A signal may be abstractly represented as an element of a Hilbert
space
ℋ
ℋ
with inner product
〈x,y〉
x
y
and induced norm
∥x∥=〈x,x〉
x
x
x
, where
x∧y∈ℋ
x
y
ℋ
.
A sequence
x
n
∈ℋ
x
n
ℋ
is said to converge strongly to
xx if the sequence of real numbers
|
x
n
−x|
x
n
x
converges to zero. A sequence
x
n
∈ℋ
x
n
ℋ
is said to converge weakly to
xx if
limit n→
∞
〈
x
n
,y〉=〈x,y〉
n
x
n
y
x
y
for every
y∈ℋ
y
ℋ
. Weak convergence implies strong convergence
in finite dimensional spaces.
A subset
𝒦⊆ℋ
𝒦
ℋ
is said to be convex if
x∧y∈𝒦
x
y
𝒦
implies that 𝒦𝒦 contains
λx+1−λy
λ
x
1
λ
y
for all
λ∈
0
1
λ
0
1
. A set is closed if it contains all its strong limit
points.
A mapping ff between normed linear
spaces is said to be nonexpansive if
|fx−fy|≤|x−y|
f
x
f
y
x
y
for all xx,
yy. In Hilbert spaces,
ff is firmly nonexpansive if and
only if
2f−I
2
f
I
is nonexpansive. A fixed point of
ff is any element in the domain of
ff satisfying
fx=x
f
x
x
.
In Hilbert space, nearest-point projection operators onto convex
sets are an important subclass of firmly nonexpansive operators.
Let 𝒦𝒦 denote any
nonempty, closed, convex subset of a Hilbert space
ℋℋ. Then there exists a
unique
y∈𝒦
y
𝒦
such that
inf
z∈𝒦
{|x−z|}=|x−y|
z
𝒦
x
z
x
y
This correspondence is denoted by
y=
P
𝒦
x
y
P
𝒦
x
, where
P
𝒦
:ℋ↦𝒦
:
P
𝒦
↦
ℋ
𝒦
is said to be the
nearest-point projection
operator, or simply the
projection, of
ℋℋ onto the closed convex set
𝒦𝒦. The projection is
uniquely characterized by the inequality
∀y,y∈𝒦:ℜ〈x−
P
𝒦
x,y−
P
𝒦
x〉≤0
y
y
𝒦
x
P
𝒦
x
y
P
𝒦
x
0
(1)
where
ℜ·
·
denotes the real part. The operator
P
𝒦
P
𝒦
is linear if and only if
𝒦𝒦 is a subspace.
Results are reported here for Hilbert spaces; most can be
directly applied in any uniformly convex Banach space.
Consideration of closed convex sets is motivated by the
existence and uniqueness of the projection. More generally, a
subset 𝒮𝒮 of any Banach space
ℬℬ is said to be a
Chebyshev set if to each point
x∈ℬ
x
ℬ
there corresponds a unique point in
𝒮𝒮 closest to
xx. Every closed convex subset of
a uniformly convex Banach space is a Chebyshev
set.
