On the relationship between norms in the spaces C(0,2\(\pi\) ) and U(0,2\(\pi\) ) (Q753057)

Let \(T_ n\) \((n=1,2,...)\) be the space of trigonometric polynomials of degree \(\leq n\). For \[ f(x)=\alpha_ 0+\sum^{n}_{p=1}(\alpha_ p \cos px+\beta_ p \sin px) \] put \[ \sigma_ q(f)(x)=\alpha_ 0+\sum^{q}_{p=1}(\alpha_ p \cos px+\beta_ p \sin px),\quad q=1,2,...,n, \] and let \(\| f\|_ C=\sup_{0\leq x\leq 2\pi}| f(x)|\) and \(\| f\|_ U=\max_{1\leq r\leq n}\| \sigma_ r(f)\|_ C\). The author proves: If \(\| f\|_ C=\| f\|_ U\) for every polynomial f from the subspace \(K\subset T_ n,\) then dim \(K\leq 10 \ln n+10\) and the estimate is of exact order for \(n\to \infty\).