Super-resolution by means of Beurling minimal extrapolation
Abstract
Let
M(
T
d
)
be the space of complex bounded Radon measures defined on the
d
-dimensional torus group
(R/Z
)
d
=
T
d
, equipped with the total variation norm
∥⋅∥
; and let
μ
^
denote the Fourier transform of
μ∈M(
T
d
)
. We address the super-resolution problem: For given spectral (Fourier transform) data defined on a finite set
Λ⊂
Z
d
, determine if there is a unique
μ∈M(
T
d
)
of minimal norm for which
μ
^
equals this data on
Λ
. Without additional assumptions on
μ
and
Λ
, our main theorem shows that the solutions to the super-resolution problem, which we call minimal extrapolations, depend crucially on the set
Γ⊂Λ
, defined in terms of
μ
and
Λ
. For example, when
Γ=0
, the minimal extrapolations are singular measures supported in the zero set of an analytic function, and when
Γ≥2
, the minimal extrapolations are singular measures supported in the intersection of
(
Γ
2
)
hyperplanes. By theory and example, we show that the case
Γ=1
is different from other cases and is deeply connected with the existence of positive minimal extrapolations. This theorem has implications to the possibility and impossibility of uniquely recovering
μ
from
Λ
. We illustrate how to apply our theory to both directions, by computing pertinent analytical examples. These examples are of interest in both super-resolution and deterministic compressed sensing. Our concept of an admissibility range fundamentally connects Beurling's theory of minimal extrapolation with Candes and Fernandez-Granda's work on super-resolution. This connection is exploited to address situations where current algorithms fail to compute a numerical solution to the super-resolution problem.