Let $f\in C^0([-1,1])$ and $P\_n(f)$ its interpolation polynomial at the Chebyshev nodes.


I would be interested to know about any existing results (positive or negative) about the convergence of $P\_n(f)$ to $f$ in $L^1([-1,1])$ (only assuming that $f$ is continuous). 


The negative results I'm aware of (existence of a continuous function $f$ for which $P\_n(f)(x)$ does not converge for any $x$ in $[-1,1]$, from Marcinkiewicz and Grünwald (1936)) do not seem to leave much hope, but still they are not strong enough to rule out convergence in $L^1$. Any additional insight would be welcome!



Convergence of Chebyshev interpolation in L^1
Here are the details (for the second definition of Chebyshev nodes). We can define the new continuous function $g$ on the circle by $g(z)=f(\frac{z+z^{-1}}2)$ and consider the interpolation by the trigonometric polynomials $Q\_n(z)=\sum\_{k=0}^n b\_k\frac{z^k+z^{-k}}{2}$ (so that $P\_n(\frac{z+z^{-1}}{2})=Q\_n(z)$) on the $2n$-th roots of unity. 
The square of the $L^2$ norm of $Q\_n$ on the circle is (up to a normalization factor) just $\sum\_{k=-n}^n|c\_k|^2$ where $c\_0=b\_0$ and $c\_k=\frac {b\_{|k|}}2$ for $1\le k\le n$ (so $Q\_n(z)=\sum\_{k=-n}^n c\_kz^k$.


Notice that $z^n= z^{-n}$ on every node $z$, so we can just as well consider the modified polynomial $\widetilde Q\_n(z)=\sum\_{k=-(n-1)}^{n-1}c\_kz^k+b\_nz^n$. It will have the same values on the nodes but now all the participating powers of $z$ will be orthogonal with respect to the counting measure on the nodes, so we will have
$$
\sum\_{k=-(n-1)}^{n-1}|c\_k|^2+2(|c\_n|^2+|c\_{-n}|^2)=\frac 1{2n}\sum\_{z:z^{2n}=1}|\widetilde Q\_n(z)|^2\\ =\frac1{2n}\sum\_{z:z^{2n}=1}|g(z)|^2\le\|g\|\_C^2=\|f\|\_C^2
$$
and the declared boundedness of the interpolation operator from $C$ to $L^2$ on the circle follows immediately.


The rest is the usual mumbo-jumbo. We have a sequence of linear interpolation operators $I\_n:g\mapsto Q\_n$ whose norms from $C$ to $L^2$ are uniformly bounded by some constant $M$. Now decompose $g$ into $g\_m+h\_m$ where $g\_m$ is a trigonometric polynomial of degree $m$ and $\|h\_m\|\to 0$ as $m\to\infty$ and notice that $I\_ng\_m=g\_m$ for $n>m$, say, so
$$
\|g-I\_ng\|\_{L^2}=\|h\_m-I\_nh\_m\|\_{L^2}\le (1+M)\|h\_m\|\_C.
$$
The rest should be clear, but feel free to ask questions if needed :-)


