Partition of unity
In mathematics, a partition of unity of a topological space
to the unit interval [0,1] such that for every point
: 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.
The existence of partitions of unity assumes two distinct forms: 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 subordinated to it, provided the space is locally compact and Hausdorff.
[1] 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, it may also be a sufficient condition.
[2] The construction uses mollifiers (bump functions), which exist in continuous and smooth manifolds, but not 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.
is a partition of unity for the cartesian product space
The tensor product of functions act as
Sometimes a less restrictive definition is used: the sum of all the function values at a particular point is only required to be positive, rather than 1, for each point in the space.
one can obtain a partition of unity in the strict sense by dividing by the sum; the partition becomes
, which is well defined since at each point only a finite number of terms are nonzero.
[3] In the field of operator algebras, a partition of unity is composed of projections[4]
Note it is not the case that in a general *-algebra that the entries of a partition of unity are pairwise-orthogonal.
, then the projections in the spectral decomposition:
[7] In the field of compact quantum groups, the rows and columns of the fundamental representation
of a quantum permutation group
[8] 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+1 linearly independent single-variable polynomials that are a partition of unity for the unit interval
The weak Hilbert Nullstellensatz asserts that if
are polynomials with no common vanishing points in
form a polynomial partition of unity subordinate to the Zariski-open cover
Partitions of unity are used to establish global smooth approximations for Sobolev functions in bounded domains.
