In mathematics, Hadamard's lemma, named after Jacques Hadamard, is essentially a first-order form of Taylor's theorem, in which we can express a smooth, real-valued function exactly in a convenient manner.
Hadamard's lemma[1] — Let be a smooth, real-valued function defined on an open, star-convexneighborhood of a point in -dimensional Euclidean space. Then can be expressed, for all in the form:
where each is a smooth function on and
Proofedit
Proof
Let Define by
Then
which implies
But additionally, so by letting
the theorem has been proven.
Consequences and applicationsedit
Corollary[1] — If is smooth and then is a smooth function on Explicitly, this conclusion means that the function that sends to
is a well-defined smooth function on
Proof
By Hadamard's lemma, there exists some such that so that implies
Corollary[1] — If are distinct points and is a smooth function that satisfies then there exist smooth functions () satisfying for every such that
Proof
By applying an invertibleaffine linear change in coordinates, it may be assumed without loss of generality that and By Hadamard's lemma, there exist such that For every let where implies Then for any
Each of the terms above has the desired properties.
See alsoedit
Bump function – Smooth and compactly supported function
Continuously differentiable – Mathematical function whose derivative existsPages displaying short descriptions of redirect targets
Smoothness – Number of derivatives of a function (mathematics)
Taylor's theorem – Approximation of a function by a truncated power series
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In mathematics Hadamard s lemma named after Jacques Hadamard is essentially a first order form of Taylor s theorem in which we can express a smooth real valued function exactly in a convenient manner Contents 1 Statement 1 1 Proof 2 Consequences and applications 3 See also 4 Citations 5 ReferencesStatement editHadamard s lemma 1 Let f displaystyle f nbsp be a smooth real valued function defined on an open star convex neighborhood U displaystyle U nbsp of a point a displaystyle a nbsp in n displaystyle n nbsp dimensional Euclidean space Then f x displaystyle f x nbsp can be expressed for all x U displaystyle x in U nbsp in the form f x f a i 1 n x i a i g i x displaystyle f x f a sum i 1 n left x i a i right g i x nbsp where each g i displaystyle g i nbsp is a smooth function on U displaystyle U nbsp a a 1 a n displaystyle a left a 1 ldots a n right nbsp and x x 1 x n displaystyle x left x 1 ldots x n right nbsp Proof edit Proof Let x U displaystyle x in U nbsp Define h 0 1 R displaystyle h 0 1 to mathbb R nbsp byh t f a t x a for all t 0 1 displaystyle h t f a t x a qquad text for all t in 0 1 nbsp Thenh t i 1 n f x i a t x a x i a i displaystyle h t sum i 1 n frac partial f partial x i a t x a left x i a i right nbsp which implies h 1 h 0 0 1 h t d t 0 1 i 1 n f x i a t x a x i a i d t i 1 n x i a i 0 1 f x i a t x a d t displaystyle begin aligned h 1 h 0 amp int 0 1 h t dt amp int 0 1 sum i 1 n frac partial f partial x i a t x a left x i a i right dt amp sum i 1 n left x i a i right int 0 1 frac partial f partial x i a t x a dt end aligned nbsp But additionally h 1 h 0 f x f a displaystyle h 1 h 0 f x f a nbsp so by lettingg i x 0 1 f x i a t x a d t displaystyle g i x int 0 1 frac partial f partial x i a t x a dt nbsp the theorem has been proven displaystyle blacksquare nbsp Consequences and applications editCorollary 1 If f R R displaystyle f mathbb R to mathbb R nbsp is smooth and f 0 0 displaystyle f 0 0 nbsp then f x x displaystyle f x x nbsp is a smooth function on R displaystyle mathbb R nbsp Explicitly this conclusion means that the function R R displaystyle mathbb R to mathbb R nbsp that sends x R displaystyle x in mathbb R nbsp to f x x if x 0 lim t 0 f t t if x 0 displaystyle begin cases f x x amp text if x neq 0 lim t to 0 f t t amp text if x 0 end cases nbsp is a well defined smooth function on R displaystyle mathbb R nbsp Proof By Hadamard s lemma there exists some g C R displaystyle g in C infty mathbb R nbsp such that f x f 0 x g x displaystyle f x f 0 xg x nbsp so that f 0 0 displaystyle f 0 0 nbsp implies f x x g x displaystyle f x x g x nbsp displaystyle blacksquare nbsp Corollary 1 If y z R n displaystyle y z in mathbb R n nbsp are distinct points and f R n R displaystyle f mathbb R n to mathbb R nbsp is a smooth function that satisfies f z 0 f y displaystyle f z 0 f y nbsp then there exist smooth functions g i h i C R n displaystyle g i h i in C infty left mathbb R n right nbsp i 1 3 n 2 displaystyle i 1 ldots 3n 2 nbsp satisfying g i z 0 h i y displaystyle g i z 0 h i y nbsp for every i displaystyle i nbsp such thatf i g i h i displaystyle f sum i g i h i nbsp Proof By applying an invertible affine linear change in coordinates it may be assumed without loss of generality that z 0 0 displaystyle z 0 ldots 0 nbsp and y 0 0 1 displaystyle y 0 ldots 0 1 nbsp By Hadamard s lemma there exist g 1 g n C R n displaystyle g 1 ldots g n in C infty left mathbb R n right nbsp such that f x i 1 n x i g i x displaystyle f x sum i 1 n x i g i x nbsp For every i 1 n displaystyle i 1 ldots n nbsp let a i g i y displaystyle alpha i g i y nbsp where 0 f y i 1 n y i g i y g n y displaystyle 0 f y sum i 1 n y i g i y g n y nbsp implies a n 0 displaystyle alpha n 0 nbsp Then for any x x 1 x n R n displaystyle x left x 1 ldots x n right in mathbb R n nbsp f x i 1 n x i g i x i 1 n x i g i x a i i 1 n 1 x i a i using g i x g i x a i a i and a n 0 i 1 n x i g i x a i i 1 n 1 x i x n a i i 1 n 1 x i 1 x n a i using x i x n x i x i 1 x n displaystyle begin alignedat 8 f x amp sum i 1 n x i g i x amp amp amp sum i 1 n left x i left g i x alpha i right right sum i 1 n 1 left x i alpha i right amp amp quad text using g i x left g i x alpha i right alpha i text and alpha n 0 amp left sum i 1 n x i left g i x alpha i right right left sum i 1 n 1 x i x n alpha i right left sum i 1 n 1 x i left 1 x n right alpha i right amp amp quad text using x i x n x i x i left 1 x n right end alignedat nbsp Each of the 3 n 2 displaystyle 3n 2 nbsp terms above has the desired properties displaystyle blacksquare nbsp See also editBump function Smooth and compactly supported function Continuously differentiable Mathematical function whose derivative existsPages displaying short descriptions of redirect targets Smoothness Number of derivatives of a function mathematics Taylor s theorem Approximation of a function by a truncated power seriesCitations edit a b c Nestruev 2020 pp 17 18 References editNestruev Jet 2002 Smooth manifolds and observables Berlin Springer ISBN 0 387 95543 7 Nestruev Jet 10 September 2020 Smooth Manifolds and Observables Graduate Texts in Mathematics Vol 220 Cham Switzerland Springer Nature ISBN 978 3 030 45649 8 OCLC 1195920718 Retrieved from https en wikipedia org w index php title Hadamard 27s lemma amp oldid 1179515315, wikipedia, wiki, book, books, library,