Reddit mentions of Who Is Fourier?: A Mathematical Adventure

I did a little research and it looks like if you're interested in the beauty then a good first read would be Who Is Fourier? Check out the reviews.

Now, if you actually want to learn fourier series, it would help a lot to be comfortable with some topics from linear algebra: vector spaces, spans, linear independence, bases, and orthogonality. You should ponder these in the context of a familiar vector space of functions (polynomials would be a good one since it is infinite-dimensional).

I always recommend Dover books if I can since they're so cheap and I happen to own a good one called Fourier Series and Orthogonal Functions by Davis. The first four chapters cover the information you're interested in. The first chapter reviews the above concepts from linear algebra and the second chapter starts exploring the concepts of orthogonality of functions and series of fuctions. The book starts getting juicy in the last section of the second chapter which applies the relatively general treatment of the five previous sections in the chapter to a specific problem of approximating a given function by a sum of sines and cosines. This motivates the material in the third chapter, the heart of the book, on Fourier series. The fourth chapter hints at the fact that Fourier series are merely a special case of a more fundamental idea and introduces series of Legendre polynomials and Bessel functions.

There are exercises at the end of every section.

This book, Who Is Fourier?: A Mathematical Adventure is the best starting-off resource I can think of. And then there's the stanford open course The Fourier Transform and its Applications. Good luck :)

I like to think of it as using different "lenses" to look at the data. Sometimes you want to use a microscope. Other times you want to use a telescope. Not to be taken literally of course, but you need the right tools.

Btw if you want to indulge that inner quant on this topic, check this book out. What I found amazing is that this is actually a kid's book in Japan.

Who Is Fourier?: A Mathematical Adventure is a great book, I suggest you look into it.

This book does a good job:

http://www.amazon.com/Who-Fourier-Mathematical-Transnational-College/dp/0964350408

Though what was perhaps even better for me was realizing that a two dimensional representation for a sine wave is actually an unnatural representation.

The most natural representation is actually more like a 3-dimensional spiral staircase or stretched out slinky,

http://www.theoryofmind.org/misc-info/Physics/spring.jpg

Forget about the cos(x) + isin(x) part for now. Just figure out why the e^(ix) part looks like the slinky.

Then from there, the rest is pretty easy to see.

For example,

cos(x) = e^(ix) - isin(x)

is pretty intuitive. cos(x) is a 2d function. We get that by starting with the 3D spiral and then subtracting off the imaginary part to get a nice 2d graph.

Obviously, I have leaving out a lot of information. But the key, IMO, is to really understand why the graph of e^(ix) looks like it does, and then the rest will fall in place.