Ряды Фурье и основы вейвлет-анализа. Фарков Ю.А. - 79 стр.

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B
B {N
m
(t)}
N
1
(t) = χ
[0,1)
(t), N
m
(t) =
t
m − 1
N
m−1
(t)+
m − t
m − 1
N
m−1
(t−1), m = 2, 3, . . . .
(1)
m = 2
N
2
(t) = t N
1
(t) + (2 − t) N
1
(t − 1)
N
2
(t) =
t, 0 ≤ t < 1,
2 − t, 1 ≤ t < 2,
0, t ∈ R \ [0, 2)
1
0
N
m
[k, k + 1] k = 0, 1, . . . , m − 1
m − 1
2
0
N
m
∈ C
m−2
(R) m ≥ 2
3
0
supp N
m
= [0, m] N
m
> 0 t ∈ (0, m)
4
0
N
m
(t) N
m
(2t − k) k = 0, 1, . . . , m
N
m
(t) = 2
−m+1
m
X
k=0
C
m
k
N
m
(2t − k), (2)
C
m
k
=
m!
k!(m − k)!
5
0
N
m
= N
1
∗ N
m−1
N
m
(t) =
Z
1
0
N
m−1
(t − τ) dτ
m ≥ 2
6
0
B
b
N
m
(ξ) = e
−imξ/2
sin(ξ/2)
ξ/2
m
.
5
0
6
0
   Ÿ 9. Íîðìàëèçîâàííûå B -ñïëàéíû è âåéâëåòû Áàòëà-Ëåìàðüå

   Ñèñòåìà íîðìàëèçîâàííûõ B -ñïëàéíîâ {Nm (t)} îïðåäåëÿåòñÿ ôîðìóëàìè
                                   t            m−t
N1 (t) = χ[0,1) (t),   Nm (t) =       Nm−1 (t)+     Nm−1 (t−1),           m = 2, 3, . . . .
                                  m−1           m−1
                                                                                       (1)
Èç ôîðìóëû (1) ïðè m = 2 ïîëó÷àåì

                          N2 (t) = t N1 (t) + (2 − t) N1 (t − 1)
è, ñëåäîâàòåëüíî,
                                    
                                     t,                0 ≤ t < 1,
                          N2 (t) =  2 − t,              1 ≤ t < 2,
                                    0,                 t ∈ R \ [0, 2)

   Ñïðàâåäëèâû ñâîéñòâà:
  10 . Nm íà êàæäîì îòðåçêå [k, k + 1], k = 0, 1, . . . , m − 1, ñîâïàäàåò ñ àëãå-
áðàè÷åñêèì ïîëèíîìîì ñòåïåíè m − 1.
  20 . Nm ∈ C m−2 (R) äëÿ m ≥ 2.
  30 . supp Nm = [0, m] è Nm > 0 äëÿ âñåõ t ∈ (0, m).
  40 . Nm (t) âûðàæàåòñÿ ÷åðåç Nm (2t − k), k = 0, 1, . . . , m, ïî ôîðìóëå
                                             m
                                             X
                                     −m+1
                          Nm (t) = 2                   Ckm Nm (2t − k),                (2)
                                                 k=0

ãäå
                                                 m!
                                    Ckm =
                                             k!(m − k)!
 áèíîìèàëüíûå êîýôôèöèåíòû.
   50 . Nm = N1 ∗ Nm−1 , ò.å.
                                        Z    1
                             Nm (t) =            Nm−1 (t − τ ) dτ
                                         0

äëÿ m ≥ 2.
  60 . Ïðåîáðàçîâàíèÿ Ôóðüå íîðìàëèçîâàííûõ B -ñïëàéíîâ íàõîäÿòñÿ ïî
ôîðìóëå                                   m
                           bm (ξ) = e−imξ/2             sin(ξ/2)
                           N                                        .
                                                           ξ/2

   Îòìåòèì, ÷òî ñâîéñòâà 50 è 60 ïðèâîäèëèñü â óïðàæíåíèè 1.8. Èç ôîðìóëû
(2) èìååì, â ÷àñòíîñòè,

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