# Polynomial interpolation, Chebyshev nodes, absolute continuity

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?

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.