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, quasi-integrable functions like Chi_q can be realized as measures, and the Fourier-Stieltjes transforms of these measures generate Fourier series which, when summed, converge frame-wise 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 point-wise 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, quasi-integrable! 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 frame-wise 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.
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Links to other episodes
Episode 1: • Episode 1 - (p,q)-adic...
Episode 2: • Episode 2 - (p,q)-adic...
Episode 3 (Part 1): • Episode 3 (Part 1) - (...
Episode 3.5: • Episode 3.5 - (Weak) C...
Episode 4 (Part 1): • Episode 4 (Part 1) - S...
Episode 5 (Part 1): • Episode 5 (Part 1) - W...
Episode 6 (Part 1): • Episode 6 (Part 1) - O...
Episode 7 (Part 1): • Episode 7 (Part 1) - F...
Episode 8 (Part 1): • Episode 8 (Part 1) - ...
Episode 8 (Part 2): • Episode 8 (Part 2) - ...
Episode 8 (Part 3): • Episode 8 (Part 3) - ...
Episode 8 (Part 4): • Episode 8 (Part 4) - ...