How to prove that for an absolutely continuous function, the Lagrange interpolation polynomial at Chebyshev nodes converges uniformly to the function as the number of nodes goes to infinity?
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The theorem appears first proved by Krylov in 1956. Johnson and Riess however give a simple proof for all functions with absolutely convergent Chebyshev series (if $f$ is continuous and of bounded variation in $[-1,1]$, this holds). The proof is reproduced below.
Take $f$ expanded in the Chebyshev polynomials $f(x)=\sum\limits_{i=0}^\infty a_i T_i(x)$ and then also expand the remainder $R_n(f)=L_n(f)(x)-f(x)$ in Chebyshev polynomials, where $L_n(f)$ is the Lagrange interpolant at the Chebyshev nodes. Set $k=2rn+\alpha$, for $-n+1\leq \alpha\leq n-1$, and note then that $$|R_n(T_k)|=|T_{k}(x)-(-1)^rT_{|\alpha|}(x)|\leq 2,\ R_n(f)=\sum\limits_{i=n}^\infty a_i R_n(T_i),$$ so that $|L_n(f)-f|\leq 2\sum\limits_{i=n}^\infty |a_i|$, finally giving the desired uniform convergence.
