Balian–Low Theorem

In mathematics, the Balian–Low theorem in Fourier analysis is named for Roger Balian and Francis E. Low.
The theorem states that there is no well-localized window function (or Gabor atom) ''g'' either in time or frequency for an exact Gabor frame (Riesz Basis).

Statement

Suppose ''g'' is a square-integrable function on the real line, and consider the so-called Gabor system

:$g_(x) = e^ g(x - n a),$

for integers ''m'' and ''n'', and ''a,b>0'' satisfying ''ab=1''. The Balian–Low theorem states that if

:$\$

is an orthonormal basis for the Hilbert space

:$L^2(\mathbb),$

then either
:$\int_^\infty x^2 , g(x), ^2\; dx = \infty \quad \textrm \quad \int_^\infty \xi^2, \hat(\xi), ^2\; d\xi = \infty.$

Generalizations

The Balian–Low theorem has been extended to exact Gabor frames.

See also

* Gabor filter (in image processing)

References

*

{{DEFAULTSORT:Balian-Low theorem
Theorems in Fourier analysis