Some extremal problems for polynomials in integral metrics (Q1922333)

The author studies the behavior of the functions: \[ \Phi_{pq} (n, \eta) : = \sup \bigl\{ |p_{n - 1} |_{q [- \eta, \eta]} \mid p_{n - 1} \in P_{n - 1},\;|p_{n - 1} |_{p [- 1, 1]} \leq 1 \bigr\} \] and \[ \Psi_{pq} (n,a) : = \sup \bigl\{ |t_{n - 1} |_{q [- \pi, \pi]} \mid t_{n - 1} \in T_{n - 1},\;|t_{n - 1} |_{p [- a, a]} \leq 1 \bigr\}, \] where \(\eta \geq 1\), \(0 < a \leq \pi\), \(1 \leq p\), \(q \leq \infty\) and \[ |f |_{p [a,b]} : = \left( \int^b_a \bigl |f(x) \bigr |^p dx \right)^{1/p}, \quad |f |_{\infty [a,b]} : = \max_{[a,b]} \bigl |f(x) \bigr |, \] furthermore, \(P_n\) and \(T_n\) denote the set of \(n\)th-order algebraic and trigonometric polynomials. Among others it is proved that for \(1 \leq p,q \leq \infty\), the following relation is valid uniformly in \(n \in \mathbb{N}\) and \(\eta \geq 1\): \[ \Phi_{pq} (n, \eta) \asymp \max \left\{ 1, {n^{2 (1/p - 1/q)} \bigl( g (\eta) \bigr)^{n - 1 + 1/p + 1/2q} \over \bigl( n \sqrt {\eta - 1} + 1 \bigr )^{2/p - 1/q}} \right\}, \] where \(g (\eta) : = \eta + \sqrt {\eta^2 - 1}\).