Interchanging Limit Processes — by Walter Rudin

As I mentioned earlier, my thesis (Trans. AMS 68, 1950, 278-363) deals with uniqueness questions for series of spherical harmonics, also known as Laplace series. In the more familiar setting of trigonometric series, the first theorem of the kind that I was looking for was proved by Georg Cantor in 1870, based on earlier work of Riemann (1854, published in 1867). Using the notations $A_{n}(x)=a_{n} \cos{nx}+b_{n}\sin{nx}$, $s_{p}(x)=A_{0}+A_{1}(x)+ \ldots + A_{p}(x)$, where $a_{n}$ and $b_{n}$ are real numbers. Cantor’s theorem says: $If \lim_{p \rightarrow \infty}s_{p}(x)=0$ at every real x, then $a_{n}=b_{n}=0$ for every n.

Therefore, two distinct trigonometric series cannot converge to the same sum. This is what is meant by uniqueness.

My aim was to prove this for spherical harmonics and (as had been done for trigonometric series) to whittle away at the hypothesis. Instead of assuming convergence at every point of the sphere, what sort of summability will do? Does one really need convergence (or summability) at every point? If not, what sort of sets can be omitted? Must anything else be assumed at these omitted points? What sort of side conditions, if any, are relevant?

I came up with reasonable answers to these questions, but basically the whole point seemed to be the justification of the interchange of some limit processes. This left me with an uneasy feeling that there ought to be more to Analysis than that. I wanted to do something with more “structure”. I could not  have explained just what I meant by this, but I found it later when I became aware of the close relation between Fourier analysis and group theory, and also in an occasional encounter with number theory and with geometric aspects of several complex variables.

Why was it all an exercise in interchange of limits? Because the “obvious” proof of Cantor’s theorem goes like this: for $p > n$, $\pi a_{n}= \int_{-\pi}^{\pi}s_{p}(x)\cos{nx}dx = \lim_{p \rightarrow \infty}\int_{-\pi}^{\pi}s_{p}(x)\cos {nx}dx$, which in turn, equals $\int_{-\pi}^{\pi}(\lim_{p \rightarrow \infty}s_{p}(x))\cos{nx}dx= \int_{-\pi}^{\pi}0 \cos{nx}dx=0$ and similarly, for $b_{n}$. Note that $\lim \int = \int \lim$ was used.

In Riemann’s above mentioned paper, the derives the conclusion of Cantor’s theorem under an additional hypothesis, namely, $a_{n} \rightarrow 0$ and $b_{n} \rightarrow 0$ as $n \rightarrow \infty$. He associates to $\sum {A_{n}(x)}$ the twice integrated series $F(x)=-\sum_{1}^{\infty}n^{-2}A_{n}(x)$

and then finds it necessary to prove, in some detail, that this series converges and that its sum F is continuous! (Weierstrass had not yet invented uniform convergence.) This is astonishingly different from most of his other publications, such as his paper  on hypergeometric functions in which mind-boggling relations and transformations are merely stated, with only a few hints, or  his  painfully brief paper on the zeta-function.

In Crelle’s J. 73, 1870, 130-138, Cantor showed that Riemann’s additional hypothesis was redundant, by proving that

(*) $\lim_{n \rightarrow \infty}A_{n}(x)=0$ for all x implies $\lim_{n \rightarrow \infty}a_{n}= \lim_{n \rightarrow \infty}b_{n}=0$.

He included the statement: This cannot be proved, as is commonly believed, by term-by-term integration.

Apparently, it took a while before this was generally understood. Ten years later, in Math. America 16, 1880, 113-114, he patiently explains the differenence between pointwise convergence and uniform convergence, in order to refute a “simpler proof” published by Appell. But then, referring to his second (still quite complicated) proof, the one in Math. Annalen 4, 1871, 139-143, he sticks his neck out and writes: ” In my opinion, no further simplification can be achieved, given the nature of ths subject.”

That was a bit reckless. 25 years later, Lebesgue’s dominated convergence theorem became part of every analyst’s tool chest, and since then (*) can be proved in a few lines:

Rewrite $A_{n}(x)$ in the form $A_{n}(x)=c_{n}\cos {(nx+\alpha_{n})}$, where $c_{n}^{2}=a_{n}^{2}+b_{n}^{2}$. Put $\gamma_{n}==\min \{1, |c_{n}| \}$, $B_{n}(x)=\gamma_{n}\cos{(nx+\alpha_{n})}$.

Then, $B_{n}^{2}(x) \leq 1$, $B_{n}^{2}(x) \rightarrow 0$ at every x, so that the D. C.Th., combined with $\int_{-\pi}^{\pi}B_{n}^{2}(x)dx=\pi \gamma_{n}^{2}$ shows that $\gamma_{n} \rightarrow 0$. Therefore, $|c_{n}|=\gamma_{n}$ for all large n, and $c_{n} \rightarrow 0$. Done.

The point of all this is that my attitude was probably wrong. Interchanging limit processes occupied some of the best mathematicians for most of the 19th century. Thomas Hawkins’ book “Lebesgue’s Theory” gives an excellent description of the difficulties that they had to overcome. Perhaps, we should not be too surprised that even a hundred years later many students are baffled by uniform convergence, uniform continuity etc., and that some never get it at all.

In Trans. AMS 70, 1961,  387-403, I applied the techniques of my thesis to another problem of this type, with Hermite functions in place of spherical harmonics.

(Note: The above article has been picked from Walter Rudin’t book, “The Way I  Remember It)) — hope it helps advanced graduates in Analysis.

More later,

Nalin Pithwa