Before I started my PDE course I heard about Fourier Transform and how useful it is (waves, heat problem, etc) but I recently finished it and all we did is solve some PDE problems where we had to apply a Fourier Transform and then solve some related ODEs.
So my question is: ¿Where is all the magic I was promised? I was told Fourier Transform can convert a complex wave into the basic ones. ¿How is this done?
I really need some insight in this topic because I'm really interested and I feel my course didn't cover any applications of the Fourier Transform.
Thanks in advance.
The Fourier transform turns the differentiation operator into multiplication, which can be viewed as a diagonalization of the differentiation operator. This eventually led to defining diagonalization of matrices through eigenvectors, which is where all that magic came from. Oddly enough, the infinite-dimensional magic came first and gave rise to the finite-dimensional magic.
Do you believe that diagonalization of Hermitian matrix is interesting, useful or even powerful? If so, then you should really appreciate the infinite-dimensional diagonalization of the differentiation operator through integral "sums" instead of finite sums. Of course you might not believe either is very interesting, in which case, a lot of modern Math will be boring to you.
The diagonalization of a Hermitian matrix with orthnormal basis $\{e_n\}$ of eigenfunctions with eigenvalues $\lambda_n$: $$ x = \sum_{n}\langle x,e_n\rangle e_n\\ \|x\|^2=\sum_{n}|\langle x,e_n\rangle|^2 \\ Ax = \sum_{n}\lambda_n\langle x,e_n\rangle e_n $$ The diagonalization of $D=\frac{1}{i}\frac{d}{dx}$ on $L^2(\mathbb{R})$ by exponentials $e_s(x) = \frac{1}{\sqrt{2\pi}}e^{isx}$ using a complex inner product: $$ f = \int_{-\infty}^{\infty} \langle f,e_{s}\rangle e_s dx \\ \|f\|^2 = \int_{-\infty}^{\infty}|\langle f,e_{s}\rangle|^2ds \\ Df = \int_{-\infty}^{\infty} s\langle f,e_s\rangle e_s $$ Diagonalization is a useful concept that came out of this infinite-dimensional context and filtered down to the finite-dimensional.