Лекции по функциональному анализу для начинающих специалистов по математической физике. Арсеньев А.А. - 467 стр.

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МГУ | Москва

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τ
S
f
n
(x
1
, . . . , x
d−1
) = f
n
(x
1
. . . , x
d−1
, 0)
H
s−1/2
(R
d−1
) τ
H
f
τ
H
f := lim
n→∞
τ
S
f
n
f ∈ H
s
(R
d
)
f
n
kτ
H
f | H
s−1/2
(R
d−1
)k ≤ C(d , s)kf | H
s
(R
d
)k.
S(R
d
) 3 f
n
(x
1
, . . . , x
d−1
, x
d
) −−−→
n→∞
f
τ
S
y
y
τ
H
S(R
d−1
) 3 f
n
(x
1
, . . . , x
d−1
, 0) −−−→
n→∞
τ
H
f
f ∈ H
s
(R
d
) , τ
H
f ∈ H
s−1/2
(R
d−1
).
τ
H
: H
s
(R
d
) 7→ H
s−1/2
(R
d−1
),
f
n
τ
H
f ∈ H
s−1/2
(R
d−1
)
f ∈ H
s
(R
d
) x
d
= 0
f ∈ S(R
d
) , F
d
f(x
1
, . . . , x
d−1
|ξ
d
) =
∞
Z
−∞
f(x
1
, . . . , x
d
) exp(−i(ξ
d
, x
d
))dx
d
.
τ
S
f(x
1
, . . . , x
d−1
) = f(x
1
, . . . , x
d−1
, 0) =
1
2π
∞
Z
−∞
F
d
f(x
1
, . . . , x
d−1
|ξ
d
)dξ
d
.
   1. Ïîñëåäîâàòåëüíîñòü

                      τS fn (x1 , . . . , xd−1 ) = fn (x1 . . . , xd−1 , 0)
ñõîäèòñÿ â ïðîñòðàíñòâå             H s−1/2 (Rd−1 )    è åå ïðåäåë τH f :

                                     τH f := lim τS fn
                                                 n→∞

çàâèñèò òîëüêî îò ðàñïðåäåëåíèÿ        f ∈ H s (Rd ) è íå çàâèñèò                    îò âûáîðà
àïïðîêñèìèðóåùåé         ïîñëåäîâàòåëüíîñòè fn â (6.128).
   2. Ñïðàâåäëèâî íåðàâåíñòâî

                  kτH f | H s−1/2 (Rd−1 )k ≤ C(d , s)kf | H s (Rd )k.                     (6.129)
   Îïèñàííóþ â òåîðåìå ñèòóàöèþ ìîæíî ïîÿñíèòü íà äèàãðàììå:
                      S(Rd ) 3 fn (x1 , . . . , xd−1 , xd ) −−−→           f
                                                             n→∞
                                                                         
                                  τS y
                                                                         τH
                                                                          y
                     S(Rd−1 ) 3 fn (x1 , . . . , xd−1 , 0) −−−→ τH f
                                                               n→∞

                         f ∈ H s (Rd ) , τH f ∈ H s−1/2 (Rd−1 ).
Òåîðåìà óòâåðæäàåò, ÷òî åñëè ñïðàâåäëèâî îáîçíà÷åííîå âåðõíåé ãîðè-
çîíòàëüíîé ñòðåëêîé ïðåäåëüíîå ñîîòíîøåíèå, òî ñïðàâåäëèâî îáîçíà-
÷åííîå íèæíåé ãîðèçîíòàëüíîé ñòðåëêîé ïðåäåëüíîå ñîîòíîøåíèå è êîð-
ðåêòíî îïðåäåëåíî îòîáðàæåíèå
                             τH : H s (Rd ) 7→ H s−1/2 (Rd−1 ),
êîòîðîå íå çàâèñèò îò âûáîðà àïïðîêñèìèðóþùåé ïîñëåäîâàòåëüíîñòè
fn .
     Ðàñïðåäåëåíèå τH f ∈ H s−1/2 (Rd−1 ) íàçûâàåòñÿ ñëåäîì ðàñïðåäåëåíèÿ
f ∈ H s (Rd ) íà ãèïåðïëîñêîñòè xd = 0.
     Äîêàçàòåëüñòâî. Ïóñòü
                                                 Z∞
  f ∈ S(Rd ) , Fd f (x1 , . . . , xd−1 |ξd ) =        f (x1 , . . . , xd ) exp(−i(ξd , xd ))dxd .
                                                 −∞

Òîãäà
                      τS f (x1 , . . . , xd−1 ) = f (x1 , . . . , xd−1 , 0) =
                           Z∞
                       1
                               Fd f (x1 , . . . , xd−1 |ξd )dξd .
                      2π
                         −∞



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