Spin spherical harmonics

In applied mathematics, Gabor atoms, or Gabor functions, are functions used in the analysis proposed by Dennis Gabor in 1946 in which a family of functions is built from translations and modulations of a generating function.

Overview

In 1946, Dennis Gabor suggested the idea of using a granular system to produce sound. In his work, Gabor discussed the problems with the Fourier analysis, and, according to him, although the mathematics is perfectly correct, it is not possible to apply it physically, mainly in usual sounds, as the sound of a siren, in which the frequency parameter is variable through time. Another problem would be the underlying supposition, as we use sine waves analysis, that the signal under concern has infinite duration – see time–frequency analysis. Gabor proposes to apply the ideas from quantum physics to sound, allowing an analogy between sound and quanta. Under a mathematical background he proposed a method to reduce the Fourier analysis into cells. His research aimed at the information transmission through communication channels. Gabor saw in his atoms a possibility to transmit the same information but using less data, instead of transmitting the signal itself it would be possible to transmit only the coefficients which represent the same signal using his atoms.

Mathematical definition

The Gabor function is defined by

$g_{\ell ,n}(x)=g(x-a\ell )e^{2\pi ibnx},-\mathrm{\infty }<\ell ,n<\mathrm{\infty },$

where a and b are constants and g is a fixed function in L²(R), such that ||g|| = 1. Depending on $a$, $b$, and $g$, a Gabor system may be a basis for L²(R), which is defined by translations and modulations. This is similar to a wavelet system, which may form a basis through dilating and translating a mother wavelet.

See also

• Fourier analysis
• Wavelet
• Gabor wavelet
• Morlet wavelet

References

• Hans G. Feichtinger, Thomas Strohmer: "Gabor Analysis and Algorithms", Birkhäuser, 1998; ISBN 0-8176-3959-4
• Hans G. Feichtinger, Thomas Strohmer: "Advances in Gabor Analysis", Birkhäuser, 2003; ISBN 0-8176-4239-0
• Karlheinz Gröchenig: "Foundations of Time-Frequency Analysis", Birkhäuser, 2001; ISBN 0-8176-4022-3

External links

• NuHAG homepage