Ряды Фурье и основы вейвлет-анализа. Фарков Ю.А. - 52 стр.

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P
0
f = P
−1
f + Q
−1
f, P
−1
f = 6
√
2ϕ
−1,0
, Q
−1
f = 2
√
2ψ
−1,0
.
f j = 1 s = 2
f(t) = 6
√
2ϕ
−1,0
(t) + 2
√
2ψ
−1,0
(t) + (ψ
00
(t) − ψ
01
(t))
f(t) = 6ϕ(t/2) + 2ψ(t/2) + (ψ(t) − ψ(t − 1)),
ϕ ψ
2
f ∈ L
2
(R)
P
j
f =
∞
X
s=1
Q
j−s
f, j ∈ Z. (16)
V
0
=
∞
M
j=1
W
−j
. (17)
L
2
(R) = V
0
M
M
j≥0
W
j
!
. (18)
ϕ = χ
[0,1)
V
0
ϕ = P
0
ϕ =
∞
X
j=1
Q
−j
ϕ =
n
X
j=1
Q
−j
ϕ + P
−n
ϕ, n ∈ N.
{a
0k
} f = ϕ
a
0k
=
1, k = 0,
0, k 6= 0.
j ≤ 0
a
j−1,k
=
a
j,0
/
√
2, k = 0,
0, k 6= 0.
n ∈ N
P
−n
ϕ = a
−n,0
ϕ
−n,0
=
1
√
2
n
ϕ
−n,0
,
Îòñþäà
                                                      √                            √
      P0 f = P−1 f + Q−1 f,      ãäå P−1 f = 6 2ϕ−1,0 ,                   Q−1 f = 2 2ψ−1,0 .
Òàêèì îáðàçîì, äëÿ äàííîé ôóíêöèè f ðàçëîæåíèå (15) â ñëó÷àå j = 1, s = 2
ïðèíèìàåò âèä
                      √              √
             f (t) = 6 2ϕ−1,0 (t) + 2 2ψ−1,0 (t) + (ψ00 (t) − ψ01 (t))
èëè
               f (t) = 6ϕ(t/2) + 2ψ(t/2) + (ψ(t) − ψ(t − 1)),
ãäå ϕ è ψ  ìàñøòàáèðóþùàÿ ôóíêöèÿ è âåéâëåò Õààðà.
                                                                                          2
  Èç (10) è (15) ñëåäóåò, ÷òî äëÿ âñåõ f ∈ L2 (R)
                                       ∞
                                       X
                              Pj f =         Qj−s f,      j ∈ Z.                               (16)
                                       s=1

Îòñþäà ïîëó÷àåì ðàâåíñòâî
                                             ∞
                                             M
                                    V0 =            W−j .                                      (17)
                                              j=1

Çíà÷èò, íàðÿäó ñ (8) èìååò ìåñòî ðàçëîæåíèå
                                                                  !
                                             M M
                           L2 (R) = V0                       Wj       .                        (18)
                                                       j≥0

  Ïðèìåð 2. Äëÿ ôóíêöèè ϕ = χ[0,1) èç ïðîñòðàíñòâà V0 â ñèëó (15) è (17)
èìååì                      ∞                 n
                           X                 X
              ϕ = P0 ϕ =         Q−j ϕ =            Q−j ϕ + P−n ϕ,            n ∈ N.
                           j=1               j=1

Ñðåäè àïïðîêñèìèðóþùèõ êîýôôèöèåíòîâ {a0k } ôóíêöèè f = ϕ òîëüêî îäèí
îòëè÷åí îò íóëÿ:            
                                              1,     k = 0,
                                 a0k =
                                              0,     k 6= 0.
Îòñþäà è èç (13) äëÿ ëþáîãî öåëîãî j ≤ 0 íàõîäèì
                                                √
                                           aj,0 / 2,         k = 0,
                           aj−1,k =
                                              0,             k 6= 0.
Ñëåäîâàòåëüíî, äëÿ âñåõ n ∈ N
                                                              n
                                                          1
                      P−n ϕ = a−n,0 ϕ−n,0 =              √          ϕ−n,0 ,
                                                           2
                                              52

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