What is the best approach when things seem hopeless?

In my experience, skipping forward is not the path to success - It may be possible to get a good grade in the course, but you definitely will not have a solid understanding of the material.

Personally, I would spend several hours working backwards from the current point in class to uncover all the dependencies. While this takes some work, it's not as difficult as slogging forward & attempting a rigorous understanding of the proofs of every theorem.

Essentially, pinpoint the material you don't understand - finding out what you don't know is easier than learning everything you don't know (especially because you'll miss things you don't know you don't know). Then, go to whatever help outside class is available - office hours, TAs, tutors, etc with your list of material and ask. Ask ask ask ask. Don't be afraid to ask the stupid questions. If you can get your professor to spare you an hour, prepare a list of the most critical things to ask him in person & get reexplained. If you're at a large university this may be difficult to do, but do you best to get some face-time with the professor. If nothing else, s/he'll know you're trying.

I cannot emphasize enough the importance of reaching out for help - be it here, or to the resources available at your university. It may be possible to catch up on your own, but it will be much easier and far less frustrating to do it with help.

I was in a similar situation when I took Fourier Analysis and PDEs my first year. I think a good way to check whether you understand the material or not is to read through the important theorems and then be able to prove them yourself. That is the best (and in my opinion...only) way to to learn. As Halmos said, for the passive reader, the proof enters one ear as easily as it leaves the other.

The reason why I think this method of learning is helpful is because it teaches you important techniques (especially so in Fourier Analysis) and approaches when trying to prove essential statements. If you are unable to prove a theorem by yourself, you will look back in the book, and be reminded of the essential technique you missed out on in your initial approach. Then the technique will stick with you better than if you had passively read the proof.

The point I am trying to make is that you really need to do mathematics to learn it. There is no other way out of it, but I sympathize with your situation. I was in a similar situation, and I think one of the factors that tends to make you feel like you will never be able to catch up is anxiety (which is a result of falling so far behind). However, if you keep pushing yourself in actively learning the material, as I have mentioned to you above, the anxiety factor will fade away. You will feel more optimistic as you will realize you are actually learning the material and not just bullshitting the assignments by piecing together different theorems you don't understand (all too common in mathematics courses these days unfortunately...)

And most importantly, you will remember most of the specifics of fourier analysis even two or three years later. If you superficially learn the material, you will remember it only up to the final exam. So, go ahead, and start reviewing the material and don't agonize too much about behind behind as long as you feel like you're fully learning the material. Of course, use good judgment and be practical too. If you have an assignment due the next day, make sure to do the necessary reading (even if it is superficial) to finish the assignment.

Best wishes!