros of \(T_{n+1}\) satisfies (*). The authors show that all projections in a certain neighbourhood of \(P^*\) also satisfy (*). Reviewer's remarks: 1) In the proof of theorem 2.3 f should be replaced by f-p at several places. 2) In theorem 3.2 the condition on the measure must be strengthened by ``absolutely continuous measure \(\nu\) with density \(0\leq c(t)\leq 1''\). 3) Theorem 3.4 is wrong. Example 3.3 in the paper is a counterexample. The authors informed the reviewer that the conjecture on page 329 must be modified. A simple counterexample to the conjecture as stated in the paper would be for \(n=0\) the projector \(Pf=f(-1)+f(0)+f(1)-f(-1/2)-f(1/2).\)
scientific article; zbMATH DE number 4077931