Trigonometric Series and the Beginnings of Set Theory

Let Image may be NSFW.
Clik here to view. be a Image may be NSFW.
Clik here to view.-periodic function. It may or may not have a representation as a trigonometric series

Image may be NSFW.
Clik here to view.

A natural question to ask is whether or not the representation of Image may be NSFW.
Clik here to view. as a trigonometric series is unique, if it has one. It was the consideration of this question that led Cantor to the invention of set theory.

There is a nice writeup of this story in the first part of this article by Alexander Kechris. I’ll give part of the story below.

Cantor solved the problem in the affirmative; i.e., he proved:

Suppose that a trigonometric series converges to zero everywhere in Image may be NSFW.
Clik here to view.. Then all the coefficients of that series are zero.

(By subtraction, this is equivalent to the problem stated above.) He was also able to show (by a very similar method) the following, which I’ll call the Isolated Points Lemma:

Suppose Image may be NSFW.
Clik here to view. and that a trigonometric series converges to zero on Image may be NSFW.
Clik here to view.. Then that series converges to zero at Image may be NSFW.
Clik here to view. as well.

From these two results, we can immediately conclude the following:

Suppose that a trigonometric series converges to zero at all but finitely many points. Then the coefficients of that series are all zero.

Call a set Image may be NSFW.
Clik here to view. a set of uniqueness if whenever a trigonometric series converges to zero on Image may be NSFW.
Clik here to view., the coefficients of that series are all zero. Then the previous result may be stated: “All finite sets are sets of uniqueness.”

But we can use the Isolated Points Lemma to show more than that. For example, we can show that the set Image may be NSFW.
Clik here to view. is a set of uniqueness. The reason is that if a trigonometric series converges to zero on Image may be NSFW.
Clik here to view., then by the Isolated Points lemma, it also converges to zero on the points in Image may be NSFW.
Clik here to view..

But, now that we know that it converges to zero on the points in Image may be NSFW.
Clik here to view., we can apply the Isolated Points Lemma again to show that it converges to zero at 0 (since we now know that at converges to zero on, e.g., Image may be NSFW.
Clik here to view.).

What we have actually shown by the above argument is the following:

Given Image may be NSFW.
Clik here to view., let Image may be NSFW.
Clik here to view. be the set of limit points of Image may be NSFW.
Clik here to view. (also known as the Cantor-Bendixson derivative of Image may be NSFW.
Clik here to view.). If Image may be NSFW.
Clik here to view. is a set of uniqueness, then Image may be NSFW.
Clik here to view. is a set of uniqueness.

For any Image may be NSFW.
Clik here to view., let Image may be NSFW.
Clik here to view. be the Image may be NSFW.
Clik here to view.th Cantor-Bendixson derivative of Image may be NSFW.
Clik here to view.. Then, by iterating the above fact, we have the following:

Suppose that for some Image may be NSFW.
Clik here to view., Image may be NSFW.
Clik here to view.. Then Image may be NSFW.
Clik here to view. is a set of uniqueness.

This is as far as we can go as long as we merely iterate the Cantor-Bendixson derivative finitely often. But, if we make the leap to iterating it transfinitely many times, we can go much further:

Theorem: All countable closed sets are sets of uniqueness.

Proof: First, define Image may be NSFW.
Clik here to view. for all ordinals and all closed sets Image may be NSFW.
Clik here to view. as follows:

1. Image may be NSFW.
   Clik here to view.
2. Image may be NSFW.
   Clik here to view..
3. Image may be NSFW.
   Clik here to view., when Image may be NSFW.
   Clik here to view. is a limit ordinal.

The Isolated Points Lemma says that if a trigonometric series converges to zero outside of Image may be NSFW.
Clik here to view., then it converges to zero outside of Image may be NSFW.
Clik here to view.. We will generalize this by showing the following lemma:

Lemma: If a trigonometric series converges to zero outside of Image may be NSFW.
Clik here to view., then it converges to zero outside of Image may be NSFW.
Clik here to view. for any Image may be NSFW.
Clik here to view..

Proof of Lemma: This is by transfinite induction. The successor step of the induction is just the Isolated Points Lemma again, so all we have to show is that, fixing a trigonometric series, if Image may be NSFW.
Clik here to view. is a limit ordinal and the series converges to zero outside of each Image may be NSFW.
Clik here to view. for Image may be NSFW.
Clik here to view., then it converges to zero outside of Image may be NSFW.
Clik here to view.. But this follows simply because every point of Image may be NSFW.
Clik here to view. must be in some Image may be NSFW.
Clik here to view. for Image may be NSFW.
Clik here to view. by definition. End of proof of Lemma.

To complete the proof of the theorem then, we just have to observe that for all countable closed Image may be NSFW.
Clik here to view., Image may be NSFW.
Clik here to view. for some Image may be NSFW.
Clik here to view.. Clearly, for all closed Image may be NSFW.
Clik here to view., there is an Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view. (this is because Image may be NSFW.
Clik here to view. is a decreasing sequence). But it is standard fact that a set Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view. is either empty or of cardinality Image may be NSFW.
Clik here to view.. (Such a set is called a perfect set and a reference for the cited fact is page 7 of David Marker’s notes on descriptive set theory.) End of proof.

As a historical note, Kechris reports that while thinking about the above issues led Cantor to discover ordinals, he never actually wrote down a proof of the above theorem; that was finally done by Lebesgue in 1903.

Further, it was later proven by Bernstein and Young independently that arbitrary countable sets are sets of uniqueness, and by Bari that countable unions of closed sets of uniqueness are sets of uniqueness.

Edit: Simplified the proof.