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Wikipedia / Partition of unity

In mathematics, a partition of unity on a topological space is a set of continuous functionss from to the unit interval:

  • there is a neighbourhood of where all but a finite number of the functions of are zero, and

  • the sum of all the function values at

Partitions of unity are useful because they often allow one to extend local constructions to the whole space. They are also important in the interpolation of data, in signal processing, and the theory of spline functions.

Existence

The existence of partitions of unity assumes two distinct forms:

  1. Given any open cover indexed over the same set such that supp Such a partition is said to be subordinate to the open cover

  2. of a locally compact indexed over a possibly distinct index set such that each has compact support and for each there is an .

Thus one chooses either to have the supports indexed by the open cover, or compact supports. If the space is compact, then there exist partitions satisfying both requirements.

A finite open cover always has a continuous partition of unity subordinate to it, provided the space is locally compact and Hausdorff. Paracompactness of the space is a necessary condition to guarantee the existence of a partition of unity subordinate to any open cover. Depending on the category to which the space belongs, this may also be a sufficient condition. In particular, a compact set in the Euclidean space admits a smooth partition of unity subordinate to any finite open cover. The construction uses mollifiers (bump functions), which exist in continuous and smooth manifolds, but not necessarily in analytic manifolds. Thus for an open cover of an analytic manifold, an analytic partition of unity subordinate to that open cover generally does not exist. See analytic continuation.

If and are partitions of unity for spaces and is a partition of unity for the cartesian product space

Example

be a bump function.

Variant definitions

.

In the field of operator algebras, a partition of unity is composed of projections. In the case of \mathrm{C}^*-algebras, it can be shown that the entries are pairwise orthogonal: Note it is not the case that in a general *-algebra that the entries of a partition of unity are pairwise orthogonal.

is a normal, and has finite spectrum, then the projections in the spectral decomposition form a partition of unity.

In the field of compact quantum groups form partitions of unity.

Applications

A partition of unity can be used to define the integral (with respect to a volume form) of a function defined over a manifold: one first defines the integral of a function whose support is contained in a single coordinate patch of the manifold; then one uses a partition of unity to define the integral of an arbitrary function; finally one shows that the definition is independent of the chosen partition of unity.

A partition of unity can be used to show the existence of a Riemannian metric on an arbitrary manifold.

Method of steepest descent employs a partition of unity to construct asymptotics of integrals.

Linkwitz–Riley filter is an example of practical implementation of partition of unity to separate input signal into two output signals containing only high- or low-frequency components.

The Bernstein polynomials of a fixed degree m are a family of m.

The weak Hilbert Nullstellensatz form a polynomial partition of unity subordinate to the Zariski-open.

Partitions of unity are used to establish global smooth approximations for Sobolev functions in bounded domains.

See also

References

  • , see chapter 13