This and the next two videos for Episode 8 deal with an issue that I've been working on intermittently for over a year. As we've seen, quasiintegrable functions like Chi_q can be realized as measures, and the FourierStieltjes transforms of these measures generate Fourier series which, when summed, converge framewise to the original function. A natural question to ask is: can the same be done for the powers of Chi_q? That is, do (Chi_q)^2, (Chi_q)^3, and the like induce measures, and do these measures' Fourier transforms generate Fourier series that allow us to recover the original functions?

Classically, one would expect the answer to be no: the pointwise product of a measure/distribution with itself (as opposed to the tensor product or convolution of a measure/distribution with itself) is generally not going to be a measure.

But, for Chi_q and friends, (Chi_q)^k *is*, in fact, again a measure and, in fact, quasiintegrable! This shows that despite its pathological properties, Chi_q really does behave more like a function than a measure.

In this video, I go through the arduous details of computing, by hand, the Fourier transform of (Chi_q)^2, and proving that the resultant Fourier series converges framewise to (Chi_q)^2. The idea is to apply the methods from Episode 4, but then give up and use purely algebraic methods to make our way to the finish line. In doing so, we will stumble upon a totally unexpected new development—the use of derivatives of formal rational functions and power series—one that will have major implications in subsequent videos.


Links to other episodes

Episode 1:    • Episode 1  (p,q)adic Analysis and t...  
Episode 2:    • Episode 2  (p,q)adic Analysis and t...  
Episode 3 (Part 1):    • Episode 3 (Part 1)  (p,q)adic Analy...  
Episode 3.5:    • Episode 3.5  (Weak) Collatz as a Con...  
Episode 4 (Part 1):    • Episode 4 (Part 1)  Shut up and Comp...  
Episode 5 (Part 1):    • Episode 5 (Part 1)  Where No Wiener ...  
Episode 6 (Part 1):    • Episode 6 (Part 1)  Offerings to Mol...  
Episode 7 (Part 1):    • Episode 7 (Part 1)  Frames and Other...  
Episode 8 (Part 1):    • Episode 8 (Part 1)   Into the Wild B...  
Episode 8 (Part 2):    • Episode 8 (Part 2)   Into the Wild B...  
Episode 8 (Part 3):    • Episode 8 (Part 3)   Into the Wild B...  
Episode 8 (Part 4):    • Episode 8 (Part 4)   Into the Wild B...