In mathematics, Borel's lemma, named after Émile Borel, 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 to ...

s and

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

Statement

Suppose ''U'' is an

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...

in the Euclidean space R^''n'', and suppose that ''f''₀, ''f''₁, ... is a sequence of

smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...

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 mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bu ...

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 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 ''ε_m'' 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.

References

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

Statement

Proof

See also

References