Лекции по функциональному анализу для начинающих специалистов по математической физике. Арсеньев А.А. - 481 стр.

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Z
exp(iη · x)h(ξ , η)dη ≡ 0,
h(ξ , η) ≡ 0.
z
i , j
(ξ , η) z
i , j
(ξ , η)
S(R
d
⊕R
d
)
a(q , p) ∈ S(R
d
⊕ R
d
) T
w
(a)
L
2
(R
d
)
∀(a ∈ S(R
d
⊕ R
d
)) : T
w
(a) = (2π)
−d
Z
F
σ
(a)(ξ ⊕ η)W (ξ , η)dξdη,
F
σ
(a)(ξ ⊕ η) = (2π)
−d
Z
exp(−iσ(ξ ⊕ η , q ⊕ p))a(q ⊕ p) dqdp.
L(L
2
(R
d
) 7→ L
2
(R
d
))
F
σ
(a)(ξ ⊕ η) ∈ S(R
d
⊕ R
d
) , kW (ξ , η) | L(L
2
(R
d
) 7→ L
2
(R
d
))k ≡ 1,
S(R
d
⊕R
d
)
1. T
w
(a)
∗
= T
w
(a
∗
).
2. ∀(a ∈ S(R
d
⊕ R
d
)) : T
w
(a) ∈ HS , kT
w
(a) | HSk
2
= (2π)
−d
ka | L
2
(R
d
⊕ R
d
)k
2
.
3. ∀(ψ ∈ S(R
d
)) : T
w
(a)ψ(x) = (2π)
−d
Z
exp(ip · (x − ξ))a
x + ξ
2
, p
ψ(ξ)dpdξ.
T
w
(a)
∗
=
(2π)
−d
Z
F
σ
(a)(ξ , η)
∗
W (ξ , η)
∗
dξdη = (2π)
−d
Z
F
σ
(a)(ξ , η)
∗
W (−ξ , −η)dξdη =
(2π)
−d
Z
F
σ
(a)(−ξ , −η)
∗
W (ξ , η)dξdη = (2π)
−d
Z
F
σ
(a
∗
)(ξ , η)W (ξ , η)dξdη.
ñëåäóåò ðàâåíñòâî          Z
                               exp(iη · x)h(ξ , η)dη ≡ 0,
è
                                    h(ξ , η) ≡ 0.
Ïîëíîòà ñèñòåìû zi , j (ξ , η) äîêàçàíà. Îðòîíîðìèðîâàííîñòü ñèñòåìû zi , j (ξ , η)
ñëåäóåò èç ïðåäûäóùåé ëåììû.
Îïðåäåëåíèå A.1.1. Íà ôóíêöèÿõ èç ïðîñòðàíñòâà Øâàðöà S(R ⊕R )               d   d

ïðåîáðàçîâàíèå Âåéëÿ îïðåäåëåíî êàê îòîáðàæåíèå, êîòîðîå ôóíêöèè
a(q , p) ∈ S(Rd ⊕ Rd ) ñòàâèò â ñîîòâåòñòâèå îïåðàòîð Tw (a) íà ïðîñòðàí-
ñòâå L2 (Rd ):
                                       Z
             d    d
  ∀(a ∈ S(R ⊕ R )) : Tw (a) = (2π)  −d
                                         Fσ (a)(ξ ⊕ η)W (ξ , η)dξdη, (A.16)

ãäå
                                             Z
                                        −d
                  Fσ (a)(ξ ⊕ η) = (2π)           exp(−iσ(ξ ⊕ η , q ⊕ p))a(q ⊕ p) dqdp.
                                                                             (A.17)
   Èíòåãðàë â ôîðìóëå (A.16) ïîíèìàåòñÿ êàê èíòåãðàë Áîõíåðà â áà-
íàõîâîì ïðîñòðàíñòâå L(L2 (Rd ) 7→ L2 (Rd )). Â ôîðìóëå (A.16)
      Fσ (a)(ξ ⊕ η) ∈ S(Rd ⊕ Rd ) , kW (ξ , η) | L(L2 (Rd ) 7→ L2 (Rd ))k ≡ 1,
ïîýòîìó ñõîäèìîñòü èíòåãðàëà ñîìíåíèé íå âûçûâàåò.
Òåîðåìà A.1.1. Íà ôóíêöèÿõ èç ïðîñòðàíñòâà Øâàðöà S(R ⊕R ) ïðå-         d   d

îáðàçîâàíèå Âåéëÿ óäîâëåòâîðÿåò óñëîâèÿì:

1. Tw (a)∗ = Tw (a∗ ).                                                       (A.18)
2. ∀(a ∈ S(Rd ⊕ Rd )) : Tw (a) ∈ HS , kTw (a) | HSk2 = (2π)−d ka | L2 (Rd ⊕ Rd )k2 .
                                                                       (A.19)
                                      Z                              
                                                            x+ξ
3. ∀(ψ ∈ S(Rd )) : Tw (a)ψ(x) = (2π)−d exp(ip · (x − ξ))a          , p ψ(ξ)dpdξ.
                                                              2
                                                                       (A.20)
      Äîêàçàòåëüñòâî. Èìååì:
Tw (a)∗ =
        Z                                       Z
     −d                ∗         ∗           −d
(2π)      Fσ (a)(ξ , η) W (ξ , η) dξdη = (2π)      Fσ (a)(ξ , η)∗ W (−ξ , −η)dξdη =
        Z                                          Z
     −d                   ∗                     −d
(2π)      Fσ (a)(−ξ , −η) W (ξ , η)dξdη = (2π)        Fσ (a∗ )(ξ , η)W (ξ , η)dξdη.

                                         469

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