Ряды Фурье и основы вейвлет-анализа. Фарков Ю.А. - 70 стр.

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bϕ ∈ C
2
(R) 2π
Φ(ξ) :=
X
k∈Z
|bϕ(ξ + 2πk)|
2
Φ(ξ) =
|bϕ(ξ)|
2
+ |bϕ(ξ + 2π)|
2
, −4π/3 ≤ ξ ≤ −2π/3,
|bϕ(ξ)|
2
, |ξ| ≤ 2π/3,
|bϕ(ξ)|
2
+ |bϕ(ξ − 2π)|
2
, 2π/3 ≤ ξ ≤ 4π/3.
(5)
Φ(ξ) = 1 |ξ| ≤ 2π/3 2π/3 ≤
ξ ≤ 4π/3
3
2π
|ξ −2π| − 1 = 1 −
3
2π
ξ −1
Φ(ξ) = cos
2
π
2
ν
3
2π
ξ −1

+ sin
2
π
2
ν
3
2π
ξ −1

= 1.
Φ(ξ) = 1 −4π/3 ≤ ξ ≤ −2π/3
X
k∈Z
|bϕ(ξ + 2πk)|
2
= 1.
V
j
⊂ V
j+1
,
\
V
j
= {0},
[
V
j
= L
2
(R)
ψ
bϕ(ξ) = H(ξ/2)bϕ(ξ/2)
H(ξ) =
bϕ(2ξ), ξ ∈ [−2π/3, 2π/3],
0, ξ ∈ [−π, −2π/3] ∪ [2π/3, π].
ϕ =
X
k∈Z
h
k
ϕ
1k
, H(ξ) =
1
√
2
X
k∈Z
h
k
e
−ikξ
h
k
=
√
2
2π
π
Z
−π
H(ξ)e
ikξ
dξ =
√
2
π
2π/3
Z
0
bϕ(2ξ) cos kξ dξ.
   Èç ôîðìóë (1) è (3) âèäíî, ÷òî ϕ
                                  b ∈ C 2 (R). Êðîìå òîãî, äëÿ 2π -
ïåðèîäè÷åñêîé íåïðåðûâíîé ôóíêöèè
                                             X
                                  Φ(ξ) :=           b + 2πk)|2
                                                   |ϕ(ξ
                                             k∈Z
èìååì
                 
                    b 2 + |ϕ(ξ
                  |ϕ(ξ)|     b + 2π)|2 ,                 −4π/3 ≤ ξ ≤ −2π/3,
          Φ(ξ) =           b 2,
                          |ϕ(ξ)|                            | ξ| ≤ 2π/3,       (5)
                    b 2 + |ϕ(ξ
                  |ϕ(ξ)|     b − 2π)|2 ,                  2π/3 ≤ ξ ≤ 4π/3.
Ó÷èòûâàÿ (1) è (3), çàìå÷àåì, ÷òî Φ(ξ) = 1 ïðè | ξ| ≤ 2π/3. Ïóñòü 2π/3 ≤
ξ ≤ 4π/3. Òîãäà                                   
                            3                               3
                              | ξ − 2π| − 1 = 1 −             ξ−1
                           2π                              2π
è, â ñèëó (3) è (5),
                                                             
                           π          3                π      3
        Φ(ξ) = cos2          ν          ξ−1   + sin2     ν      ξ−1   = 1.
                           2         2π                2     2π
Àíàëîãè÷íî, Φ(ξ) = 1 ïðè −4π/3 ≤ ξ ≤ −2π/3. Òàêèì îáðàçîì, äëÿ ôóíêöèè
(3) âûïîëíåíî óñëîâèå îðòîíîðìèðîâàííîñòè:
                                     X
                                            b + 2πk)|2 = 1.
                                           |ϕ(ξ
                                     k∈Z

Ïîëüçóÿñü òåîðåìîé 1 è ïðåäëîæåíèåì 2, âèäèì, ÷òî ñâîéñòâà
                                       \                  [
                   Vj ⊂ Vj+1 ,              Vj = {0},         Vj = L2 (R)
òàêæå âûïîëíåíû.
   Òàêèì îáðàçîì, ôóíêöèÿ (4) ÿâëÿåòñÿ ìàñøòàáèðóþùåé ôóíêöèåé. Ñîîò-
âåòñòâóþùèé âåéâëåò ψ áóäåì èñêàòü ñ ïîìîùüþ ôîðìóëû (4.34). Ïîëüçóÿñü
(3), èç ðàâåíñòâà
                                     ϕ(ξ)
                                     b = H(ξ/2)ϕ(ξ/2)
                                               b
íàõîäèì                    
                                 ϕ(2ξ),        ξ ∈ [−2π/3, 2π/3],
               H(ξ) =
                                 b
                                  0,         ξ ∈ [−π, −2π/3] ∪ [2π/3, π].
Ñîîòâåòñòâåííî, êîýôôèöèåíòû ðàçëîæåíèé
                           X                               1 X
                  ϕ=             hk ϕ1k ,    è     H(ξ) = √       hk e−ikξ
                           k∈Z
                                                            2 k∈Z
âû÷èñëÿþòñÿ ïî ôîðìóëå
                    √ Zπ            √ 2π/3
                                       Z
                     2        ikξ    2
               hk =      H(ξ)e dξ =       ϕ(2ξ)
                                           b    cos kξ dξ.
                    2π              π
                            −π                            0

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