AbstractWe show that a real-valued function f in the shift-invariant space generated by a totally positive function of Gaussian type is uniquely determined, up to a sign, by its absolute values $$\{|f(\lambda )|: \lambda \in \Lambda \}$$
{
|
f
(
λ
)
|
:
λ
∈
Λ
}
on any set $$\Lambda \subseteq {\mathbb {R}}$$
Λ
⊆
R
with lower Beurling density $$D^{-}(\Lambda )>2$$
D
-
(
Λ
)
>
2
.We consider a totally positive function of Gaussian type, i.e., a function $$g \in L^2({\mathbb {R}})$$
g
∈
L
2
(
R
)
whose Fourier transform factors as $$\begin{aligned} \hat{g}(\xi )= \int _{{\mathbb {R}}} g(x) e^{-2\pi i x \xi } dx = C_0 e^{- \gamma \xi ^2}\prod _{\nu =1}^m (1+2\pi i\delta _\nu \xi )^{-1}, \quad \xi \in {\mathbb {R}}, \end{aligned}$$
g
^
(
ξ
)
=
∫
R
g
(
x
)
e
-
2
π
i
x
ξ
d
x
=
C
0
e
-
γ
ξ
2
∏
ν
=
1
m
(
1
+
2
π
i
δ
ν
ξ
)
-
1
,
ξ
∈
R
,
with $$\delta _1,\ldots ,\delta _m\in {\mathbb {R}}, C_0, \gamma >0, m \in {\mathbb {N}} \cup \{0\}$$
δ
1
,
…
,
δ
m
∈
R
,
C
0
,
γ
>
0
,
m
∈
N
∪
{
0
}
, and the shift-invariant space$$\begin{aligned} V^\infty (g) = \Big \{ f=\sum _{k \in {\mathbb {Z}}} c_k\, g(\cdot -k): c \in \ell ^\infty ({\mathbb {Z}}) \Big \}, \end{aligned}$$
V
∞
(
g
)
=
{
f
=
∑
k
∈
Z
c
k
g
(
·
-
k
)
:
c
∈
ℓ
∞
(
Z
)
}
,
generated by its integer shifts within $$L^\infty ({\mathbb {R}})$$
L
∞
(
R
)
. As a consequence of (1), each $$f \in V^\infty (g)$$
f
∈
V
∞
(
g
)
is continuous, the defining series converges unconditionally in the weak$$^*$$
∗
topology of $$L^\infty $$
L
∞
, and the coefficients $$c_k$$
c
k
are unique [6, Theorem 3.5].
A concise and universal method for deriving arbitrary paraxial and d’Alembertian cylindrical Gaussian-type light modes