Decomposition of the one

Four functions that form a decomposition of one

A decomposition of the one (also: division of the one or decomposition of the unit ) is a construction from mathematics . Under certain circumstances, mathematics has to distinguish between a local and a global perspective. For example:

• In order to define the surface integral in analysis , or to integrate it in general using manifolds , coordinates must be chosen, which is only possible locally. The integrand must therefore be broken down in such a way that it remains locally integrable, but becomes zero outside the scope of the coordinate system.

• In differential geometry , vector fields are constructed on surfaces or manifolds . Often there are only locally valid constructions, but these should be combined to form a global one. For example, the normal field of a submanifold should be extended to the entire manifold, ...

• In the solution theory of partial differential equations , the solution of a partial differential equation in any area can often be put together with the help of the decomposition of the one by solving the equation on the whole space and the (disturbed) half space (so-called localization).

definition

A (continuous) decomposition of one over a topological space is a family of continuous functions of in the space of real numbers , so that for every point : ${\displaystyle E}$ ${\displaystyle (f_{i})_{i\in I}}$ ${\displaystyle E}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle x\in E}$

• The functions map into the interval , that is, it applies . ${\displaystyle f_{i}}$ ${\displaystyle [0,1]}$ ${\displaystyle 0\leq f_{i}(x)\leq 1}$
• The (possibly infinite ) sum of all function values ​​at point x is 1, that is, it holds . ${\displaystyle \textstyle \sum _{i\in I}f_{i}(x)=1}$

One speaks of a locally finite decomposition of one if the following condition is also met:

• Every point has a neighborhood in which only a finite number of functions have a function value different from 0. ${\displaystyle x}$ ${\displaystyle f_{i}}$

If , in addition , there is an open cover of and if it is also true , then a decomposition of the one with respect to the cover is called . denote the carrier of . A decomposition of the one with respect to a locally finite cover is always locally finite. ${\displaystyle {\mathcal {X}}=\{X_{i}\}_{i\in I}}$ ${\displaystyle E}$ ${\displaystyle \operatorname {supp} (f_{i})\subset X_{i}}$ ${\displaystyle (f_{i})_{i\in I}}$ ${\displaystyle {\mathcal {X}}}$ ${\displaystyle \operatorname {supp} (f)}$ ${\displaystyle f}$

In the topology

In every normal space there is a decomposition of the one with respect to every locally finite open cover. The consequence of this is that for every locally finite open coverage of a closed subset of normal space there is a family of continuous functions that are restricted to a locally finite decomposition of one, and the sum of which is outside the open coverage, i.e. outside zero. For this one simply add the open coverage with the complement of the closed set to an open coverage of the whole space, choose a decomposition of the one with respect to this coverage and add all these functions with the exception of the function whose carrier lies in the complement of . Is as compact provided, then the result is transferred to any subspaces normal spaces (these are just all completely regular spaces ), because cortical and environments remain as an element of a larger space construed Kompakta or environments as embeddings are continuous and open. In particular, for every compact subset of a completely regular space with an open environment, there is a continuous function in the unit interval, which is one on the compact unit and zero outside the environment. If the space is also locally compact , then such a family of functions even exists if the requirement is that their carriers are compact. To do this, construct a refinement of from relatively compact sets that still cover, and choose a finite partial cover. ${\displaystyle {\mathcal {X}}}$ ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle \bigcup {\mathcal {X}}}$ ${\displaystyle F}$ ${\displaystyle \bigcup {\mathcal {X}}}$ ${\displaystyle F}$ ${\displaystyle {\mathcal {X}}}$ ${\displaystyle F}$

Conversely, the existence of a decomposition of the one with respect to every cover from two open sets already implies Urysohn's lemma and thus the normality of space. In a paracompact Hausdorff space there are decompositions of the one with respect to any open cover, this results from the fact that such a paracompact space by definition has a locally finite refinement and every paracompact Hausdorff space is normal.

In analysis

In analysis it is usually still required that the functions are differentiable and have a compact support . A function g can then be converted into functions

${\displaystyle g_{i}=g\cdot f_{i}}$

be disassembled, which all have a compact carrier. Then

${\displaystyle \sum _{i\in I}g_{i}=\sum _{i\in I}g\cdot f_{i}=g\cdot \sum _{i\in I}f_{i}=g\cdot 1=g.}$

If, on the other hand, a family is given, the h _i only being defined and differentiable on the respective carriers of the f _i , then the sum is ${\displaystyle (h_{i})_{i\in I}}$

${\displaystyle \sum h_{i}\cdot f_{i}}$

a convex linear combination , defined and differentiable everywhere.

Every paracompact - manifold ( ) also has a -decomposition of one. ${\displaystyle C^{k}}$ ${\displaystyle 1\leq k\leq \infty }$ ${\displaystyle C^{k}}$

Analytical decompositions of the one are not possible, however, since an analytical function that is constant 0 in a non-empty, open set (such as the complement of its carrier) is already constant 0 everywhere.

example

The function

${\displaystyle r(x)={\begin{cases}\exp \left(-x^{-2}\right),&x>0,\\0,&x\leq 0,\end{cases}}}$

can be differentiated any number of times. The function s with

${\displaystyle s(x)=r(x+1)\cdot r(1-x)}$

is then also differentiable any number of times, strictly positive in the interval (−1; 1) and zero outside of it. The functions with ${\displaystyle f_{i},i\in \mathbb {Z} ,}$

${\displaystyle f_{i}(x)={\frac {s(x-i)}{\sum _{k\in \mathbb {Z} }s(x-k)}}}$

form an arbitrarily often differentiable decomposition of the one on the real axis, which is subordinate to the open coverage ; so at every point x it applies : ${\displaystyle (i-1;i+1),i\in \mathbb {Z} ,}$

${\displaystyle \sum _{i\in \mathbb {Z} }f_{i}(x)=1.}$

Note that in the definition of at every point x there are always at least one summand and at most two summands in the denominator not equal to zero (only the integers k adjacent to x can deliver a positive summand at all). ${\displaystyle f_{i}}$

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 .

Individual evidence

1. ^ John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer-Verlag, New York NY et al. 2003, ISBN 0-387-95448-1 , p. 54.
2. ^ Nicolas Bourbaki : Topologie Générale (=  Éléments de mathématique ). Springer , Berlin 2007, ISBN 3-540-33936-1 , chap. 9 , p. 46 ff .
3. ^ Gerald B. Folland : Real Analysis . Modern Techniques and Their Applications. 2nd Edition. John Wiley & Sons, New York 1999, ISBN 0-471-31716-0 , pp. 134 .
4. ↑ Bourbaki, p. 49