Let $$f\in C[a,b]$$ A triangular system is a series of numbers \begin{matrix} x_{11}\\ x_{21}&x_{22}\\ x_{31}&x_{32}&x_{33}\\ \cdots \end{matrix} that $$a<x_{n1}<x_{n2}<\cdots<x_{nn}<b$$ Consider the Lagrange interpolation polynomial $p_{n}$ by the points on n-th "layer". I have following questions:
1) Is there a continuous function $f$ such that for any triangular system, we have $$\lim\limits_{n\to\infty}||p_{n}-f||_{\infty}=0$$ Present results: polynomials are definitly what we want. But I want to generalize this results to analytic functions (which I guess is ture). Is there a proof or a counterexample?
2)Similarly, is there a function that is bad enough that any triangular system dose not give a converging approximation?
This one is solved that the Chebyshev equioscillation theorem implies that the polynomial of best approximation is actually an interpolation. And the Weierstrass' theorem implies the convergence.
3)Is there a triangular system that gives approximation to any continuous function?
The present results show that the Chebyshev nodes gives a perfect approximation to $BV[a,b]\bigcap C[a,b]$. What about other conditions? (A diverging example for Chebyshev nodes is equally wanted.)
4) Is there a triangular system and function $f$ such that $$\forall x\in[a,b]\quad\lim\limits_{n\to\infty}(p_{n}(x)-f(x))\neq 0$$
And welcome any other related results or questions.
