Abstract
In this paper, we discuss how to partially determine the Fourier transform
F
(
z
)
=
∫
-
1
1
f
(
t
)
e
i
z
t
𝑑
t
,
z
∈
ℂ
,
F(z)=\int_{-1}^{1}f(t)e^{izt}\,dt,\quad z\in\mathbb{C},
given the data
|
F
(
z
)
|
{\lvert F(z)\rvert}
or
arg
F
(
z
)
{\arg F(z)}
for
z
∈
ℝ
{z\in\mathbb{R}}
. Initially, we assume
[
-
1
,
1
]
{[-1,1]}
to be the convex hull of the support of the signal f. We start with reviewing the computation of the indicator function and indicator diagram of a finite-typed complex-valued entire function, and then connect to the spectral invariant of
F
(
z
)
{F(z)}
. Then we focus to derive the unimodular part of the entire function up to certain non-uniqueness. We elaborate on the translation of the signal including the non-uniqueness associates of the Fourier transform. We show that the phase retrieval and magnitude retrieval are conjugate problems in the scattering theory of waves.
Smoothness and Function Spaces Generated by Homogeneous Multipliers
