Uncertainty principles for generating systems
{
e
n
}
n
=
1
∞
⊂
L
2
(
R
)
\{e_n\}_{n=1}^{\infty } \subset L^2(\mathbb {R})
are proven and quantify the interplay between
ℓ
r
(
N
)
\ell ^r(\mathbb {N})
coefficient stability properties and time-frequency localization with respect to
|
t
|
p
|t|^p
power weight dispersions. As a sample result, it is proven that if the unit-norm system
{
e
n
}
n
=
1
∞
\{e_n\}_{n=1}^{\infty }
is a Schauder basis or frame for
L
2
(
R
)
L^2(\mathbb {R})
, then the two dispersion sequences
Δ
(
e
n
)
\Delta (e_n)
,
Δ
(
e
n
^
)
\Delta (\widehat {e_n})
and the one mean sequence
μ
(
e
n
)
\mu (e_n)
cannot all be bounded. On the other hand, it is constructively proven that there exists a unit-norm exact system
{
f
n
}
n
=
1
∞
\{f_n\}_{n=1}^{\infty }
in
L
2
(
R
)
L^2(\mathbb {R})
for which all four of the sequences
Δ
(
f
n
)
\Delta (f_n)
,
Δ
(
f
n
^
)
\Delta (\widehat {f_n})
,
μ
(
f
n
)
\mu (f_n)
,
μ
(
f
n
^
)
\mu (\widehat {f_n})
are bounded.