On the size of the polynomials orthonormal on the unit circle with respect to a measure which is a sum of the Lebesgue measure and \(P\) point masses (Q2790179)

The following variational problem from the theory of orthogonal polynomials on the unit circle is considered in the present paper.NEWLINENEWLINEGiven two positive integers \(n\) and \(p<n/2\), and \(\delta\in (0,1)\), define a class of probability measures on the unit circle \(\mathbb{T}\) NEWLINE\[NEWLINE P_{\delta, p}:= \bigl\{\sigma=\delta \mu(dt)+\sum_{j=1}^p m_j\delta(t_j)\bigr\}, NEWLINE\]NEWLINE where \(\mu\) is the normalized Lebesgue measure on \(\mathbb{T}\), \(m_j>0\), \(\{t_j\}\) are different points on \(\mathbb{T}\), and \(\delta(t)\) is the Dirac measure at the point \(t\in\mathbb{T}\). The problem is to find NEWLINE\[NEWLINE M_{n,p,\delta}:=\max_{\sigma\in P_{\delta,p}} \|\varphi_n(\cdot,\sigma)\|_\infty, NEWLINE\]NEWLINE where \(\varphi_n(\cdot,\sigma)\) is the \(n\)-th orthonormal polynomial for the measure \(\sigma\), and \(\|\cdot\|_\infty\) is the uniform norm on the unit circle. The main result claims that for every \(k>1.5\) NEWLINE\[NEWLINE M_{n,p,\delta}\leq C(\delta,k)\,\min(\log^k(n/p)\sqrt{p}, p). NEWLINE\]