Uploaded August 2021 | Updated September 2026, 2 days ago
This video goes over a fun little problem from A Hilbert Space Problem Book by Paul Halmos. It's a simple problem to understand, and the solution to it is surprisingly deep. It allows us to introduce the idea of a Reproducing Kernel Hilbert Space through what are called Generating Functions. The prerequisite for this is primarily a knowledge of power series and some linear algebra. I try to build up what I can in the first few minutes, which lets us get to some more advanced ideas. Ultimately we are talking about a connection between sequences, Hilbert Spaces, generating functions, complex analysis, and the Hardy space.
Since 3blue1brown is running his Summer of Mathematical Exposition, I thought I'd go with a bit more of an accessible topic this time. That's also the plan for the future, where I will mix in research, graduate, and undergraduate work into my channel content. As I said before, this one is aimed at someone who has had Calculus 2 and some Linear Algebra. Experience with complex analysis will help, but all we need from there are power series (analytic functions).
Music:
Go on - Cacti
Come 2gether - Ooyy
Try Once More - Ben Elson
Video Call from Los Angeles - Trevor Kowalski
Disque Magique - Dusty Decks
I Know Now - Cacti
Wrong - Dan Henig
Guardians + Tek - Craig Hardgrove
#summerofmathexposition
This video goes over a fun little problem from A Hilbert Space Problem Book by Paul Halmos. It's a simple problem to understand, and the solution to it is surprisingly deep. It allows us to introduce the idea of a Reproducing Kernel Hilbert Space through what are called Generating Functions. The prerequisite for this is primarily a knowledge of power series and some linear algebra. I try to build up what I can in the first few minutes, which lets us get to some more advanced ideas. Ultimately we are talking about a connection between sequences, Hilbert Spaces, generating functions, complex analysis, and the Hardy space.
Since 3blue1brown is running his Summer of Mathematical Exposition, I thought I'd go with a bit more of an accessible topic this time. That's also the plan for the future, where I will mix in research, graduate, and undergraduate work into my channel content. As I said before, this one is aimed at someone who has had Calculus 2 and some Linear Algebra. Experience with complex analysis will help, but all we need from there are power series (analytic functions).
Music:
Go on - Cacti
Come 2gether - Ooyy
Try Once More - Ben Elson
Video Call from Los Angeles - Trevor Kowalski
Disque Magique - Dusty Decks
I Know Now - Cacti
Wrong - Dan Henig
Guardians + Tek - Craig Hardgrove
#summerofmathexposition
