Partition of unity

In mathematics, a partition of unity on a topological space ⁠$X$⁠ is a set ⁠$R$⁠ of continuous functions from ⁠$X$⁠ to the unit interval [0,1] such that for every point $x\in X$:

• there is a neighbourhood of ⁠$x$⁠ where all but a finite number of the functions of ⁠$R$⁠ are zero, and
• the sum of all the function values at ⁠$x$⁠ is 1, i.e., ${\textstyle \sum _{\rho \in R}\rho (x)=1.}$

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 functions.