On the distribution of the points of Chebyshev alternation with applications to strong unicity constants (Q1813369)

Let \(C_{2\pi}\) be the set of all \(2\pi\)-periodic real valued continuous functions with the sup-norm \(\|\cdot\|_{2\pi}\) and let \(\pi_ n^*\subset C_{2\pi}\) be the subspace of all trigonometric polynomials of degree \(\leq n\). For \(f\in C_{2\pi}\), let \(T_ n^* f\) be the best uniform approximation with respect to \(\pi_ n^*\). Let there be \(2n+2\) points \(-\pi<t_{-n}^{(n)}<t_{- n+1}^{(n)}<\dots<t_{n+1}^{(n)}\leq\pi\) such that (1) \((f-T_ n^* f)(t_ k^{(n)})=\varepsilon(-1)^ k \rho_ n^*(f)\) for \(k=-n,- n+1,\dots,n+1\) where \(\varepsilon=1\) or \(\varepsilon=-1\) and \(\rho_ n^*=\| f-T_ n^* f\|_{2\pi}\) is the minimal error if \(f\) is approximated by elements of \(\pi_ n^*\). Let \(\Delta_ n^{(f)}=\max_{-n\leq k\leq n+1} | t_ k^{(n)}-{{k\pi} \over {n+1}}|\) and \(d_ n(f)=\min_{-n\leq k\leq n+1} (t_{k+1}^{(n)}- t_ k^{(n)})\). One of the two theorems proved is: Theorem. Let \(f\in C_{2\pi} \setminus \bigcup_{n=0}^ \infty \pi_ n^*\) be even. Then for \(\varepsilon>0\), there exists a subsequence \(\{x_ j\}_{j=1}^ \infty\) of the natural numbers such that: (2) \(\rho_ n^*(f)>\rho_{n+1}^*(f)\), (3) \(\Delta_ n(f)<8\pi(\log n/n)^{1/2}\), (4) \(d_ n(f)=1/n^{2+\varepsilon}\), for \(n=n_ j\), \(j=1,2,3,\dots\;\).