Editing Partition of unity (section)

{{short description|Set of functions from a topological space to [0,1] which sum to 1 for any input}}

In [[mathematics]], a '''partition of unity''' on a [[topological space]] {{tmath|X}} is a [[Set (mathematics)|set]] {{tmath|R}} of [[continuous function (topology)|continuous function]]s from {{tmath|X}} to the [[unit interval]] [0,1] such that for every point <math>x\in X</math>:
* there is a [[neighbourhood (mathematics)|neighbourhood]] of {{tmath|x}} where all but a [[finite set|finite]] number of the functions of {{tmath|R}} are non zero<ref>Lee, John M., and John M. Lee. Smooth manifolds. Springer New York, 2003.</ref>, and
* the sum of all the function values at {{tmath|x}} is 1, i.e., <math display="inline">\sum_{\rho\in R} \rho(x) = 1.</math>

[[Image:Partition of unity illustration.svg|center|thumb|500px|A partition of unity on a circle with four functions. The circle is unrolled to a line segment (the bottom solid line) for graphing purposes. The dashed line on top is the sum of the functions in the partition.]]

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 function]]s.