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. 



For every table of interpolating nodes, there is a positive continuous function whose interpolating polynomials are not positive infinitely often
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 $M=\max\_{[-1,1]}f$. 


