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.
- Discrete Hartley transform
- Hadamard transform
- Spherical harmonic transform (What is this again?)
- MDCT (TODO - understand how this works for audio resynthesis and psychoacoustic purposes)

- Legendre transform (into Legendre polynomials)
- Bases of disjoint support; the orthogonal bases for non-negative/non-positive data

For related ideas, see also

- functional regression,
- sparse basis dictionaries (which are not necc orthogonal),
- Hilbert space miscellanea.