Fourier Analysis: Pointwise vs. L^2 Convergence

[r4nd0m]

I'm just taking Calculus 4 this semester, where part of it is also Fourier analysis.

When I was browsing a little bit about the subject I found out that there are several different approaches and so I'm a bit confused now.

So this is how I understand it, correct me if I'm wrong:

There is the approach of pointwise convergence (which we are taking) and the approach of convergence in L^2 (is this called the weak convergence or is it something completely different?). Now the Fourier series can converge in the L^2 space but not pointwise and vice-versa.

Did I get it correct?

Another question is: What is a good book on Fourier analysis from the pointwise point-of-view on the level like Rudin's Principles of math. analysis?

Thanks

 

[HallsofIvy]

No, that is not "weak convergence", that is "convergence in L²" or "convergence in the mean". You aren't going to find a book on Fourier Analysis from the "pointwise-point-of-view" because "pointwise convergence" is useless with Fourier series. For Fourier series to make any sense, you MUST use Lebesque integration and convergence in the norm.

 

[r4nd0m]

Well, we are not using anything from the Lebesgue integration theory, we didn't even mention L^2 spaces.
Does that mean, that what we're learning ther is useless ?

 

[HallsofIvy]

As far as the theory is concerned, yes! You might well be learning to calculate Fourier series for simple functions and use them in basic applications- so I guess that is useful. But you can't be learning much of the theory. Most functions that have Fourier series aren't even Riemann integrable.