It was shown by P. Erdős and P. Vertesi [Acta Math. Acad. Sci. Hung. 36, 71–89 (1980; Zbl 0463.41002)] that for every system of nodes
\[ 0\leq x_1^n < x_2^n < \cdots < x_n^n \leq 1,\qquad n\in \mathbb{N}, \tag{1} \] there exists a continuous function \(f\) such that the corresponding sequence of interpolating polynomials \((p_n)_{n\in \mathbb{N}}\) on (1) diverges almost everywhere. In this paper the author shows that for every \(p\in [1,\infty)\), there exists an infinitely differentiable function \(f: [0, 1]\to \mathbb{R}\) and a system of nodes (1) with \[ \lim_{n\to \infty} \max\{x_1^n - 0, x_2^n - x_1^n, \ldots, x_n^n - x_{n-1}^n, 1 - x_n^n\} = 0, \] such that the polynomials \(p_n\) interpolating \(f\) on (1) have the following property: For every function \(g\in L_p\) there exists a subsequence \((n_k)_{k\in \mathbb{N}}\) of natural numbers with \(\lim_{k\to \infty} p_{n_k} = g\).