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, 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 the domain or representation of points in it explicitly, you are
doing the kernel trick.

The *discrete* qualifier is ambiguous; Here what I mean is that we are finding orthogonal bases for signal vectors of countable, usually 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 \(\mathbb{R}_+^{640\times 480 \times 3}\).

- Discrete Fourier transform and cousins
- Discrete Hartley transform
- Hadamard transform
- Spherical harmonic transform
- MDCT (TODO - understand how this works for audio resynthesis and psychoacoustic purposes)
- Sparse FFT.
- nonuniform FFT.

- Legendre transform (into Legendre polynomials)
- z-transform
- arbitrary orthogonal bases

For related ideas, see also

- functional regression,
- sparse basis dictionaries (which are not necc orthogonal, though it doesn’t hurt),
- Hilbert space miscellanea.