Partition of unity

A decomposition of the fuel (including: subdivision of the unit or the division of one) is a construction of mathematics. Under certain conditions must be distinguished in mathematics between a local and a global perspective. For example:

• To define the surface integral in calculus, or in general to integrate over manifolds, coordinates must be chosen, which is possible only locally. Thus, the integrand must be dismantled, that it remains locally integrable, outside the scope of the coordinate system but becomes zero.

• In the differential geometry of vector fields are constructed on surfaces or manifolds. There are often only locally valid structures that are to be joined together but to a global. For example, the normal field of a submanifold to be continued to the whole manifold, ...

• In the solution theory of partial differential equations, the solution of a partial differential equation in any field can often be composed using the partition of unity by solutions of the equation on the whole space and the ( disturbed ) half-space (so-called localization).

Definition

A ( continuous) partition of unity on a topological space is a family of continuous functions, such that in the space of real numbers for each point:

• The function maps in the interval, that is, it is.
• The ( possibly infinite ) the sum of all function values ​​in the point x is 1, that is, it is.

One speaks of a locally finite partition of unity if in addition the following condition is satisfied:

• Each point has a neighborhood in which only a finite number of functions have a function different from 0 value.

Is also an open cover of and is in addition, it means a partition of unity with respect to the coverage. call it the support of. A partition of unity with respect to a locally finite covering is always locally finite.

In topology

In any normal room exists for each locally finite open covering a partition of unity with respect to this. This has the consequence that for every locally finite open cover of a closed subset of a normal space, a family of continuous functions exist, which is restricted to a locally finite partition of unity, and their sum is outside the open covering, so outside is zero. For this, you simply complete the open cover with the complement of the closed set to an open covering of the entire space, choose a partition of unity with respect to this coverage and add all these features except the function whose support lies in the complement of. Is even compact, so the result transfers to arbitrary subspaces of normal spaces ( these are just all completely regular spaces ), for cortical and environments remain as an element of a larger room construed compacta or environments since embeddings are continuous and open. In particular, there exists a continuous function from the unit interval, which is on the compact set one outside and around zero for each compact subset of a completely regular space with an open environment. If, in addition, the space locally compact, then such a family of functions exists even if one makes the claim that their institutions are compact. For this purpose we construct a refinement of relatively compact sets that are still covered, and choose a finite subcover.

The existence of a partition of unity with respect to each overlap of two open sets implies conversely, already the lemma of Urysohn and thus the normality of the room. In a paracompact Hausdorff space decompositions exist of one relation to any open covering, this results from the fact that such a by definition of a paracompact space has a locally finite refinement there and also any paracompact Hausdorff space is normal.

In the Analysis

In calculus is usually still requires that the functions are differentiable and have compact support. This can then be a function g in functions

Be disassembled, all of which have compact support. Then

If, however, a family set, the hi are defined only on the respective supports of fi and differentiable, then the sum

A convex linear combination, everywhere defined and differentiable.

Each paracompact manifold () also has a - partition of unity.

However Analytical partitions of unity are not possible, since an analytic function in a non- empty open set (such as the complement of the wearer ) is constant 0, already everywhere is constantly 0.

Example

The function

Is infinitely differentiable. The function s with

Is then also infinitely differentiable, strictly positive in the interval ( -1, 1) and zero outside. The functions with

Now form an infinitely differentiable partition of unity on the real axis, which is subordinate to the open covering; so it applies at each point x:

Note that in the definition of at each point x is always at least one and at most two addend addend in the denominator is not equal to zero (only the can adjacent integers x k at all a addend positive supply ).

Swell

• Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu: Manifolds, Tensor Analysis and Applications (= global analysis, Pure and Applied 2). Addison -Wesley, Reading MA 1983 ISBN 0-201-10168-8.