Interesting Theorem Regarding Polynomials
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
Theorem: Let be fixed and define the sequence of polynomials by for some real numbers . Suppose that there exists points such that converges for . Then, converges uniformly to a polynomial of degree less than or equal to on every compact subspace of .
Proof: We first show that if is compact then the convergence of the coefficient sequences implies the uniform convergence of to the polynomials whose coefficients are the limits of the coefficient sequences. Indeed, suppose that is compact. Then, if then . To see this merely let . Then, given choosing large enough so that implies that
from where it follows by the arbitrariness of that as desired. We now show that since converges for each then . Indeed, consider that this fact is equivalent to saying that
where clearly . Or, if we define , , and then this can be rephrased as . But, since is invertible (it’s a Vandermonde matrix) and since (being a linear operator on a finite dimensional vector space) is continuous we may conclude that
and thus converges and thus each sequence converges. The fact then that converges uniformly to a polynomial of degree less than or equal to then follows from the first part of the proof.
Remark: A simpler proof for the second part would be to recall that all norms on finite dimensional vector spaces induce the same topology. In particular note that with the infinity norm is a finite dimensional vector space in which converges. That said, is a norm on and thus by prior comment (since both norms induce the same topology) it follows that converges in this norm. But, this clearly implies that each converges.
1. Apostol, Tom M. Mathematical Analysis. Reading, MA: Addison-Wesley Pub., 1974. Print.
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