Ряды Фурье и основы вейвлет-анализа. Фарков Ю.А. - 104 стр.

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ω ∈ [0, 1/2) m
ϕ
b
l
ϕ p = 3 n = 2
b
0
= 1, b
1
= a, b
2
= α, b
3
= 0, b
4
= b, b
5
= β, b
6
= 0, b
7
= c, b
8
= γ,
|a|
2
+ |b|
2
+ |c|
2
= |α|
2
+ |β|
2
+ |γ|
2
= 1.
ϕ 3 L
2
(R
+
)
a 6= 0, α 6= 0 a = 0, c 6= 0, α 6= 0 a 6= 0, α = 0, β 6= 0
ϕ
ϕ(x) =
8
X
α=0
c
k
ϕ(3x k)
c
0
=
1
3
(1 + a + b + c + α + β + γ),
c
1
=
1
3
(1 + a + α + (b + β)ε
2
3
+ (c + γ)ε
3
),
c
2
=
1
3
(1 + a + α + (b + β)ε
3
+ (c + γ)ε
2
3
),
c
3
=
1
3
(1 + (a + b + c)ε
2
3
+ (α + β + γ)ε
3
),
c
4
=
1
3
(1 + c + β + (a + γ)ε
2
3
+ (b + α)ε
3
),
c
5
=
1
3
(1 + b + γ + (a + β)ε
2
3
+ (c + α)ε
3
),
c
6
=
1
3
(1 + (a + b + c)ε
3
+ (α + β + γ)ε
2
3
),
c
7
=
1
3
(1 + b + γ + (a + β)ε
3
+ (c + α)ε
2
3
),
c
8
=
1
3
(1 + c + β + (a + γ)ε
3
+ (b + α)ε
2
3
),
ε
3
= exp(2πi/3).
äëÿ ω ∈ [0, 1/2). Ñëåäîâàòåëüíî, ìàñêà m óäîâëåòâîðÿåò ìîäèôèöèðîâàííî-
ìó óñëîâèþ Êîýíà. Ñîîòâåòñòâóþùàÿ ìàñøòàáèðóþùàÿ ôóíêöèÿ ϕ ìîæåò
áûòü çàäàíà ðàçëîæåíèåì (14), êîýôôèöèåíòû êîòîðîãî ïî ïàðàìåòðàì bl
îïðåäåëÿþòñÿ îäíîçíà÷íî (ñ òî÷íîñòüþ äî åäèíè÷íûõ ïî ìîäóëþ ìíîæèòå-
ëåé).
   4.13. Ïóñòü ϕ îïðåäåëåíà ïî ôîðìóëå (14) ïðè p = 3, n = 2 è
      b0 = 1, b1 = a, b2 = α, b3 = 0, b4 = b, b5 = β, b6 = 0, b7 = c, b8 = γ,
ãäå
                    | a|2 + | b|2 + | c|2 = | α|2 + | β|2 + |γ|2 = 1.
Ñ ïîìîùüþ 4.7 ïðîâåðÿåòñÿ, ÷òî ϕ ãåíåðèðóåò 3-ÊÌÀ â L2 (R+ ) â ñëåäóþùèõ
òðåõ ñëó÷àÿõ: 1) a 6= 0, α 6= 0, 2) a = 0, c 6= 0, α 6= 0, 3) a 6= 0, α = 0, β 6= 0.
Ôóíêöèÿ ϕ óäîâëåòâîðÿåò óðàâíåíèþ
                                       8
                                       X
                              ϕ(x) =         ck ϕ(3x   k)
                                       α=0

ñ êîýôôèöèåíòàìè
                            1
                        c0 = (1 + a + b + c + α + β + γ),
                            3
                      1
                  c1 = (1 + a + α + (b + β)ε23 + (c + γ)ε3 ),
                      3
                      1
                  c2 = (1 + a + α + (b + β)ε3 + (c + γ)ε23 ),
                      3
                      1
                  c3 = (1 + (a + b + c)ε23 + (α + β + γ)ε3 ),
                      3
                      1
                  c4 = (1 + c + β + (a + γ)ε23 + (b + α)ε3 ),
                      3
                      1
                  c5 = (1 + b + γ + (a + β)ε23 + (c + α)ε3 ),
                      3
                      1
                  c6 = (1 + (a + b + c)ε3 + (α + β + γ)ε23 ),
                      3
                      1
                  c7 = (1 + b + γ + (a + β)ε3 + (c + α)ε23 ),
                      3
                      1
                  c8 = (1 + c + β + (a + γ)ε3 + (b + α)ε23 ),
                      3
ãäå ε3 = exp(2πi/3).




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