Ряды Фурье и основы вейвлет-анализа. Фарков Ю.А. - 97 стр.

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b
s
= m(sp
−n
) m
p n
p
n
−1
X
α=0
a
α
w
α
(sp
−n
) = b
s
, 0 ≤ s ≤ p
n
− 1. (7)
a
α
=
1
p
n
p
n
−1
X
s=0
b
s
w
α
(s/p
n
), 0 ≤ α ≤ p
n
− 1. (8)
ϕ ∈ L
2
(R
+
)
p
n
−1
X
α=0
a
α
= 1 supp ϕ ⊂ [ 0, p
n−1
].
{ϕ(· k)|k ∈ Z
+
} L
2
(R
+
)
m(0) = 1, |m(ω)|
2
+|m(ω+1/p)|
2
+···+|m(ω+(p−1)/p)|
2
= 1, ω ∈ [ 0, 1/p ).
(9)
b
0
= 1, |b
l
|
2
+ |b
l+p
n−1
|
2
+ ··· + |b
l+(p−1)p
n−1
|
2
= 1, 0 ≤ l ≤ p
n−1
− 1, (10)
b
l
= m(lp
−n
) p = n = 2
b
0
= 1, b
2
= 0, |b
1
|
2
+ |b
3
|
2
= 1,
p = 3, n = 2
b
0
= 1, b
3
= b
6
= 0, |b
1
|
2
+ |b
4
|
2
+ |b
7
|
2
= |b
2
|
2
+ |b
5
|
2
+ |b
8
|
2
= 1.
n
b
p
n−1
= b
2 p
n−1
= ··· = b
(p−1)p
n−1
= 0.
M ⊂ [0, 1)
T
p
M =
p−1
[
l=0
n
l/p + ω/p | ω ∈ M
o
.
  4.4. Ïóñòü bs = m(sp −n )  çíà÷åíèÿ, ïðèíèìàåìûå ìàñêîé m óðàâíåíèÿ
(4) íà p -àäè÷åñêèõ èíòåðâàëàõ ðàíãà n, ò.å.
                       n
                      pX −1
                              aα wα (sp−n ) = bs ,       0 ≤ s ≤ p n − 1.                       (7)
                       α=0

Òîãäà
                                 n
                           p −1
                         1 X
                    aα = n      bs wα (s/p n ),            0 ≤ α ≤ p n − 1.                     (8)
                        p s=0
è îáðàòíî, èç ôîðìóë (8) ñëåäóþò (7).
   Äëÿ ðåàëèçàöèè ïðåîáðàçîâàíèé (7) è (8) ìîæíî ïðèìåíÿòü áûñòðûå àë-
ãîðèòìû Âèëåíêèíà  Êðåñòåíñîíà (ñì. 2.7 - 2.10 è 3.3).
  4.5. Åñëè ôóíêöèÿ ϕ ∈ L2 (R+ ) èìååò êîìïàêòíûé íîñèòåëü, óäîâëåòâîðÿ-
åò óðàâíåíèþ (4) è óñëîâèþ (5), òî
                         n
                        pX −1
                                aα = 1 è       supp ϕ ⊂ [ 0, p n−1 ].
                        α=0

Êðîìå òîãî, åñëè ñèñòåìà {ϕ(· k)| k ∈ Z+ } îðòîíîðìèðîâàíà â L2 (R+ ), òî
ìàñêà (6) óäîâëåòâîðÿåò óñëîâèÿì


m(0) = 1,      |m(ω)|2 +|m(ω+1/p)|2 +· · ·+|m(ω+(p−1)/p)|2 = 1,                    ω ∈ [ 0, 1/p ).
                                                                                              (9)
  Óñëîâèÿ (9) ýêâèâàëåíòíû ðàâåíñòâàì
 b0 = 1,    |bl |2 + |bl+p n−1 |2 + · · · + |bl+(p−1)p n−1 |2 = 1,      0 ≤ l ≤ p n−1 − 1, (10)
ãäå bl = m(lp −n ).  ÷àñòíîñòè, äëÿ p = n = 2 ïîëó÷àþòñÿ òðè ðàâåíñòâà
                         b0 = 1,       b2 = 0,        |b1 |2 + |b3 |2 = 1,
à äëÿ p = 3, n = 2 èìååì
     b0 = 1,     b3 = b6 = 0,        |b1 |2 + |b4 |2 + |b7 |2 = |b2 |2 + |b5 |2 + |b8 |2 = 1.
Êðîìå òîãî, èç (10) ïðè ëþáîì n ñëåäóþò ðàâåíñòâà
                         bp n−1 = b2 p n−1 = · · · = b(p−1)p n−1 = 0.

  Ïóñòü M ⊂ [0, 1) è
                                      p−1
                                      [n                      o
                           Tp M =            l/p + ω/p | ω ∈ M .
                                       l=0

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