Find $p$ s.t. there is a sequence of nodes in $[0,1]$ s.t. sequence of interpolating polynomials of every continuous function converges in $p$-norm

Question

Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $f(a)=p_{f,A,n}(a),\forall a\in A$. Such a polynomial is called the interpolating polynomial of $f$ with nodes $A$. Due to this, given $A\subseteq [a,b]$ with $n+1=|A|$, we can consider a linear map $X_{A,n} : C[a,b]\to C[a,b]$ such that $X_{A,n}(f)=p_{A,f,n}$ . Then $X_{A,n}$ is a continuous map and also a projection i.e. $X^2_{A,n}=X_{A,n}$. It can be shown that there is a constant $c$ such that $||X_{A,n}||_{\infty} \ge c \ln (n)$ for any set $A\subseteq [a,b]$ of size $n+1$, where $||.||_{\infty}$ is the $L^{\infty}$ operator norm induced from the $L^{\infty}$ norm on $C[a,b]$. Since $(C[a,b],||.||_{\infty})$ is a Banach space, it follows from Uniform boundedness principle, that given any sequence of subsets $\{A_n\}_{n=1}^\infty$ of $[a,b]$ with each $A_n$ having size $n+1$, $\exists f\in C[a,b]$ such that $||X_{A_n,n}(f) - f||_{\infty} $ does not converge to $0$.

This can be rephrased as: Given any sequence $\{A_n\}_{n=1}^\infty$ of subsets of $[a,b]$, with each $A_n$ of size $n+1$, there exists a continuous function $f$ on $[a,b]$ such that the sequence of interpolating polynomials $p_{f,A_n,n}(x)$ doesn't converge to $f$ uniformly on $C[a,b]$.

My question is the following: Does there exist a $p\in [1,\infty)$ such that the following holds: There exists a sequence $\{A_n\}_{n=1}^\infty$ of subsets of $[0,1]$, with each $A_n$ of size $n+1$, such that for every continuous function $f$ on $[0,1]$ , the sequence of interpolating polynomials $p_{f,A_n,n}(x)$ converge to $f$ in $p$-norm, on $[0,1]$ i.e. $||X_{A_n,n}(f)-f||_p\to 0,\forall f \in C[0,1]$ ?

Answer 1

The answer to the question is yes. The property even holds for any $0<p<\infty$.

The first such result is due to Erdös and Turan (1936) : let $f$ be a continuous function and $w(x)$ a weight on $[-1,1]$. Denote by $L_{n}(f,w)$ the Lagrange interpolant to $f$ at the zeros of the orthogonal polynomials with respect to $w$. They proved that $$\left\| f - L _ { n } ( f , w ) \right\| _ { 2 , w } \rightarrow 0 , \quad\text{ as }n \rightarrow \infty.$$ For the Chebyshev weight $w(x)=(1-x^{2})^{-1/2}$, the result holds true for any $1<p<\infty$.

The most general result is as follows. Let $w$ be a Jacobi weight of the form $(1-x)^{\alpha}(1+x)^{\beta}$, $\alpha,\beta>-1$, on $[-1,1]$ (actually $w$ can be more general than that). Let $0<p<\infty$ and $u\in(L\log^{+}L)^{p}$, $u\geq0$, positive on a set of positive measure. Then, $$ \lim _ { n \rightarrow \infty } \left\| u \left[ f - L _ { n } ( f , w ) \right] \right\| _ { p } = 0,\quad f\in C([-1,1]), $$ if and only if $u(w\sqrt{1-x^{2}})^{-1/2}\in L^{p}$, see

P. Nevai, Mean convergence of Lagrange interpolation. III, Trans. Amer. Math. Soc. 282 (1984) 669-698; and J. Approx. Theory 60 (1990) 360-363.

All that is discussed in

J. Szabados, P. Vertesi, A survey on mean convergence of interpolatory processes. Orthogonal polynomials and numerical methods. J. Comput. Appl. Math. 43 (1992), 3-18.