Ряды Фурье и основы вейвлет-анализа. Фарков Ю.А. - 24 стр.

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{e
ikt
| k Z } (13)
L
2
[π, π]
π
Z
π
e
ikt
· e
ilt
dt = 2πδ
k,l
.
f L
2
[π, π]
1
2π
π
Z
π
|f(t)|
2
dt =
X
kZ
|c
k
|
2
f L
2
[π, π]
f(t) = e
it/2
c
k
=
1
2π
π
Z
π
f(t)e
i(k+1/2)t
dt =
1
2π
π
Z
π
cos (k + 1/2)t dt =
=
1
π
π
Z
0
cos (k + 1/2)t dt =
sin (kπ + π/2)
kπ + π/2
=
2(1)
k
(2k + 1)π
.
t (π, π)
e
it/2
=
2
π
X
kZ
(1)
k
2k + 1
e
ikt
. (14)
cos
t
2
i sin
t
2
=
2
π
X
kZ
(1)
k
2k + 1
(cos kt + i sin kt).
cos
t
2
=
2
π
X
kZ
(1)
k
2k + 1
cos kt =
2
π
+
4
π
X
k=1
(1)
k+1
4k
2
1
cos kt (15)
sin
t
2
=
2
π
X
kZ
(1)
k+1
2k + 1
sin kt =
8
π
X
k=1
(1)
k+1
k
4k
2
1
sin kt. (16)
    Ñèñòåìà ýêñïîíåíò
                               { eikt | k ∈ Z }                                          (13)
îðòîãîíàëüíà è ïîëíà â L2 [−π, π]. Ïðè ýòîì
                                   Zπ
                                         eikt · e−ilt dt = 2πδk,l .
                                   −π

Îòñþäà âèäíî, ÷òî êîýôôèöèåíòû (10) ÿâëÿþòñÿ êîýôôèöèåíòàìè Ôóðüå
ôóíêöèè f ∈ L2 [−π, π] ïî ñèñòåìå (13). Ðàâåíñòâî Ïàðñåâàëÿ äëÿ ñèñòåìû
(13) çàïèñûâàåòñÿ òàê:
                                        Zπ
                                    1                        X
                                             |f (t)|2 dt =         | ck |2
                                   2π
                                        −π                   k∈Z

è èìååò ìåñòî äëÿ ëþáîé ôóíêöèè f ∈ L2 [−π, π].
    Ïðèìåð 2. Äëÿ ôóíêöèè f (t) = e−it/2 êîýôôèöèåíòû Ôóðüå (10) âû÷èñ-
ëÿþòñÿ òàê:
                        Zπ                                    Zπ
                    1                               1
              ck =           f (t)e−i(k+1/2)t dt =                 cos (k + 1/2)t dt =
                   2π                              2π
                        −π                                   −π

                  Zπ
              1                            sin (kπ + π/2)    2(−1)k
            =          cos (k + 1/2)t dt =                =           .
              π                               kπ + π/2      (2k + 1)π
                  0
Ïîêàæåì, ÷òî äëÿ âñåõ t ∈ (−π, π) ñïðàâåäëèâî ðàâåíñòâî

                                    −it/2      2 X (−1)k ikt
                                   e         =            e .                            (14)
                                               π   2k + 1
                                                  k∈Z

Ïîëüçóÿñü ôîðìóëîé Ýéëåðà, çàïèøåì åãî â âèäå
                       t        t  2 X (−1)k
                cos      − i sin =            (cos kt + i sin kt).
                       2        2 π    2k + 1
                                               k∈Z

Îòäåëÿÿ äåéñòâèòåëüíûå è ìíèìûå ÷àñòè, âèäèì, ÷òî ðàâåíñòâî (14) âåðíî
òîãäà è òîëüêî òîãäà, êîãäà
                                                                      ∞
              t  2 X (−1)k          2 4 X (−1)k+1
           cos =            cos kt = +             cos kt                                (15)
              2 π    2k + 1         π π   4k 2 − 1
                         k∈Z                                        k=1
è                                                              ∞
                 t  2 X (−1)k+1          8 X (−1)k+1 k
              sin =             sin kt =               sin kt.                           (16)
                 2 π     2k + 1          π    4k 2 − 1
                             k∈Z                               k=1

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