# Spectral methods and orhtogonal basis decompositions

### Fourier and friends

Transforming something (let us assume, discrete experimental data rather than abstract mathematical objects) into a new domain, as in the Fourier or Laplace or z-transforms, or a wavelet basis of some kind, 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.

Todo: discuss orthogonal measurements in terms of isometry properties.

## Discrete setting

Here I am especially interested in the discrete orthogonal case, by which I mean 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 $\mathbb{R}_+^{640\times 480 \times 3}$. Apparently this case can be related to the continuous case by considering general transforms on Locally Compact Abelian groups, via Pontryagin_duality.

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. 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.

See the following course notes

or the texts Rudi17, ReSt00.

## Orthogonal basis zoo

• Cousins of the discrete Fourier transform. 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
• Spherical harmonic transform
• MDCT, which is nearly the DCT. TODO: undertand “nearly”.
• Legendre transform (into Legendre polynomials)
• Bases of disjoint support; the orthogonal bases for non-negative/non-positive data
• Practical: Welch periodograms, and Tukey DTFT

For related ideas, see also

## Wavelets

Not (usually) orthogonal. See sparse basis dictionaries.

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