The proximum is a term used mainly in numerical mathematics from the theory of metric spaces . The Proximum to a point within a not containing amount is from that point , to be the minimum distance from.
x
{\ displaystyle x}
x
{\ displaystyle x}
Y
{\ displaystyle Y}
Y
{\ displaystyle Y}
x
{\ displaystyle x}
definition
Be a metric space , a subset, and arbitrary. The distance between the element and the subset is defined by
means of the distance function
(
X
,
d
)
{\ displaystyle (X, d)}
Y
⊂
X
{\ displaystyle Y \ subset X}
x
∈
X
{\ displaystyle x \ in X}
x
{\ displaystyle x}
Y
{\ displaystyle Y}
dist
{\ displaystyle \ operatorname {dist}}
dist
(
x
,
Y
)
: =
inf
y
∈
Y
d
(
x
,
y
)
.
{\ displaystyle \ operatorname {dist} (x, Y): = \ inf _ {y \ in Y} d (x, y) \ ,.}
Now exists with:
p
∈
Y
{\ displaystyle p \ in Y}
d
(
x
,
p
)
=
dist
(
x
,
Y
)
{\ displaystyle d (x, p) = \ operatorname {dist} (x, Y) \,}
this is called the proximum or best approximation to in .
p
{\ displaystyle p}
x
{\ displaystyle x}
Y
{\ displaystyle Y}
If there is a proximum, it need not be unique.
Usually one has to do with a normalized space in approximation theory . A proximum to in is then - if it exists - characterized by the equation
(
X
,
‖
⋅
‖
)
{\ displaystyle (X, \ lVert \ cdot \ rVert)}
p
{\ displaystyle p}
x
∈
X
{\ displaystyle x \ in X}
Y
⊂
X
{\ displaystyle Y \ subset X}
‖
x
-
p
‖
=
inf
y
∈
Y
‖
x
-
y
‖
{\ displaystyle \ lVert xp \ rVert = \ inf _ {y \ in Y} \ lVert xy \ rVert}
To the existence of a proximum
Be a metric space . be a compact subset. Then each has a proximum in .
(
X
,
d
)
{\ displaystyle (X, d) \,}
A.
⊂
X
{\ displaystyle A \ subset X}
x
∈
X
{\ displaystyle x \ in X}
A.
{\ displaystyle A}
Be a normalized space. be a finite-dimensional subspace and a closed subset. Then each has a proximum in .
(
X
,
‖
⋅
‖
)
{\ displaystyle (X, \ lVert \ cdot \ rVert)}
V
⊂
X
{\ displaystyle V \ subset X}
Y
⊂
V
{\ displaystyle Y \ subset V}
x
∈
X
{\ displaystyle x \ in X}
Y
{\ displaystyle Y}
Uniqueness of the proximal in Chebyshev systems
Be a Chebyshev system . Then the proximum for off is uniquely determined.
f
∈
C.
[
a
,
b
]
,
U
⊂
C.
[
a
,
b
]
{\ displaystyle f \ in C [a, b], U \ subset C [a, b]}
f
{\ displaystyle f}
U
{\ displaystyle U}
Let be a finite-dimensional subspace of . If the proximum of each is clearly determined, then there is a Chebyshev system .
U
{\ displaystyle U}
C.
[
a
,
b
]
{\ displaystyle C [a, b]}
f
∈
C.
[
a
,
b
]
{\ displaystyle f \ in C [a, b]}
U
{\ displaystyle U}
U
{\ displaystyle U}
Alternant criterion in Chebyshev systems
Be a -dimensional Chebyshev system . is a proximum for out if and only if there are places with such that
f
∈
C.
[
a
,
b
]
,
U
⊂
C.
[
a
,
b
]
{\ displaystyle f \ in C [a, b], U \ subset C [a, b]}
n
{\ displaystyle n}
u
0
∈
U
{\ displaystyle u_ {0} \ in U}
f
{\ displaystyle f}
U
{\ displaystyle U}
n
+
1
{\ displaystyle n + 1}
x
i
{\ displaystyle x_ {i}}
a
≤
x
0
<
x
1
<
⋯
<
x
n
≤
b
{\ displaystyle a \ leq x_ {0} <x_ {1} <\ cdots <x_ {n} \ leq b}
|
f
(
x
i
)
-
u
0
(
x
i
)
|
=
Max
x
∈
[
a
,
b
]
|
f
(
x
)
-
u
0
(
x
)
|
{\ displaystyle | f (x_ {i}) - u_ {0} (x_ {i}) | = \ max _ {x \ in [a, \, b]} | f (x) -u_ {0} ( x) |}
, (Extreme point)
i
=
0
,
...
,
n
{\ displaystyle i = 0, \ ldots, n}
sign
(
f
(
x
i
-
1
)
-
u
0
(
x
i
-
1
)
)
=
-
sign
(
f
(
x
i
)
-
u
0
(
x
i
)
)
{\ displaystyle \ operatorname {sign} \ left (f (x_ {i-1}) - u_ {0} (x_ {i-1}) \ right) = - \ operatorname {sign} (f (x_ {i} ) -u_ {0} (x_ {i}))}
, (alternating)
i
=
1
,
...
,
n
{\ displaystyle i = 1, \ ldots, n}
This follows from the Kolmogorow criterion from approximation theory . The Remez algorithm for the numerical determination of the proximum in Chebyshev systems is based on this criterion .
Proximum in the Hilbert space
If a Hilbert space and a closed convex non-empty subset, then the proximum is unique, that is, there is exactly one with
for each
X
{\ displaystyle X}
Y
⊂
X
{\ displaystyle Y \ subset X}
x
∈
X
{\ displaystyle x \ in X}
p
∈
Y
{\ displaystyle p \ in Y}
‖
x
-
p
‖
≤
‖
x
-
y
‖
∀
y
∈
Y
{\ displaystyle \ lVert xp \ rVert \ leq \ lVert xy \ rVert \, \, \ forall \, y \ in Y}
.
If a closed sub-vector space , the proximum is obtained as an orthogonal projection of on .
Y
{\ displaystyle Y}
p
{\ displaystyle p}
x
{\ displaystyle x}
Y
{\ displaystyle Y}
See also
literature
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">