Proximum

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. ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle Y}$ ${\displaystyle Y}$ ${\displaystyle x}$

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 ${\displaystyle (X,d)}$ ${\displaystyle Y\subset X}$ ${\displaystyle x\in X}$ ${\displaystyle x}$ ${\displaystyle Y}$ ${\displaystyle \operatorname {dist} }$

${\displaystyle \operatorname {dist} (x,Y):=\inf _{y\in Y}d(x,y)\,.}$

Now exists with: ${\displaystyle p\in Y}$

${\displaystyle d(x,p)=\operatorname {dist} (x,Y)\,}$

this is called the proximum or best approximation to in . ${\displaystyle p}$ ${\displaystyle x}$ ${\displaystyle Y}$

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 ${\displaystyle (X,\lVert \cdot \rVert )}$ ${\displaystyle p}$ ${\displaystyle x\in X}$ ${\displaystyle Y\subset X}$

${\displaystyle \lVert x-p\rVert =\inf _{y\in Y}\lVert x-y\rVert }$

To the existence of a proximum

• Be a metric space . be a compact subset. Then each has a proximum in . ${\displaystyle (X,d)\,}$ ${\displaystyle A\subset X}$ ${\displaystyle x\in X}$ ${\displaystyle A}$

• Be a normalized space. be a finite-dimensional subspace and a closed subset. Then each has a proximum in . ${\displaystyle (X,\lVert \cdot \rVert )}$ ${\displaystyle V\subset X}$ ${\displaystyle Y\subset V}$ ${\displaystyle x\in X}$ ${\displaystyle Y}$

Uniqueness of the proximal in Chebyshev systems

Be a Chebyshev system . Then the proximum for off is uniquely determined. ${\displaystyle f\in C[a,b],U\subset C[a,b]}$ ${\displaystyle f}$ ${\displaystyle U}$

Let be a finite-dimensional subspace of . If the proximum of each is clearly determined, then there is a Chebyshev system . ${\displaystyle U}$ ${\displaystyle C[a,b]}$ ${\displaystyle f\in C[a,b]}$ ${\displaystyle U}$ ${\displaystyle U}$

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 ${\displaystyle f\in C[a,b],U\subset C[a,b]}$ ${\displaystyle n}$ ${\displaystyle u_{0}\in U}$ ${\displaystyle f}$ ${\displaystyle U}$ ${\displaystyle n+1}$ ${\displaystyle x_{i}}$ ${\displaystyle a\leq x_{0}<x_{1}<\cdots <x_{n}\leq b}$

• ${\displaystyle |f(x_{i})-u_{0}(x_{i})|=\max _{x\in [a,\,b]}|f(x)-u_{0}(x)|}$, (Extreme point) ${\displaystyle i=0,\ldots ,n}$
• ${\displaystyle \operatorname {sign} \left(f(x_{i-1})-u_{0}(x_{i-1})\right)=-\operatorname {sign} (f(x_{i})-u_{0}(x_{i}))}$, (alternating) ${\displaystyle i=1,\ldots ,n}$

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 ${\displaystyle X}$ ${\displaystyle Y\subset X}$ ${\displaystyle x\in X}$ ${\displaystyle p\in Y}$

${\displaystyle \lVert x-p\rVert \leq \lVert x-y\rVert \,\,\forall \,y\in Y}$.

If a closed sub-vector space , the proximum is obtained as an orthogonal projection of on . ${\displaystyle Y}$ ${\displaystyle p}$ ${\displaystyle x}$ ${\displaystyle Y}$

See also

• Alternant set
• Minimal solution

literature

• Arnold Schönhage: Approximation Theory. de Gruyter, Berlin 1971, ISBN 3-11-001982-5 ( limited preview in the Google book search).