Gabor Function

AKA: Gabor Atom.
See: Gabor System, Wavelet, Gaussian Function, Sinusoid Function, Gabor Filter, Gabor Transform, Dennis Gabor.

References

(Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Gabor_atom Retrieved:2014-7-26.
- 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.

(Wikipedia, 2014) ⇒ http://wikipedia.org/wiki/Gabor_atom#Mathematical_definition Retrieved:2014-7-26.
- The Gabor function is defined by :[math]\displaystyle{ g_{\ell,n}(x) = g(x - a\ell)e^{2\pi ibnx}, \quad -\infty \lt \ell,n \lt \infty, }[/math] where a and b are constants and g is a fixed function in L²(R), such that ||g|| = 1. Depending on [math]\displaystyle{ a }[/math], [math]\displaystyle{ b }[/math], and [math]\displaystyle{ g }[/math], 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.