Ряды Фурье и основы вейвлет-анализа. Фарков Ю.А. - 23 стр.

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S
2n
(t) S
2n1
(t)
g
n
(t) = 2
n
X
k=1
sin(2k 1)t
2k 1
. (8)
g
0
n
(t) =
sin 2nt
sin t
,
π/2n g
n
(t)
g
n
(π/2n) =
n
X
k=1
sin(2k 1)(π/2n)
(2k 1)(π/2n)
·
π
n
π
Z
0
sin t
t
dt 1, 8519,
f(0 + 0) = π/2 1, 5707 n
g
n
(π/2n) f(π/2n)
2
X
k=1
1
(2k 1)
2
=
π
2
8
.
X
kZ
c
k
e
ikt
. (9)
e
ikt
= cos kt + i sin kt, cos kt = (e
ikt
+ e
ikt
)/2, sin kt = i(e
ikt
+ e
ikt
)/2.
c
k
=
1
2π
π
Z
π
f(t)e
ikt
dt (10)
c
0
= a
0
/2, c
k
= (a
k
ib
k
)/2, c
k
= (a
k
+ ib
k
)/2, k N. (11)
S
0
(t) = c
0
, S
n
(t) =
n
X
k=n
c
k
e
ikt
, n Z. (12)
  ×àñòè÷íûå ñóììû S2n (t) è S2n−1 (t) ðÿäà (7) ñîâïàäàþò ñ íå÷åòíîé ôóíê-
öèåé
                                          n
                                          X sin(2k − 1)t
                             gn (t) = 2                              .                             (8)
                                                     2k − 1
                                           k=1
Ïðîèçâîäíàÿ ýòîé ôóíêöèè âû÷èñëÿåòñÿ ïî ôîðìóëå
                                                 sin 2nt
                                     gn0 (t) =           ,
                                                   sin t
è, ñëåäîâàòåëüíî, â òî÷êå π/2n ôóíêöèÿ gn (t) èìååò ëîêàëüíûé ìàêñèìóì.
Èç ôîðìóëû (8) ïîëó÷èì
                    n                                                π
                      sin(2k − 1)(π/2n) π
                                           Z
                    X                        sin t
        gn (π/2n) =                    · →         dt ≈ 1, 8519,
                        (2k − 1)(π/2n) n       t
                       k=1                                       0

â òî âðåìÿ êàê f (0 + 0) = π/2 ≈ 1, 5707. Òàêèì îáðàçîì, ïðè áîëüøèõ n
çíà÷åíèÿ gn (π/2n) îòëè÷àþòñÿ îò f (π/2n) ïðèáëèçèòåëüíî íà 18 %.
                                                                   2
   Îòìåòèì, ÷òî èç ðàâåíñòâà Ïàðñåâàëÿ (5) è ðàçëîæåíèÿ (7) ñëåäóåò ðàâåí-
ñòâî                        ∞
                           X       1       π2
                                         = .
                               (2k − 1)2    8
                                    k=1

  Êîìïëåêñíàÿ ôîðìà òðèãîíîìåòðè÷åñêîãî ðÿäà Ôóðüå
                            X
                               ck eikt .                                                           (9)
                                          k∈Z

ïîëó÷àåòñÿ èç (2) è (3) ñ ïîìîùüþ ôîðìóë Ýéëåðà
 eikt = cos kt + i sin kt,   cos kt = (eikt + e−ikt )/2,                 sin kt = i(e−ikt + eikt )/2.
Êîýôôèöèåíòû ðÿäà (9) âû÷èñëÿþòñÿ ïî ôîðìóëå
                                           Zπ
                                     1
                               ck =              f (t)e−ikt dt                                    (10)
                                    2π
                                           −π

è ñâÿçàíû ñ êîýôôèöèåíòàìè (2) ðàâåíñòâàìè

       c0 = a0 /2, ck = (ak − ibk )/2, c−k = (ak + ibk )/2,                         k ∈ N.        (11)
  Ñîîòâåòñòâåííî, ÷àñòè÷íûå ñóììû (4) ïðèìóò âèä
                                                     n
                                                     X
                    S0 (t) = c0 ,     Sn (t) =           ck eikt ,       n ∈ Z.                   (12)
                                                 k=−n

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