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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
( R e d i r e c t e d f r o m B o r e l ’ s l e m m a )
Statement [ edit ]
Suppose U is an open set in the Euclidean space R n , and suppose that f 0 , f 1 , ... is a sequence 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
∂
k
F
∂
t
k
|
(
0
,
x
)
=
f
k
(
x
)
,
{\displaystyle \left.{\frac {\partial ^{k}F}{\partial t^{k}}}\right|_{(0,x)}=f_{k}(x ),}
for k ≥ 0 and x in U .
Proofs of Borel's lemma can be found in many text books on analysis, including Golubitsky & Guillemin (1974) and Hörmander (1990) , 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 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
)
=
t
m
m
!
⋅
ψ
(
t
ε
m
)
⋅
f
m
(
x
)
,
{\displaystyle F_{m}(t,x)={t^{m} \over m!}\cdot \psi \left({t \over \varepsilon _{m}}\right)\cdot f_{m}(x ),}
where εm is chosen sufficiently small that
‖
∂
α
F
m
‖
∞
≤
2
−
m
{\displaystyle \|\partial ^{\alpha }F_{m}\|_{\infty }\leq 2^{-m}}
for |α | < m . These estimates imply that each sum
∑
m
≥
0
∂
α
F
m
{\displaystyle \sum _{m\geq 0}\partial ^{\alpha }F_{m}}
is uniformly convergent and hence that
F
=
∑
m
≥
0
F
m
{\displaystyle F=\sum _{m\geq 0}F_{m}}
is a smooth function with
∂
α
F
=
∑
m
≥
0
∂
α
F
m
.
{\displaystyle \partial ^{\alpha }F=\sum _{m\geq 0}\partial ^{\alpha }F_{m}.}
By construction
∂
t
m
F
(
t
,
x
)
|
t
=
0
=
f
m
(
x
)
.
{\displaystyle \partial _{t}^{m}F(t,x)|_{t=0}=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 [ edit ]
References [ edit ]
Erdélyi, A. (1956), Asymptotic expansions , Dover Publications, pp. 22–25, ISBN 0486603180
Golubitsky, M. ; Guillemin, V. (1974), Stable mappings and their singularities , Graduate Texts in Mathematics , vol. 14, Springer-Verlag, ISBN 0-387-90072-1
Hörmander, Lars (1990), The analysis of linear partial differential operators, I. Distribution theory and Fourier analysis (2nd ed.), Springer-Verlag, p. 16, ISBN 3-540-52343-X
This article incorporates material from Borel lemma on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Borel%27s_lemma&oldid=1037606018 "
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● S h o r t d e s c r i p t i o n i s d i f f e r e n t f r o m W i k i d a t a
● W i k i p e d i a a r t i c l e s i n c o r p o r a t i n g t e x t f r o m P l a n e t M a t h
● T h i s p a g e w a s l a s t e d i t e d o n 7 A u g u s t 2 0 2 1 , a t 1 5 : 5 1 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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