Ряды Фурье и основы вейвлет-анализа. Фарков Ю.А. - 51 стр.

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РГГРУ | Москва

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P
j
f
@
@
@
@
@R
-
P
j−1
f
Q
j−1
f
@
@
@
@
@R
-
P
j−2
f
Q
j−2
f
@
@
@
@
@R
-
. . . P
j−s
f
. . . Q
j−s
f
f
f(t) = 9χ
[0,1/2)
(t) + 7χ
[1/2,1)
(t) + 3χ
[1,3/2)
(t) + 5χ
[3/2,2)
(t), t ∈ R,
f(t) = 9 ϕ(2t) + 7 ϕ(2t − 1) + 3 ϕ(2t − 2) + 5 ϕ(2t − 3).
ϕ
1k
(t) =
√
2ϕ(2t − k),
f = P
1
f =
3
X
k=0
a
1k
ϕ
1k
,
a
10
= 9/
√
2, a
11
= 7/
√
2, a
12
= 3/
√
2, a
13
= 5/
√
2.
a
00
=
a
10
+ a
11
√
2
= 8, d
00
=
a
10
− a
11
√
2
= 1,
a
01
=
a
12
+ a
13
√
2
= 4, d
01
=
a
12
− a
13
√
2
= −1.
f = P
0
f + Q
0
f, P
0
f = 8ϕ
00
+ 4ϕ
01
, Q
0
f = ψ
00
− ψ
01
.
a
−1,0
=
a
00
+ a
01
√
2
= 6
√
2, d
−1,0
=
a
00
− a
01
√
2
= 2
√
2.
  Ñõåìàòè÷íî:


                              -                        -                         -
           Pj f                   Pj−1 f                     Pj−2 f                    . . . Pj−s f
                     @                      @                             @
                      @                      @                             @
                          @                   @                             @
                           @                         @                         @
                            R
                            @                         R
                                                      @                         R
                                                                                @
                                  Qj−1 f                   Qj−2 f                     . . . Qj−s f




                                                    Ðèñ. 1




  Ïðèìåð 1. Ôóíêöèÿ f , çàäàííàÿ ðàâåíñòâîì
      f (t) = 9χ[0,1/2) (t) + 7χ[1/2,1) (t) + 3χ[1,3/2) (t) + 5χ[3/2,2) (t),         t ∈ R,
âûðàæàåòñÿ ÷åðåç ìàñøòàáèðóþùóþ ôóíêöèþ Õààðà ïî ôîðìóëå
          f (t) = 9 ϕ(2t) + 7 ϕ(2t − 1) + 3 ϕ(2t − 2) + 5 ϕ(2t − 3).
                    √
Ïîñêîëüêó ϕ1k (t) = 2ϕ(2t − k), òî
                                                3
                                                X
                                   f = P1 f =         a1k ϕ1k ,
                                                k=0
ãäå               √            √                            √                  √
          a10 = 9/ 2, a11 = 7/ 2,                   a12 = 3/ 2,        a13 = 5/ 2.
Ïîëüçóÿñü ôîðìóëàìè (13), íàõîäèì
                               a10 + a11             a10 − a11
                      a00 =       √       = 8, d00 =    √       = 1,
                                    2                     2
                              a12 + a13             a12 − a13
                     a01    =    √       = 4, d01 =    √       = −1.
                                   2                     2
Çíà÷èò,
          f = P0 f + Q0 f,        ãäå P0 f = 8ϕ00 + 4ϕ01 ,            Q0 f = ψ00 − ψ01 .
Äàëåå, ïîâòîðíî ïðèìåíÿÿ (13), ïîëó÷àåì
                            a00 + a01    √                    a00 − a01    √
                  a−1,0 =      √      = 6 2,        d−1,0 =      √      = 2 2.
                                 2                                 2
                                               51

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