Re: First Year Uni Calculus Marathon
Here's a cool one to think about:
_{k=0}^\infty$ of real numbers, does there exist a a smooth function $f\in\mathcal{C}^\infty(\mathbb{R})$ such that\\ \\ $\frac{d^kf}{dx^k}(0)=a_k$ for each $k$?$\\ \\ )
Bump.
Fun fact, am currently proving a predictable generalisation of this theorem's big brother for use in attacking a thesis problem. Will give a brief outline to show how easier problems can lead to pretty strong and useful results.
So classically, we define the notion of differentiability on open sets, that is sets where we can find small balls about any given point that are contained in our set.
Given a set K that is instead compact (bounded and with open complement if we are talking about Euclidean space R^d), and continuous functions f_alpha on K, when can we find a smooth function g on R^d such that the (alpha)-th derivative of g is f_alpha? (Here alpha ranges over multi-indices if d > 1. In the d=1 case alpha is just a non-negative integer.) The theorem that resolves this question is called the Whitney extension theorem, and the question I posed earlier is the Borel theorem, which is the special case where K is just a single point. The Borel theorem can be used as a stepping stone to a proof of the Whitney theorem.
We can ask the same question with "smooth" replaced by a stricter notion of regularity. In my case this is something called the Gevrey class. (Gevrey functions can be thought of as a sliding scale between functions that are merely smooth and functions that are analytic, which is a strong property indeed). In this case, we also have a version of the Whitney extension theorem, although of course the functions f_alpha have to obey stronger properties for an extension to exist.
My version involves functions f(x,y) where x and y live in compact sets in euclidean spaces, but f has different orders of Gevrey regularity in x and y (we call these kinds of function spaces with differing regularity anisotropic). The question is then whether or not there exists an extension of the same anisotropic Gevrey regularity. It is not super hard to answer given the results already established in the non-anisotropic case, but some of the estimates are a little tedious.
:=\sum_{k\leq n}\frac{a_kx^k}{k!})
for
but the formal expression
=\frac{cos(\frac{(n+1)x}{2})sin(\frac{nx}{2}) } {sin\frac{x}{2}} )
 $ to $ \frac{\pi}{8n} $ (Line 2 to Line 3) $ )