Bound for the max value of a Lagrange polynomial Given some Lagrange polynomial $L$(x) that interpolates over the points $x$0, $x$1,..., $x$n with values in the set  $A$ = {$a$0, $a$1,..., $a$n } on some interval [a,b], show that the max value that $L$(x) takes on the interval is bounded as such:
||$P$||∞  ≤  M||$A$||2
where M is a function of "n" (the number of interpolation points).
My Ideas:
||$P$||∞ is just the magnitude of some point $P$(xi) (maybe not unique) for $P$(x) on [a,b]. So I started by representing this maximum as the linear combination of the base Lagrange polynomials:
$P$(xi) = aoqo(xi) + ... + anqn(xi)   
I then squared both sides:
($P$(xi))2 = (aoqo(xi) + ... + anqn(xi))2
Next, I used the Cauchy inequality to say:
($P$(xi))2 = (aoqo(xi) + ... + anqn(xi))2 ≤ (ao2 +...+ an2)(qo(xi)2 +...+ qn(xi)2). 
This means:
$P$(xi)) ≤ ||$A$||2 (qo(xi)2 +...+ qn(xi)2)1/2
But I don't know how to turn (qo(xi)2 +...+ qn(xi)2)1/2 = M into something that's only based on "n".
 A: You are very close. $q_i(x)^2$ is a polinomial that equals $1$ at $x=x_i$ and has a double root at $x=x_j$ for any $j\neq i$, hence
$$ q_i(x)^2 \leq \frac{K}{K+(x-x_i)^2} \tag{1}$$
where $K$ just depends on the mesh $\delta$ of the partition given by $x_0,x_1,\ldots,x_n$. 
A continuous analogue is given by the inequality:
$$\left(\frac{\sin (\pi x)}{\pi x}\right)^2 \leq \frac{\frac{3}{\pi^2}}{\frac{3}{\pi^2}+x^2} $$
where the LHS is a function that equals $1$ at $x=0$ and has a double root at any other integer.
Now, since the function $\frac{K}{K+x^2}$ is integrable over $\mathbb{R}$ and its integral equals $\pi \sqrt{K}$, it is not difficult to provide a uniform bound for $\sum_{j=0}^{n}q_j(x)^2$ that depends on $n$ only.
Anyway, I am waiting to know if some extra information on the partition given by $x_0,x_1,\ldots x_n$ is available. I can give more explicit bounds for uniform or Chebyshev partitions. I am expecting that something has to be known: for instance, by taking $n=2$, $[a,b]=[0,1]$ and $x_i=\frac{5+i}{12}$ we have the following wild behaviour of $q_0(x)^2+q_1(x)^2+q_2(x)^2$:
$\hspace{1in}$
At least we have to know that $x_0$ is close to $a$ and $x_n$ is close to $b$: otherwise, $(1)$ may fail, as depicted above.
