Stone-Weierstrass on a disk

A couple weeks ago I wrote about a sort of paradox, that Weierstrass' approximation theorem could seem to contradict Morera's theorem. Weierstrass says that the uniform limit of polynomials can be an arbitrary continuous function, and so may have sharp creases. But Morera's theorem says that the uniform limit of polynomials is analytic and thus very smooth.

Both are true, but they involve limits in different spaces. Weierstrass' theorem applies to convergence over a compact interval of the real line, and Morera's theorem applies to convergence over compact sets in the complex plane. The uniform limit of polynomials is better behaved when it has to be uniform in two dimensions rather than just one.

This post is a sort of variation on the post mentioned above, again comparing convergence over the real line and convergence in the complex plane, this time looking at what happens when you conjugate variables [1].

There's an abstract version of Weierstrass' theorem due to Marshall Stone known as the Stone-Weierstrass theorem. It generalizes the real interval of Weierstrass' theorem to any compact Hausdorff space [2]. The compact Hausdorff space we care about for this post is the unit disk in the complex plane.

There are two versions of the Stone-Weierstrass theorem, one for real-valued functions and one for complex-valued functions. I'll state the theorems below for those who are interested, but my primary interest here is a corollary to the complex Stone-Weierstrass theorem. It says that any continuous complex-valued function on the unit disk can be approximated as closely as you like with polynomials in z and the conjugate of z with complex coefficients, i.e. by polynomials of the form

When you throw in conjugates, things change a lot. The uniform limit of polynomials in z alone must be analytic, very well behaved. But the uniform limit of polynomials in z and z conjugate is merely continuous. It can have all kinds of sharp edges etc.

Conjugation opens up a lot of new possibilities, for better or for worse. As I wrote about here, an analytic function can only do two things to a tiny disk: stretch it or rotate it. It cannot flip it over, as conjugation does.

By adding or subtracting a variable and its conjugate, you can separate out the real and imaginary parts. The the parts are no longer inextricably linked and this allows much more general functions. The magic of complex analysis, the set of theorems that seem too good to be true, depends on the real and imaginary parts being tightly coupled.

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Now for those who are interested, the statement of the Stone-Weierstrass theorems.

Let X be a compact Hausdorff space. The set of real or complex valued functions on X forms an algebra. That is, it is closed under taking linear combinations and products of its elements.

The real Stone-Weierstrass theorem says a subalgebra of the continuous real-valued functions on X is dense if it contains a non-zero constant function and if it separates points. The latter condition means that for any two points, you can find functions in the subalgebra that take on different values on the two points.

If we take the interval [0, 1] as our Hausdorff space, the Stone-Weierstrass theorem says that the subalgebra of polynomials is dense.

The complex Stone-Weierstrass theorem says a subalgebra of he continuous complex-valued functions on X is dense if it contains a non-zero constant function, separates points, and is closed under taking conjugates. The statement above about polynomials in z and z conjugate follows immediately.

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[1] For a complex variable z of the form a + bi where a and b are real numbers and i is the imaginary unit, the conjugate is a - bi.

[2] A Hausdorff space is a general topological space. All it requires is that for any two distinct points in the space, you can find disjoint open sets that each contain one of the points.

In many cases, a Hausdorff space is the most general setting where you can work without running into difficulties, especially if it is compact. You can often prove something under much stronger assumptions, then reuse the proof to prove the same result in a (compact) Hausdorff space.

See this diagram of 34 kinds of topological spaces, moving from strongest assumptions at the top to weakest at the bottom. Hausdorff is almost on bottom. The only thing weaker on that diagram is T₁, also called a Fri(C)chet space.

In a T₁ space, you can separate points, but not simultaneously. That is, given two points p₁ and p₂, you can find an open set U₁ around p₁ that does not contain p₂, and you can find an open set U₂ around p₂ that does not contain p₁. But you cannot necessarily find these sets in such a way that U₁ and U₂ are disjoint. If you could always choose these open sets to be distinct, you'd have a Hausdorff space.

Here's an example of a space that is T₁ but not Hausdorff. Let X be an infinite set with the cofinite topology. That is, open sets are those sets whose complements are finite. The complement of p₁ is an open set containing p₂ and vice versa. But you cannot find disjoint open sets separating two points because there are no disjoint open sets. The intersection of any pair of open sets must contain an infinite number of points.