Extremal problems in the theory of polynomial operators (Q1207387)

Let \(C\) be the space of all continuous \(2\pi\)-periodic functions \(f\) with the ``sup'' norm. Let \(\Pi_ n\) be the space of all trigonometric polynomials of order \(\leq n\). Finally, let \(\Omega\) be a set of linear operators \(A:\) \(C\to C\), such that \(Af=f\) for all \(f\in\Pi_ n\). If \(E_ n(f)\) is the best approximation of \(f\) by elements of \(\Pi_ n\), then \[ \| Af-f\|\leq(1+\| A\|)E_ n(f). \] This initiates the problem: find the number \(\rho=\inf\{\| A\|:A\in\Omega\}\) and the operators \(A\) such that \(\rho=\| A\|\). The author solves this problem for some interesting classes of operators.