The Fourier Operator - L18 - Frederic Schuller

[@active285]

12:00 Typo: Facts (i) should read L^2(IR^d). In fact, it even holds: S(IR^d) c L^p(IR^n) für p≥1. 26:50 To be honest: Normally, there is no problem, since we usually define F[f](x) := \hat{f}(x), i.e. the hat does not meet the variable x in round brackets. Hence, \hat{f} can be understood as a shorthand for F[f]. Is the notation presented here maybe physicist notation?

[@jwp4016]

Thanks! It seems worth to remark on your comment. Typo in your typo report : That last n should be d. :) For your second point, his notation has a distinctive value, as you can see at 44:40. To use the "hat(f)" notation, we should, for example, define a new function f'(x) := f(x-a), and then write it as hat(f')(p). But the "hat(f(x))" notation allows us to express it directly in terms of f and variables. Actually, we can fully be rigorous while sticking to variables. Extend our first order language to include lambda calculus. Then, formulas of the form f = λx.f(x) are a well-formed formula . Thus we are perfectly justified to use expressions like hat(λx.f(x-a)).

[@stefaanmelis]

1:20:00 Herr Schuller was at some point somewhat confused, as he confessed himself. Therefore I believe he gave a wrong explanation why we get y^2 shifted by x into (y-x)^2. According to his explanation you have to arrange this by applying a binomial manipulation of (p^2+ipx). In my opinion, the position shift (y-x) after the Fourier operator is simply caused by the phase shift prior to applying the operator (according to the proposition that Herr Schuller showed earlier in the lecture).

[@mbivert]

Not sure this is exactly what you meant, but for future references, in the lectures notes (Simon Rea, Richie Dadhley), the change of variables is performed in two steps (z

[@millerfour2071]

21:21, 31:44, 1:13:27

[@ndmath]

Does anyone know a good resource for extra information on the Fourier operator as discussed here?

[@ernestomamedaliev4253]

Its just Fourier analysis, there is a lot of bibliography around it. I studied it in spanish, so I dont know what book might be suitable for you. In any case, there are a lot of books that might be helpful, even if the information is too extense. For the use in QM, maybe you can find enough information in "Quantum Mechanics - Cohen-Tannoudji, Vol 1". You can get it through this link: www.zuj.edu.jo/download/quantum-mechanics-vol-1-cohen-tannoudji-pdf/ Look for the index somewhere, but the book is good in order to explain the mathematical stuff usually used when one deals with QM. Good luck!

[@ernestomamedaliev4253]

PS: he just gave some theorems and results that you can easily find on wikipedia I guess.

[@hassaannaeem4374]

48:03 cool, if true!

[@jimnewton4534]

Schuller proved that the inverse Fourier transform of the Fourier transform of f is f. Is it necessary to show that the Fourier transform of the inverse Fourier transform of f is again f ? Because if you don't know they're inverses then you don't know they commute, right?

[@mbivert]

I think this is correct: we've proved in this lecture that F^{-1} is the left inverse of F; it remains to prove that F^{-1} is the right inverse of F. However, this likely has been omitted because the proofs are very symmetrical: with a similar regulator, executing the Fourier transform and its inverse, we're still able to shift limit and integrals with dominated convergence & Fubini.