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Spectra [of sequences]
The spectra shown are given by Abs[Fourier[data]] , where the symmetrical second half of this list is dropped in the pictures. … These are related to the autocorrelation function according to
Fourier[list] 2 Fourier[ListConvolve[list, list, {1, 1}]]/Sqrt[Length[list]]
(See also page 1074 .)

Fourier transforms
In a typical Fourier transform, one uses basic forms such as Exp[ π r x/n] with r running from 1 to n . The weights associated with these forms can be found using Fourier , and given these weights the original data can also be reconstructed using InverseFourier . … Fourier[data] can be thought of as multiplication by the n × n matrix Array[Exp[2 π #1 #2/n] &, {n, n}, 0] .

2D spectra
The pictures below give the 2D Fourier transforms of the nested patterns shown on page 583 .

Wavelets
Each basic form in an ordinary Walsh or Fourier transform has nonzero elements spread throughout. … Sharp edges have fewer wiggles than with Fourier transforms.

JPEG compression
In common use since the early 1990s JPEG compression works by first assigning color values to definite bins, then applying a discrete Fourier cosine transform, then applying Huffman encoding to the resulting weights.

Digit reversal
Sequences of the form
Table[FromDigits[ Reverse[IntegerDigits[n, k, m]], k], {n, 0, k m - 1}]
shown below appear in algorithms such as the fast Fourier transform and, with different values of k for different coordinates, in certain quasi-Monte Carlo schemes.

Many sine functions
Adding many sine functions yields a so-called Fourier series (see page 1074 ). … Other so-called Fourier series in which the coefficient of Sin[m x] is a smooth function of m for all integer m yield similarly simple results.

Diffraction patterns
X-ray diffraction patterns give Fourier transforms of the spatial arrangement of atoms in a material.

At a mathematical level, following work by Joseph Fourier around 1810 it became clear by the mid-1800s how any sufficiently smooth function could be decomposed into sums of sine waves with frequencies corresponding to successive integers.

As discovered by Joseph Fourier around 1810, this is satisfied for basis functions such as Sin[2 n π x]/ √ 2 .