Ряды Фурье и основы вейвлет-анализа. Фарков Ю.А. - 85 стр.

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ω
(m)
j
(f) := α
(m)
j
(f) − β
(m)
j
(f),
m j 0 ≤ j ≤ 2
m
− 1
f ∈ C
b
(∆) N m, l ∈ Z
+
N − 1 = 2
m
+ l 0 ≤ l ≤ 2
m
− 1 g
N
g
N
(t) =
(
(α
(m+1)
j
(f) − β
(m+1)
j
(f))/2 t ∈ I
(m+1)
j
, 0 ≤ j ≤ 2l + 1,
(α
(m)
j
(f) − β
(m)
j
(f))/2 t ∈ I
(m)
j
, l + 1 ≤ j ≤ 2
m
− 1,
f ∆
D
N
kf − g
N
k
∆
= inf
g∈D
N
kf − gk
∆
.
kf − g
N
k
∆
=
1
2
max
max
0≤j≤2l+1
ω
(m+1)
j
(f), max
l<j<2
m
ω
(m)
j
(f)
.
r
0
: R → {−1, 1} r
0
(x) = 1
x ∈ [0, 1/2) r
0
(x) = −1 x ∈ [1/2, 1) r
0
(x + 1) = r
0
(x)
x ∈ R. {r
n
| n ∈ Z
+
}
r
n
(x) = r
0
(2
n
x), x ∈ R.
r
n
1/2
n
n+1
−1
r
n
(x) = (−1)
k
, x ∈ I
(n+1)
k
, k ∈ Z ,
k/2
n+1
k ∈ Z
Z
I
(n)
k
r
n
(x) dx = 0, n ∈ Z
+
, k ∈ Z .
{r
n
| n ∈ Z
+
} L
2
(∆)
P
∞
n=0
|a
n
|
2
< +∞ a
n
∞
X
n=0
a
n
r
n
(x)
∆ f
L
q
(∆) q ∈ (0, +∞) q
A
q
∞
X
n=0
|a
n
|
2
!
1/2
≤
Z
∆
|f(x)|
q
dx
1/q
≤ B
q
∞
X
n=0
|a
n
|
2
!
1/2
,
                                   (m)               (m)           (m)
                         ωj (f ) := αj (f ) − βj (f ),
ãäå m è j  öåëûå ÷èñëà, 0 ≤ j ≤ 2m − 1.
   1.8. Ïóñòü f ∈ Cb (∆), N  íàòóðàëüíîå ÷èñëî è ïóñòü m, l ∈ Z+ îïðåäåëå-
íû èç óñëîâèé N − 1 = 2m + l, 0 ≤ l ≤ 2m − 1. Òîãäà ïîëèíîì gN , çàäàííûé
ôîðìóëîé
           (
                (m+1)         (m+1)                  (m+1)
              (αj     (f ) − βj     (f ))/2 äëÿ t ∈ Ij     , 0 ≤ j ≤ 2l + 1,
  gN (t) =      (m)         (m)                  (m)
              (αj (f ) − βj (f ))/2 äëÿ t ∈ Ij , l + 1 ≤ j ≤ 2m − 1,
îñóùåñòâëÿåò íàèëó÷øåå ðàâíîìåðíîå ïðèáëèæåíèå ôóíêöèè f íà ∆ ñðåäè
âñåõ ïîëèíîìîâ Õààðà èç DN , ò.å. äëÿ íåãî âûïîëíåíî ðàâåíñòâî
                                 kf − gN k∆ = inf kf − gk∆ .
                                                      g∈DN

Ïðè ýòîì
                                                                                            
                       1                                    (m+1)                (m)
           kf − gN k∆ = max                    max         ωj     (f ),    maxm ωj (f )          .
                       2                      0≤j≤2l+1                    l

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