Convergence of polynomial interpolation

[Milky]

Homework Statement

Let f be a contintuous real valued function on a closed interval [a, b]. The sup norm of such a function is max_x|f(x)|

Let pn be a polynomial interpolation of f determined by n points [xi, f(xi)] where xi Ε [a, b] for every i.
Suppose that the sequence {pn} is Cauchy with respect to the sup norm. Can you prove that pn --> f in sup norm? What can you say about this problem.

Homework Equations

The Attempt at a Solution

Not yet given, I am unsure how to even approach this problem and was hoping someone had ideas?

 

[mighty2000]

Hello,

Thank you for your forum post. This is an interesting problem and I would be happy to offer some guidance.

First, let's define what it means for a sequence of functions to be Cauchy with respect to a norm. A sequence of functions {fn} is said to be Cauchy with respect to a norm if for any given ε > 0, there exists an integer N such that for all n, m > N, ||fn - fm|| < ε, where ||·|| represents the norm.

In this case, our norm is the sup norm, which is defined as the maximum absolute value of a function on a given interval. So, given a function f, the sup norm is max|f(x)| for x in the interval [a, b].

Now, let's consider the given sequence {pn} that is Cauchy with respect to the sup norm. This means that for any given ε > 0, there exists an integer N such that for all n, m > N, ||pn - pm|| < ε. We want to prove that this sequence converges to f in the sup norm, which means that for any given ε > 0, there exists an integer N such that for all n > N, ||pn - f|| < ε.

To prove this, we can use the fact that {pn} is Cauchy and the continuity of f. Since f is continuous on the closed interval [a, b], we know that it is uniformly continuous. This means that for any given ε > 0, there exists a δ > 0 such that for all x, y in [a, b], if |x - y| < δ, then |f(x) - f(y)| < ε.

Now, let's choose ε > 0 and let N be the integer such that for all n, m > N, ||pn - pm|| < ε. Since {pn} is Cauchy, we know that for any given x in [a, b], |pn(x) - pm(x)| < ε for all n, m > N. So, for any given x in [a, b], we have |f(x) - pn(x)| < ε and |f(x) - pm(x)| < ε. This means that |pn(x) - pm(x)| < 2ε for all n, m > N.

Now, let's