# Orthogonal basis decompositions

### Fourier and friends

Transforming something (let us assume, discrete experimental data rather than abstract mathematical objects) into a new basis, as in the Fourier or Laplace or z-transforms, usually with a goal of handling some inconveniently curvy data as linear data plus a non-local curvy transformation. This is big in, e.g. signal processing and time series analysis.

Other names: Spectral methods, Hilbert space methods, basis expansions, orthogonal projection methods.

An orthogonal basis requires you to have a domain and inner product specified. If you choose a new inner product, you need a different base to be orthogonal in general. If you use the machinery of an implicit Hilbert space for a given inner product without troubling to define an explicit form for the “feature space”, you are doing the kernel trick.

Todo: discuss orthogonal measurements in terms of isometry properties.

The discrete qualifier is ambiguous; Here what I mean is that we are finding orthogonal bases for signal vectors of countable, often finite, length, e.g. if we are looking at spaces of 1 second of audio at 44kHz, then we consider spaces $\mathbb{R}^{44100}$. If we are looking at RGB webcam images we might consider $\mathbb{R}_+^{640\times 480 \times 3}$.

This all gets weird and irrelevant for my current purposes if we start considering transforms of arbitrary objects over uncountable groups.

## Orthogonal basis zoo

• Discrete Fourier transform and cousins! Mostly a list of things to look up later, since I can’t remember what any of them are right now, and I ran across them when reading about something else. Idk.
• Legendre transform (into Legendre polynomials)
• Bases of disjoint support; the orthogonal bases for non-negative/non-positive data

For related ideas, see also