Editorially prepared introduction for this redesigned journal. This is new explanatory content, separate from Pablo's original notes.
1. Fourier transforms
Fourier analysis decomposes functions into oscillatory components, connecting time or space to frequency.
Definition. For \(f\in L^1(\mathbb R)\), one normalization of the Fourier transform is \[\widehat f(\xi)=\int_{-\infty}^{\infty}f(x)e^{-2\pi i x\xi}\,dx.\]
2. Linearity and translation
The transform is linear. Translating a function by \(a\) multiplies its transform by a phase: \[\widehat{f(\cdot-a)}(\xi)=e^{-2\pi i a\xi}\widehat f(\xi).\]
3. Convolution
For integrable functions, the convolution is \((f*g)(x)=\int f(x-y)g(y)\,dy\). Its Fourier transform satisfies \(\widehat{f*g}=\widehat f\,\widehat g\).
4. Plancherel
The Fourier transform extends to an isometry on \(L^2(\mathbb R)\), in the sense that \(\|\widehat f\|_2=\|f\|_2\). This links frequency analysis to Hilbert-space geometry.