# 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 ${\ displaystyle (X, \ lVert \ cdot \ rVert)}$ ${\ displaystyle p}$ ${\ displaystyle x \ in X}$ ${\ displaystyle Y \ subset X}$ ${\ displaystyle \ lVert xp \ rVert = \ inf _ {y \ in Y} \ lVert xy \ rVert}$ ## To the existence of a proximum

• Be a metric space . be a compact subset. Then each has a proximum in .${\ displaystyle (X, d) \,}$ ${\ displaystyle A \ subset X}$ ${\ displaystyle x \ in X}$ ${\ displaystyle A}$ • 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} • ${\ displaystyle | f (x_ {i}) - u_ {0} (x_ {i}) | = \ max _ {x \ in [a, \, b]} | f (x) -u_ {0} ( x) |}$ , (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${\ displaystyle X}$ ${\ displaystyle Y \ subset X}$ ${\ displaystyle x \ in X}$ ${\ displaystyle p \ in Y}$ ${\ displaystyle \ lVert xp \ rVert \ leq \ lVert xy \ rVert \, \, \ forall \, y \ in Y}$ .

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