Download Analyse fonctionnelle : une introduction pour physiciens by N. Boccara. PDF

By N. Boccara.

Dans ce chapitre sont construits les outils indispensables a l'élaboration des théories qui seront développées par los angeles suite. los angeles concept de mesure у joue un rôle essentiel. Les résultats les plus importants sont le théorème de los angeles convergence dominée de Lebesgue et le théorème de Fubini relatif a l'interversion des ordres d’intégration dans les intégrales multiples.
Plus encore dans ce chapitre que dans les suivants, il est recommande de se familiariser avec ces théorèmes, en étudiant les différents exemples et en essayant de faire les exercices, avant d'en aborder los angeles démonstration.

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2) ~h e r e ~ K(x,y):::- Jf(x,y,deicr(X'Y'~) dl'; r ang es ov er of ~1 -an d ~ 2 ~T R~ an d x ,y ov er op en n 8~ b s e t s ~ L 1 r esp e ctivel y. Thc f un c t i on an d 52 2 has bounde d i maginary part an d i s ca l l ed a phas e f unc t i sn ~hil e f (x ,y ,~ ) i s c o~ pl e x and c alled an ampli t u de . Opera tors Cu(x) = ~c(x, ; )U ( Y )dY - 57 - L . G~rding defined by C~kernels o(x,y) are said to be smoot h and in t h e calculus that we are going to develop, su ch op erators are disregarded.

Gltrding (Pf,g)=f,~g) when f and ~ are then have the property that sufficiently smooth and their p~oduct has c ompact Gu~po rt . g. A=P wi t h D(A) = C o~. Then A is P- densely defined and AW = W wi t h D(A ) = all f i n H su ch t hat P-f belongs to H when taken in the s en s e of ditributions . ( has constant degree m and it s pr i n c i pal part Pm (x,d = L:. a~xk· Itt/=m never vanishes wh en ~ Io is real, t he s i t ua t i on i s simple r . 1t° (=> u ~ 'St m where ~k is t~e space of distribu ti ons wh os e de rivative s of order ~k are locally square integrable.

2 A = D1 \+D2 2 2 +D and l et H be the o 3 i 2 2 3 corresponding selfadjoint operator on L =L (R ). 1) Fou(~) = Jre-iX~u(X)dX -1 then F oH0 F 0 is multiplicati on by - 39 - L. ( 1. that Ho• Fou is integrable and h ence u(x) = Jeixl',;F 0 u(l;)dl; I bounded and unifo rmly c ontinu ou s when uE: D(Ho). 3) wher e K and L are co n centr ic bal l s of rad i i 1 an d 2. Her e 2 ( 2 /Iu, KI/ = ) K'u (x)! dx and t he l eft s i d e of (1. 3) i s def ined an al ogou sly. 5) ::: ::(" X) . 0 ,gn ' 1'1- 3/ 2jI,-i(X-r)2/ 4' f (y )dY wh e r e c i s a c ons tant.

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