Even further, some authors drop the requirement that the supports be locally finite, requiring only that formula_21 for all formula_22. 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 polynomials that are a partition of unity for the unit interval formula_23. Partition of unity is used to establish global smooth approximations for Sobolev functions in bounded domains.