Ряды Фурье и основы вейвлет-анализа. Фарков Ю.А. - 102 стр.

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c
3
= c
7
= (1 − b + β)/4, c
4
= (−1 + b + β)/4, c
5
= (−1 + b − β)/4.
p = 2 n = 3
b
0
= 1, b
1
= a, b
2
= b, b
3
= c, b
4
= 0, b
5
= α, b
6
= β, b
7
= γ,
|a|
2
+ |α|
2
= |b|
2
+ |β|
2
= |c|
2
+ |γ|
2
= 1.
γ(1, 0, 0) = a, γ(0, 1, 0) = b, γ(1, 1, 0) = c,
γ(1, 0, 1) = α, γ(0, 1, 1) = β, γ(1, 1, 1) = γ
c
l
[m] = γ(µ
k
, 0, 0)γ(µ
k−1
, µ
k
, 0) . . . γ(µ
0
, µ
1
, µ
2
),
l =
k
X
i=0
µ
i
2
i
, µ
k
= 1, µ
i
∈ {0, 1}.
ϕ(x) = (1/4)χ
[0,1)
(y)(1 +
X
l∈N(2,3)
c
l
[m]w
l
(y))
= (1/4)χ
[0,1)
(y)(1 + a w
1
(y) + ab w
2
(y) + ac w
3
(y) + abα w
5
(y)
+ acβ w
6
(y) + acγ w
7
(y) + ab
2
α w
10
(y) + abcα w
11
(y)+
+ acβγ w
14
(y) + acγ
2
w
15
(y) + ab
2
α
2
w
21
(y) + abcαβ w
22
(y) + abαβγ w
23
(y)
+ abcαβ w
26
(y) + ac
2
αβ w
27
(y) + acαβγ w
29
(y) + acβγ
2
w
30
(y) + . . . ),
y = x/4 a c ϕ L
2
(R
+
)
abc 6= 0 E = [0, 1),
a 6= 0, b = 0, c 6= 0 E = [0, 1/2) ∪ [3/4, 1) ∪
[3/2, 7/4) ϕ
ϕ(x) =
7
X
k=0
c
k
ϕ(2x k)
c
0
=
1
4
(1 + a + b + c + α + β + γ),
c
1
=
1
4
(1 + a + b + c − α − β − γ),
c
2
=
1
4
(1 + a − b − c + α − β − γ),
      c3 = c7 = (1 − b + β)/4,        c4 = (−1 + b + β)/4,             c5 = (−1 + b − β)/4.
    4.11. Ïóñòü p = 2, n = 3 è
          b0 = 1, b1 = a, b2 = b, b3 = c, b4 = 0, b5 = α, b6 = β, b7 = γ,
ãäå | a|2 + |α|2 = | b|2 + |β|2 = | c|2 + |γ|2 = 1. Òîãäà

                   γ(1, 0, 0) = a,       γ(0, 1, 0) = b,        γ(1, 1, 0) = c,
                   γ(1, 0, 1) = α,       γ(0, 1, 1) = β,         γ(1, 1, 1) = γ
è
                  cl [m] = γ(µk , 0, 0)γ(µk−1 , µk , 0) . . . γ(µ0 , µ1 , µ2 ),
ãäå
                              k
                              X
                         l=         µi 2i ,   µk = 1,         µi ∈ {0, 1}.
                              i=0
Ðàçëîæåíèå (14) ïðèíèìàåò âèä
                                                         X
                    ϕ(x) = (1/4)χ[0,1) (y)(1 +                     cl [m]wl (y))
                                                        l∈N(2,3)

          = (1/4)χ[0,1) (y)(1 + a w1 (y) + ab w2 (y) + ac w3 (y) + abα w5 (y)
              + acβ w6 (y) + acγ w7 (y) + ab2 α w10 (y) + abcα w11 (y)+
    + acβγ w14 (y) + acγ 2 w15 (y) + ab2 α2 w21 (y) + abcαβ w22 (y) + abαβγ w23 (y)
      + abcαβ w26 (y) + ac2 αβ w27 (y) + acαβγ w29 (y) + acβγ 2 w30 (y) + . . . ),
ãäå y = x/4. Åñëè a è c íåíóëåâûå, òî ôóíêöèÿ ϕ ãåíåðèðóåò 2-ÊÌÀ â L2 (R+ )
(ìîäèôèöèðîâàííîå óñëîâèå Êîýíà â ñëó÷àå abc 6= 0 âûïîëíåíî äëÿ E = [0, 1),
à â ñëó÷àå a 6= 0, b = 0, c 6= 0 îíî èìååò ìåñòî äëÿ E = [0, 1/2) ∪ [3/4, 1) ∪
[3/2, 7/4)). Ôóíêöèÿ ϕ óäîâëåòâîðÿåò ìàñøòàáèðóþùåìó óðàâíåíèþ
                                              7
                                              X
                                 ϕ(x) =             ck ϕ(2x      k)
                                              k=0

ñ êîýôôèöèåíòàìè
                              1
                          c0 = (1 + a + b + c + α + β + γ),
                              4
                              1
                          c1 = (1 + a + b + c − α − β − γ),
                              4
                              1
                          c2 = (1 + a − b − c + α − β − γ),
                              4

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