New answers tagged set-theory
1
Here are the references to the original papers of Cantor:
On a theorem concerning trigonometric series. (Ueber einen die trigonometrischen Reihen betreffenden Lehrsatz.)
Borchardt J. LXXII, 130-138 (1870).
Proof that a function given for every real value of by a trigonometric series has only one representation in this form. (Beweis, dass eine für jeden ...
17
It was Fourier series rather than Fourier transform. Considering that the sets where Fourier series converge can be very intricate it is not that surprising that they led Cantor to develop set theory for subsets of real numbers. But at some point he took a turn into the abstract (for which he is best known today) that was not really motivated by the initial ...
3
paul garrett has the idea. $E \subseteq \mathbb R$ is a set of uniqueness
if: given a trigonometric series $\sum_{n=-\infty}^\infty c_n e^{int}$, if it converges to $0$ except possibly on $E$, then $c_n = 0$ for all $n$.
Here is a description.
The empty set is a set of uniqueness. This is just a fancy way to say that if a trigonometric series converges to ...
4
Unfortunately, I do not have a concrete reference... but I seem to recall that Cantor's earliest work was about "sets of uniqueness" for Fourier series (I think not Fourier transforms, but I could easily be mistaken).
This would be similar to other late 19th century "constructive" analysis projects, where limits of limits of ... ...
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