# Polynomial interpolation

1. Apr 10, 2010

### math8

Let $$x_{0}, x_{1}, \cdots , x_{n}$$ be distinct points in the interval [a,b] and $$f \in C^{1}[a,b]$$.

We show that for any given $$\epsilon >0$$ there exists a polynomial p such that

$$\left\| f-p \right\|_{\infty} < \epsilon$$ and $$p(x_{i}) = f(x_{i})$$ for all $$i=1,2, \cdots , n$$

I know $$\left\| f\right\| _{\infty}= max_{x \in [a,b]}|f(x)|$$ and I wonder if the polynomial they are asking for is the Lagrangian polynomial interpolating f at the nodes $$x_{0}, x_{1}, \cdots , x_{n}$$. If yes, I am not sure how to prove the problem.