On mean convergence of Lagrange interpolation for general arrays (Q1577919)

Let \(\{x_{jn}\}_{j=1}^n, n\geq 1\), be \(n\) distinct points in a compact set \(K\subset \mathbb R\) and let \(L_n[\cdot]\) denote the corresponding Lagrange interpolation operator. Let \(v\) be a suitable restricted function on \(K\). What conditions on the array \(\{x_{jn}\}_{1\leq j\leq n,n\geq 1}\) ensure the existence of a number \(p>0\) such that \(\lim_{n\to\infty}\|(f-L_n[f])v\|_{L_p(K)}=0\) holds for every continuous \(f:K\to\mathbb R\)? This paper shows that it is necessary and sufficient that there exists a number \(r>0\) satisfying \(\sup_{n\geq 1}\|\pi_nv\|\sum_{j=1}^n[1/|\pi'_n(x_{jn})|]<\infty\). Here \(\pi_n(x)=\prod_{j=1}^n(x-x_{jn})\). The necessity of this condition is due to the reviewer [Acta Math. Hungar. 70, No. 1-2, 27-38 (1996; Zbl 0861.41004)].