Upper estimates for the coefficients of algebraic polynomials via their \(L_p\)-norms on intervals (Q2714713)

Let \(\tau_n(\beta)\) be the smallest possible constant in the inequality NEWLINE\[NEWLINE\left(\sum^n_{k=0} |\alpha_k |^2\right)^{1/2} \leq\tau_n (\beta)\left( \int^\beta_0 \left|\sum^n_{k=0} a_kx^k \right|^2 dx \right)^{1/2}.NEWLINE\]NEWLINE The author proves the asymptotic \((n\to\infty)\) relation \(\tau_n(\beta) \asymp(n+1)^{-1/4} [\sqrt{\beta^{-1}} +\sqrt{1+\beta^{-1}}]^{2n}\) and a similar relation in the case when the interval \((0,\beta)\) is replaced by \((-\beta,\beta)\).NEWLINENEWLINEFor the entire collection see [Zbl 0952.00006].