Borel's Lemma
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Borel's lemma, named after
Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French people, French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Biograp ...
, is an important result used in the theory of
asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation ...
s and
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s.


Statement

Suppose ''U'' is an
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
in the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
R''n'', and suppose that ''f''0, ''f''1, ... is a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of smooth functions on ''U''. If ''I'' is any open interval in R containing 0 (possibly ''I'' = R), then there exists a smooth function ''F''(''t'', ''x'') defined on ''I''×''U'', such that :\left.\frac\_ = f_k(x), for ''k'' ≥ 0 and ''x'' in ''U''.


Proof

Proofs of Borel's lemma can be found in many text books on analysis, including and , from which the proof below is taken. Note that it suffices to prove the result for a small interval ''I'' = (−''ε'',''ε''), since if ''ψ''(''t'') is a smooth
bump function In mathematical analysis, a bump function (also called a test function) is a function f : \Reals^n \to \Reals on a Euclidean space \Reals^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supp ...
with compact support in (−''ε'',''ε'') equal identically to 1 near 0, then ''ψ''(''t'') ⋅ ''F''(''t'', ''x'') gives a solution on R × ''U''. Similarly using a smooth
partition of unity In mathematics, a partition of unity on a topological space is a Set (mathematics), set of continuous function (topology), continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood (mathem ...
on R''n'' subordinate to a covering by open balls with centres at ''δ''⋅Z''n'', it can be assumed that all the ''f''''m'' have compact support in some fixed closed ball ''C''. For each ''m'', let :F_m(t,x)= \cdot \psi\left(\right)\cdot f_m(x), where ''εm'' is chosen sufficiently small that :\, \partial^\alpha F_m \, _\infty \le 2^ for , ''α'', < ''m''. These estimates imply that each sum :\sum_ \partial^\alpha F_m is uniformly convergent and hence that :F=\sum_ F_m is a smooth function with :\partial^\alpha F=\sum_ \partial^\alpha F_m. By construction :\partial_t^m F(t,x), _=f_m(x). Note: Exactly the same construction can be applied, without the auxiliary space ''U'', to produce a smooth function on the interval ''I'' for which the derivatives at 0 form an arbitrary sequence.


See also

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References

* * * {{PlanetMath attribution, title=Borel lemma, id=6185 Partial differential equations Lemmas in mathematical analysis Asymptotic analysis