I would like to collect the applications of partition of unity theorem in math, for example manifold, topology, fixed point theory, differential forms, differential geometry, vector analysis, algebraic topology, differential topology , and any other related fields.
see the following references
http://booksc.org/book/1821531/8ed028
http://booksc.org/book/1821779/1a8af3
http://booksc.org/book/1821534/86ef06
http://booksc.org/book/1821500/d30160
http://booksc.org/book/30309/720152
They are basic in differential geometry, where given a manifold you can define geometric objects (such as metrics) locally on the sets of an open covering by pulling back the corresponding objects on open subsets of the Euclidean space R^n and then, in the absence of obstructions, proceed to glue them together with the aid of the partition of unity.
They are also used to prove the fundamental results that the Lie algebra of derivations of the ring of smooth functions on a manifold is isomorphic to the Lie algebra of vector fields (endowed with the canonical Lie bracket).
In distribution theory, partitions of unity are used to show that a distribution is uniquely determined by its localizations.
Hi everybody
The notion of partition of unity is used in in many areas in Mathematics. It is used in the framework of Nachbin weighted spaces to characterize the closure of s subspace (see M. Klilou, L. Oubbi , Multiplication operators on generalized weighted spaces of continuous functions, Mediterr. J. Math. (2016) 13: 3265. https://doi.org/10.1007/s00009-016-0684-x)
It is also used in the theory of Hyers-Ulam stability of functional equation when one deals with the space $C(K)$ of continuous functions on a compact set $K$.
Usually, the existence of partitions of unity is proved by means of Urysohn's lemma. In case of a locally compact space, one applies the partition of unity in order to prove the subadditivity of a positive Borel measure (as function of set), starting from a linear positive functional defined on the vector space of continuous compactly supported functions. In particular, the subadditivity of Lebesgue measure starting from the functional defined by Riemann's integral follows (see W. Rudin, "Real and complex analysis. Third edition", Chapter 2, Riesz representation theorem). Since Urysohn's lemma for C^(infinity) functions holds too, it follows that C^(infinity) partitions of unity can be constructed on subsets of R^n, and, maybe, in a more general framework.
Partition of unity is useful in defining so many concepts in Differential Topology.
In analysis, for example, you can split a function $f$ as $\sum \phi_n f$ where $(\phi_n)$ is a partition of unity. In this way you can manage the singularities of $f$ locally. I used this idea in my article "Asymptotics of Individual Eigenvalues of a Class of Large Hessenberg Toeplitz Matrices" (section 4 page 83).
A partition of unity can be used to prove that given any metric continuum X and any epsilon > 0, X can be epsilon mapped onto a finite polyhedron. The partition of unity can be used directly to get the barycentric coordinates of the image of a given point; the polyhedron is just the nerve of the covering to which the partition of unity is subordinate.
This is the only thing I recall having used one for.
Thank you david , can you also provide a reference to that application please?
Not offhand. I saw it mentioned once and created a proof for myself. I suspect it dates to the 1930's, but I could be wrong. Look at the papers of Mardesic and Segal; as I recall, they make use of it, so they likely have a reference.
A partition of unity induces periodicity in transforms, by the rules of cyclic convolution. This has application in semi-discrete Hilbert space models for example.
An important application of the "partition of unity" is the classical sampling theorem.
Multiplying the function that is constantly 1 with a Dirac comb III yields (trivially) the Dirac comb. Its coefficients are ..1,1,1,1,1.. which can be thought of being the "sampled" constant function 1.
The reversal of this operation ("discretization of unity") amounts to the question: What is the function f such that III * f = 1 where * is convolution. In other words, which function f can be used to compose 1 from shifted versions (periodization) of it? This construction is called "partition of unity" because 1 falls into shifted variants of f. The operation "III *" applied to f can be understood as the "periodization of f".
A trivial function satisfying this equation is the rect function because III * rect = 1. We may think of this equation as the partition of the real axis into intervals of lengths equal to unity. Another function satisfying a trivial "partition of unity" is the sinc function because III * sinc = 1. However, the latter equality must be understood asymptotically. One may recall, the sinc-integral merely exists as an "improper integral". Replacing unity (which is a band-limited contant function) by another (band-limited) function yields the "classical sampling theorem", i.e., the sinc function is used as an ideal interpolator in order to restore the original function from its samples.
Beside rect and sinc there are many more such functions which compose unity. They are known as Lighthill's unitary functions. We may think of these functions as the convolution inverses of the Dirac comb. More:
Article On Inverses of the Dirac Comb
Please don't forget about many(!) applications of partitions of unity in differential inclusions. It can be found in any book on the subject.
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