Theorem of Balian-Low

In Mathematical, the theorem of Balian-Low is a result of Analyze of Fourier of to the physicists Roger Balian and Francis Low, respectively French and American.

Theorem of Balian-Low

That is to say G a summable function of Square on the real line. Let us pose for any couple of entireties m and N :

$g_ {m, N} \ left (X \ right) \ = \ e^ {2 \ pi I m X} \ G \ left (X - N \ right),$

If the whole of the $\ {g_ {m, N}: m, N \ in \ mathbb {Z} \}$ form a orthonormée Base of the Space of Hilbert $L^2 \ left (\ mathbb {R} \ right)$ , then one a:

$\ int_ {- \ infty} ^ \ infty x^2 \ | G \ left (X \ right)|^2 \ dx \ = \ \ infty$

or:

$\ int_ {- \ infty} ^ \ infty \ xi^2 \ |\ hat {G} \ left (\ xi \ right)|^2 \ D \ xi \ = \ \ infty$

with $\ hat {G}$ the Transformed of Fourier of the function G .

Statement are equivalent

Family of Gabor

One calls family of Gabor any whole of the form:

$f_ {m, N} \ left (T \ right) \ = \ e^ {2 \ pi I m F_0 T} \ F \ left (T - N T_0 \ right),$

with F a summable function of Square on the real line, called prototype function ; $F_0$ and $T_0$ two constants real, and (m, N) a couple of entireties.

One calls density of the family the real number:

d \ = \ \ frac {1} {F_0 \, T_0}