For every table of interpolating nodes, there is a positive continuous function whose interpolating polynomials are not positive infinitely often

Question

Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that the sequence of interpolating polynomials $p_n(x)$ of $f$ at those points satisfy $(-\infty,0]\cap p_n([a,b])\ne \phi$ for infinitely many $n$ ?

NOTE: If this is true, then it would imply that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there is a continuous function $f:[a,b]\to (0,\infty)$ such that the sequence of interpolating polynomials $p_n(x)$ of $f$ does not converge uniformly to $f$ on $[a,b]$. i.e. a positive answer to my question implies Faber's theorem.

Answer 1

Yes, it follows from the following result of S. Bernstein (Quelques remarques sur l'interpolation, Math. Ann. 79 (1918), 1-12):

For an arbitrary scheme $X=\cup_{n} A_{n}$ of points in $[-1,1]$, there exist a continuous function $f$ and a point $x_{0}$ in $[-1,1]$ such that $$ { \limsup _ { n \rightarrow \infty } } \left| L _ { n } \left( f, X , x _ { 0 } \right) \right| = \infty. $$

If there is a subsequence $n_{k}$ with $$ { \lim_ { n_{k} \rightarrow \infty } } ~L _ { n_{k} } \left( f, X , x _ { 0 } \right) = -\infty,$$ consider the positive function $g(x)=f(x)-m+1>0$, where $m=\min_{[-1,1]}f$.

If there is a subsequence $n_{k}$ with $$ { \lim_ { n_{k} \rightarrow \infty } } ~L _ { n_{k} } \left( f, X , x _ { 0 } \right) = \infty,$$ consider the positive function $g(x)=-f(x)+M+1>0$, where $M=\max_{[-1,1]}f$.