Some extremal properties of the integral of Legendre polynomials (Q5957985)

Let \(1=x_1>x_2>\ldots>x_n=-1\), \(\omega(x) = (x-x_1)(x-x_2)\ldots(x-x_n)\), \(\ell_i(x) = \omega(x)/(\omega'(x_i)(x-x_i))\). \textit{P. Erdős} [Acta Math. Acad. Sci. Hung. 12, 235-244 (1961; Zbl 0098.04103)] set a question to find \((x_1,\ldots,x_n)\) which minimizes \(f(x_1,\ldots,x_n) = \int_{-1}^{1}\sum_{i=1}^{n}\ell_i(x)^2 dx\). In connection with this problem, the authors solve several related extremal problems where the set of the roots of the integral of Legendre polynomials turns to deliver the extremum.