Abstract
AbstractIn this paper we solve an open problem concerning the characterization of those measurable sets $$\Omega \subset {\mathbb {R}}^{2d}$$
Ω
⊂
R
2
d
that, among all sets having a prescribed Lebesgue measure, can trap the largest possible energy fraction in time-frequency space, where the energy density of a generic function $$f\in L^2({\mathbb {R}}^d)$$
f
∈
L
2
(
R
d
)
is defined in terms of its Short-time Fourier transform (STFT) $${\mathcal {V}}f(x,\omega )$$
V
f
(
x
,
ω
)
, with Gaussian window. More precisely, given a measurable set $$\Omega \subset {\mathbb {R}}^{2d}$$
Ω
⊂
R
2
d
having measure $$s> 0$$
s
>
0
, we prove that the quantity $$\begin{aligned} \Phi _\Omega =\max \Big \{\int _\Omega |{\mathcal {V}}f(x,\omega )|^2\,dxd\omega : f\in L^2({\mathbb {R}}^d),\ \Vert f\Vert _{L^2}=1\Big \}, \end{aligned}$$
Φ
Ω
=
max
{
∫
Ω
|
V
f
(
x
,
ω
)
|
2
d
x
d
ω
:
f
∈
L
2
(
R
d
)
,
‖
f
‖
L
2
=
1
}
,
is largest possible if and only if $$\Omega $$
Ω
is equivalent, up to a negligible set, to a ball of measure s, and in this case we characterize all functions f that achieve equality. This result leads to a sharp uncertainty principle for the “essential support” of the STFT (when $$d=1$$
d
=
1
, this can be summarized by the optimal bound $$\Phi _\Omega \le 1-e^{-|\Omega |}$$
Φ
Ω
≤
1
-
e
-
|
Ω
|
, with equality if and only if $$\Omega $$
Ω
is a ball). Our approach, using techniques from measure theory after suitably rephrasing the problem in the Fock space, also leads to a local version of Lieb’s uncertainty inequality for the STFT in $$L^p$$
L
p
when $$p\in [2,\infty )$$
p
∈
[
2
,
∞
)
, as well as to $$L^p$$
L
p
-concentration estimates when $$p\in [1,\infty )$$
p
∈
[
1
,
∞
)
, thus proving a related conjecture. In all cases we identify the corresponding extremals.
Publisher
Springer Science and Business Media LLC
Cited by
10 articles.
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