Bernstein \cite{Be31} proved that for any choice of $a_i^n$ there exists $x_0\in [-1,1]$ such that\[\limsup_{n\to \infty} \sum_{1\leq i\leq n}\lvert p_{i}^n(x)\rvert=\infty.\]Erd\H{o}s and V\'{e}rtesi \cite{ErVe80} proved that for any choice of $a_i^n$ there exists a continuous $f:[-1,1]\to \mathbb{R}$ such that\[\limsup_{n\to \infty} \lvert \mathcal{L}^nf(x)\rvert=\infty\]for almost all $x\in [-1,1]$.


References


[Be31] S. Bernstein, Sur la limitation des valeurs d'un polynome $P_n(x)$ de degr\'{e} n sur tout un segment par ses valeurs en $(n+1)$ points du segment. Izv. Akad. Nauk. SSSR (1931), 1025-1050.

[ErVe80] Erd\H{o}s, P. and V\'{e}rtesi, P., On the almost everywhere divergence of Lagrange
interpolatory polynomials for arbitrary system of nodes. Acta Math. Acad. Sci. Hungar. (1980), 71-89.