Balian–Low theorem - Alchetron, The Free Social Encyclopedia

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).

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

g m , n ( x ) = e 2 π i m b x g ( x − n a ) ,

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

{ g m , n : m , n ∈ Z }

is an orthonormal basis for the Hilbert space

L 2 ( R ) ,

then either

∫ − ∞ ∞ x 2 | g ( x ) | 2 d x = ∞ or ∫ − ∞ ∞ ξ 2 | g ^ ( ξ ) | 2 d ξ = ∞ .

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

References

Balian–Low theorem Wikipedia

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