Point of post: This post proves that the uniform limit of polynomials of a fixed degree is a polynomial of degree less than or equal to that fixed degree.
Surely most of us are aware of the Stone-Weierstrass theorem (see here for a more generalized version for compact Hausdorff spaces) which states that for any one has that the polynomials are dense in the space of continuous functions with the infinity norm. Another interpretation of this is that every continuous function can be arbitrarily and uniformly approximated by a sequence of polynomials is the sense that . One may begin to wonder “what kinds of polynomials do I need to approximate these functions?” in the sense that could I approximate with polynomials of only even degree? What about odd degree? What bout prime degree? This seems like a valid question since, either the Stone-Weierstrass theorem is done by constructing the polynomials (see the first link and the notion of the Bernstein polynomials) or it’s proven in full generality (as in the second link). The first way leaves one wondering if there isn’t a ‘better’ choice of polynomials to approximate , I mean, it’s conceivable that using Bernstein polynomials while a valid way to prove the theorem is overly complicated. The second proof gives the reader absolutely zero idea what the polynomials used to approximate the functions look like. So with this in mind, one may ask an an example of such a thought experiment (although a naive one) whether continuous functions can be approximate by polynomials which all have a fixed degree . Well, in this post we will show that not only can you not approximate all the continuous functions with such sequences of polynomials but you can really only approximate other polynomials of degree less than or equal to ! In other words, we’ll prove that the space of all polynomials of degree less than or equal to is closed in with the infinity norm. In fact, we’ll prove something MUCH stronger
Point of post: This post is a continuation of
Point of post: This is a literal continuation of this post. I just hate when that dark grey to light grey thing happens. Just pretend that there was no lapse in the posts
And so we continue…
This post is the first in a line of posts which deal with theorems of approximation due to Stone and/or Weierstrass. The simplest and first to be discussed is that the set of all polynomials defined on (denoted ) is dense in . From there we will make a generalization to where is compact Hausdorff. Finally, after a brief discussion of locally compact Hausdorff spaces we shall make our final statement regarding on such spaces. But, we begin with baby steps.
Theorem: The subspace is dense in .
Proof: To do this we must merely show that given any and there exists some such that . We take a constructionist approach.
We first note that it suffices to prove the case when .
So, define to be the th Bernstein Polynomial given by
We begin by noting some identities. By the binomial theorem
Differentiating with respect to and doing some rearrangement gives
Differentiating with respect to again gives us
Splitting the sum over the addition and apply the first identity to the first sum gives
Lastly, multiplying by gives
Now, using the first identity we see that
Since is a continuous map on a compact metric space it follows that is uniformly continuous and so we may find a such that
From here we break our sum into two terms
Clearly the first sum is less than (just replace the term with by , yank it out, and apply the first identity). So, we must merely show the second sum can be made less than independent of . We know appealing to ‘s continuity again that it is bounded and so for all . It clearly follows that
Thus, we merely need to make the right hand sum less than . Note that the last identity in the above shows that
But, it is easily verified that for and so
Taking larger than implies that
It follows that
The conclusion follows. .
With this theorem we may prove some very cool stuff. For example:
Theorem: is separable.
Proof: Let be the set of all polynomials with rational coefficients. We prove that is dense in . To do this let and be arbitrary. Since is dense in there exists some such that .
Now, the mapping given by is continuous and so it attains a maximum on . So, choose numbers such that . Then, clearly and
It follows that and so
Thus, is dense in . It remains to show that it is countable. To see this define by . Clearly is surjective and since is the countable union of countable sets it’s countable. The conclusion follows. .
The next problem is a particular favorite of mine, for no particular reason.
Theorem: Define a moment of by . If have the same moments for then .
Proof: We prove (mainly for fun) a lemma from real analysis.
Lemma: Let be continuous. Then, only if .
Proof: Suppose that then for some , call that point . Since we clearly have for some . So, by ‘s continuity we see that there exists some such that . Taking the concentric closed ball of radius we see that for every . Thus, by ‘s continuity and the compactness of we have that for some . It follows that
Which of course contradicts the assumption that the integral is zero. The conclusion follows. .
Using this lemma we may assume that otherwise we would have that
and so we’d be done.
So, under this assumption define the functional by
We claim that is continuous. To see this and be given. Then, choosing such that
we see that
‘s continuity follows. We next note that that is linear and so
The last part gotten by noticing that
It follows that and since the right hand side is closed we see that
It follows that for every . In particular
The conclusion follows. .
We now prove a nice little extension of the Weierstrass theorem.
Theorem: Let be closed and bounded. Prove that is dense in .
Proof: Note that since is closed and bounded it is a subset of for some . Now, given any there exists by Tietze’s extension theorem an extension such that . Now, for every there exists some such that . Let . Furthermore, let
and finally let . Clearly, . So, it remains to show that given any and there exists some such that . To see this let and be as above and let . Then, clearly and
The conclusion follows.
Corollary: Using the exact same idea except taking one may prove that is separable for every closed and bounded .