Теория приближенных методов решения операторных уравнений. Габдулхаев Б.Г. - 81 стр.

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α
k
∈ R
m
X
k=0
α
k
d
r
(AQ
k
) = d
r
(y), r = 0, m,
d
r
(f) =
b
Z
a
ρ(t)f(t)Q
r
(t) dt, f ∈ L
2
(ρ);
d
r
Ã
A
m
X
k=0
α
k
Q
k
!
= d
r
(y), r = 0, m.
P
n
= P
m+1
: L
2
(ρ) −→ L({Q
k
(t)}
m
0
) ⊂ L
2
(ρ)
P
n
(f; t) =
m
X
r=0
c
r
(f) Q
r
(t) ≡ P
m+1
(f; t),
c
r
(f) = d
r
(f)/kQ
r
(t)k
L
2
(ρ)
P
2
n
= P
n
, P
∗
n
= P
n
, kP
n
k = 1 (n = 0, 1, . . .).
H = L
2
(ρ) E
n
(f)
f ∈ L
2
(ρ)
m (m = 0, 1, . . .)
n = m + 1 ∈ N
11.3.3.
H = L
2
[a, b] ≡ L
2
ρ = ρ(t) ≡ 1
ϕ
k
= ϕ
k,n
(t), k = 1, n, t ∈ [a, b]
t
k
= t
k,n
= a + k
b − a
n
, k = 0, n, n ∈ N. (11.11)
êîýôôèöèåíòû αk ∈ R êîòîðîãî áóäåì îïðåäåëÿòü â ëèíåéíîì ñëó÷àå èç
ÑËÀÓ             m    X
                              αk dr (AQk ) = dr (y),              r = 0, m,
                      k=0
ãäå
                                  Zb
                   dr (f ) =             ρ(t)f (t)Qr (t) dt,       f ∈ L2 (ρ);
                                   a
â íåëèíåéíîì ñëó÷àå íåèçâåñòíûå êîýôôèöèåíòû îïðåäåëÿþòñÿ èç ÑÍÀÓ
                          Ã       m
                                                  !
                                  X
                     dr       A          αk Qk        = dr (y),    r = 0, m.
                                  k=0

      Îïåðàòîð ïðîåêòèðîâàíèÿ

                Pn = Pm+1 : L2 (ρ) −→ L({Qk (t)}m
                                                0 ) ⊂ L2 (ρ)

çäåñü îïðåäåëèì ïî ôîðìóëå
                                         m
                                         X
                    Pn (f ; t) =                cr (f ) Qr (t) ≡ Pm+1 (f ; t),
                                         r=0

ãäå cr (f ) = dr (f )/kQr (t)kL                . Íåòðóäíî ïîêàçàòü, ÷òî
                                       2 (ρ)


               Pn2 = Pn ,         Pn∗ = Pn ,          kPn k = 1 (n = 0, 1, . . .).

Ïîýòîìó äëÿ ðàññìàòðèâàåìîé ñõåìû ìåòîäà ðåäóêöèè ñïðàâåäëèâà òåî-
ðåìà 11.3, â êîòîðîé H = L2 (ρ) , à En (f )  íàèëó÷øåå âåñîâîå ñðåä-
íåêâàäðàòè÷åñêîå ïðèáëèæåíèå ôóíêöèè f ∈ L2 (ρ) âñåâîçìîæíûìè àë-
ãåáðàè÷åñêèìè ìíîãî÷ëåíàìè ñòåïåíè íå âûøå m (m = 0, 1, . . .) , ãäå
n = m + 1 ∈ N.

           11.3.3. Ìåòîä ñïëàéíïîäîáëàñòåé íóëåâîãî ïîðÿäêà

     Ïóñòü H = L2 [a, b] ≡ L2 ñ óêàçàííûìè âûøå ñêàëÿðíûì ïðîèç-
âåäåíèåì è íîðìîé ïðè ρ = ρ(t) ≡ 1 . Çà êîîðäèíàòíóþ ñèñòåìó ôóíê-
öèé â ýòîì ïðîñòðàíñòâå âîçüìåì ñèñòåìó ôóíäàìåíòàëüíûõ ñïëàéíîâ
ϕk = ϕk,n (t), k = 1, n, t ∈ [a, b] , íóëåâîé ñòåïåíè ïî ñåòêå óçëîâ
                                                 b−a
                  tk = tk,n = a + k                  ,     k = 0, n,    n ∈ N.       (11.11)
                                                  n

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