# Fourier Analysis: Pointwise vs. L^2 Convergence

• r4nd0m
In summary, the conversation discusses the different approaches to Fourier analysis, specifically pointwise convergence and convergence in L^2. The speaker is looking for a book on Fourier analysis from the pointwise perspective but is informed that this approach is not useful without using Lebesgue integration. The conversation ends with the realization that what they are learning may not be useful in terms of theory, but can still be applied in basic applications.

#### 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

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.

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 ?

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.