Fourier series and transforms: a primer

A no-nonsense introduction to Fourier series and transforms, followed by a more interesting question: why do frequency coordinates repeatedly become useful representations for learning systems?

The deeper issue is geometric: a representation changes which differences appear locally meaningful. That question continues in the geometry of statistical distinguishability.

Questions this note must answer

• What changes—and what does not—when a signal moves from space or time into frequency?
• Why are locality and sparsity represented differently in the two domains?
• What exactly do convolutional models inherit from the Fourier view of translation?
• Where does the analogy between Fourier bases and learned latent spaces break?

This queued outline will become a visual essay with executable image experiments, derivations, and primary references.