Logismoc II, Seirèc Fourieryanath/teaching/calculus2/...An ìmwc èpairna peiroucìrouc sto parap...
Transcript of Logismoc II, Seirèc Fourieryanath/teaching/calculus2/...An ìmwc èpairna peiroucìrouc sto parap...
Logismoc II, Seirèc Fourier
A. N. Giannakìpouloc
Tm ma Statistik c
O.P.A
Earinì Ex�mhno 2017
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 1 / 43
Periodikèc sunart seic
MÐa sun�rthsh f onom�zetai periodik me perÐodo T an
f (x + T ) = f (x), ∀ x ∈ R.
An mia sun�rthsh èqei perÐodo T tìte ja eqei ka perÐodo nT gia k�je
n ∈ N.
Autì sumbaÐnei epeid
f (x + nT ) = f (x + (n − 1)T + T ) = f (x + (n − 1)T )
= f (x + (n − 2)T + T ) = f (x + (n − 2)T )
= · · · = f (x).
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 2 / 43
Oi peperasmènoi grammikoÐ sundiasmoÐ periodik¸n sunart sewn me thn
Ðdia perÐodo mac dÐnoun epÐshc periodikèc sunart seic.
Pr�gmati, an oi sunart seic f1, · · · , fn eÐnai periodikec me perÐodo Ttìte kai h sunarthsh
f =n∑
i=1
ci fi , ci ∈ R
eÐnai epÐshc periodik me thn Ðdia perÐodo.
f (x + T ) =n∑
i=1
ci fi (x + T )
=n∑
i=1
ci fi (x) = f (x).
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 3 / 43
An ìmwc èpairna �peirouc ìrouc sto parap�nw �jroisma (kai to
�jroisma autì mporoÔse na oristeÐ!) tìte ti ja sunèbaine?
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 4 / 43
Oi sunart seic cos(nx) kai sin(nx), n ∈ N.
Oi sunarthseic autèc eÐnai periodikec me periodo Tn = 2πn .
'Eqoun thn idiìthta∫ π
−πsin(nx) sin(mx)dx =
∫ π
−πcos(nx) cos(mx)dx = π δn,m,∫ π
−πcos(nx) sin(mx)dx = 0
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 5 / 43
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 6 / 43
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 7 / 43
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 8 / 43
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 9 / 43
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 10 / 43
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 11 / 43
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 12 / 43
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 13 / 43
Seirèc Fourier
An f kat� tm mata suneqhc sto [−π, π] h seir� Fourier thc f eÐnai h
seir�
a02
+∞∑n=1
(an cos(nx) + bn sin(nx)),
me suntelestèc,
a0 =1
π
∫ π
−πf (x)dx ,
an =1
π
∫ π
−πf (x) cos(nx)dx ,
bn =1
π
∫ π
−πf (x) sin(nx)dx ,
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 14 / 43
Par�deigma
f (x) =
{0 an −π ≤ x < 0,1 an 0 ≤ x ≤ π,
me f (x + 2π) = f (x).
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 15 / 43
MporoÔme na upologÐsoume touc suntelestèc thc seir�c
a0 = 1,
an = 0,
bn = − 1
nπ(cos(nπ)− 1) =
{0 an n artio2nπ an n perittì
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 16 / 43
H seir� Fourier thc f eÐnai h
1
2+
2
πsin(x) +
2
3πsin(3x) +
2
5πsin(5x) + · · ·
=1
2+∞∑k=1
2
(2k − 1)πsin((2k − 1)x)
ìpou anaparast same touc perittoÔc me n = 2k − 1, k = 1, 2, · · ·
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 17 / 43
Ta merika ajroÐsmata thc seir�c Fourier gia thn f proseggÐzoun thn
sun�rthsh f .
An
SK (x) =1
2+
K∑k=1
2
(2k − 1)πsin((2k − 1)x), x ∈ [−π, π],
tìte ìso pio meg�lo ginetai to K tìso pio mikrì to sf�lma
|SK (s)− f (x)|
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 18 / 43
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 19 / 43
Je¸rhma
'Estw f : [−π, π]→ R mia sunarthsh kat� tm mata auneqhc kai
F (x) :=a02
+∞∑n=1
(an cos(nx) + bn sin(nx)),
h seir� Fourier pou par�getai apo thn sun�rthsh aut .
An to shmeio x ∈ [−π, π] einai shmeio sunèqeiac thc f tìte
F (x) = f (x).
An to shmeÐo x ∈ [−π, π] einai shmeÐo asunèqeiac thc f tote
F (x) = 12(f (x
+) + f (x−)).
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 20 / 43
H tautìthta tou Parseval
An f : [−π, π]→ R sun�rthsh gia thn opoÐa isqÔei ìti∫ π−π |f (x)|
2dx <∞, kai
F (x) =a02
+∞∑n=1
(an cos(nx) + bn sin(nx)),
h seir� Fourier gia thn sun�rthsh f , tìte
1
π
∫ π
−π|f (x)|2dx =
1
2a20 +
∞∑n=1
(a2n + b2n).
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 21 / 43
'Artiec kai perittèc sunart seic
Orismìc
MÐa sunarthsh f : I → R onomazetai
�rtia an f (−x) = f (x) gia k�je x ∈ I ,
peritt an f (−x) = −f (x) gia k�je x ∈ I .
Par�deigma
Oi sunart seic cos(nx) eÐnai �rtiec sunart seic enw oi sin(nx)eÐnai perittèc.
Oi sunart seic x2m eÐnai �rtiec sunart seic en¸ oi x2m+1 eÐnai
perittèc.
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 22 / 43
Ac p�roume mia sunarthsh f : [−π, π]→ R.
An f �rtia tìte h seir� Fourier thc f ja perilamb�nei mono ìrouc
cos(nx) kai ton stajerì oro!
An f peritt tìte h seir� Fourier thc f ja perilamb�nei mono ìrouc
sin(nx)
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 23 / 43
Par�deigma
DÐnetai h sun�rthsh f : [−π, π]→ R, h opoÐa orÐzetai wc f (x) = x2.
Upologiste thn seir� Fourier kai qrhsimopoi ste thn gia na deixete oti
1− 1
22+
1
32− · · · =
∞∑n=1
(−1)n+1
n2=π2
12,
kai oti
1 +1
22+
1
32+ · · · = π2
6.
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 24 / 43
H sunarthsh f eÐnai �rtia sunep¸c h seir� Fourier ja perièqei mìno
sunhmÐtona.
a0 =1
π
∫ π
−πf (x)dx =
1
π
∫ π
−πx2dx =
2
3π2,
an =1
π
∫ π
−πf (x) cos(nx)dx =
1
π
∫ π
−πx2 cos(nx)dx
=1
π
[x2
nsin(nx)
]π−π− 1
π
∫ π
−π
2x
nsin(nx)dx
= 0 +1
π
[2x
n2cos(nx)
]π−π− 2
πn
∫ π
−π
1
ncos(nx)dx
=4
n2(−1)n.
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 25 / 43
H seir� Fourier gia thn f einai h
F (x) =π2
3+∞∑n=1
4
n2(−1)n cos(nx).
Gia x = 0 h seir� Fourier F (0) sugklÐnei sthn tim f (0) = 0, sunep¸c
0 =π2
3+∞∑n=1
4
n2(−1)n cos(n0),
kai epeid cos(n0) = 1 èqoume oti
π2
12=∞∑n=1
(−1)n+1
n2
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 26 / 43
Gia x = π h seir� Fourier F (0) sugklÐnei sthn tim f (π) = π2, sunep¸c
π2 =π2
3+∞∑n=1
4
n2(−1)n cos(nπ),
kai epeid cos(nπ) = (−1)n èqoume oti
π2
6=∞∑n=1
1
n2.
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 27 / 43
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 28 / 43
Ti sumbaÐnei an f : [−L, L]→ R?O metasqhmatismìc y = π
Lx apeikonÐzei to di�sthma [−L, L] stodi�sthma [−π, π].SÔmfwna me thn parat rhsh aut h seir� Fourier gia mia periodik
sunarthsh me perÐodo 2L, me basikì pedÐo orismoÔ f : [−L, L]→ R ja
èqei thn morf ,
F (x) =a02
+∞∑n=1
(an cos
(nπLx)+ bn sin
(nπLx))
gia
a0 =1
L
∫ L
−Lf (x)dx ,
an =1
L
∫ L
−Lf (x) cos
(nπLx)dx ,
bn =1
L
∫ L
−Lf (x) sin
(nπLx)dx ,
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 29 / 43
'Artiec epekt�seic sunart sewn kai sunhmitonoeideÐc
seirèc
Ac upojèsoume oti mia sunarthsh orÐzetai mìno sto [0, L] kai ìqi seìlo to [−L, L], dhlad f : [0, L]→ R.
Orismìc
H �rtia epèktash thc f : [0, L]→ R eÐnai mia sun�rthsh
fe : [−L, L]→ R h opoÐa orÐzetai wc
fe(x) =
{f (−x) gia x ∈ [L, 0),f (x) gia x ∈ [0, L],
H sunarthsh aut epekteinetai periodika gia x 6∈ [−L, L] sÔmfwna me
ton kanìna f (x + 2L) = f (x) gia k�je x ∈ R.
H fe eÐnai mia �rtia sunarthsh h opoÐa tautÐzetai me thn arqik
sunarthsh f sto di�sthma [0, L]
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 30 / 43
H seira Fourier gia thn fe ja eÐnai mia seir� pou ja perièqei mìno
ìrouc cos(nπL x)kai ton stajerì ìro kai onomazetai seir� sunhmitìnwn
( half range cosine series)H seir� aut apoteleÐ mia prosèggish thc arqikhc sunarthsh f sto
di�sthma [0, L], mìno apo sunhmÐtona kai ton stajerì ìro,
Fc = a0 +∞∑n=1
an cos(nπ
Lx),
a0 =2
L
∫ L
−Lf (x)dx ,
an =2
L
∫ L
−Lf (x) cos
(nπLx)dx .
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 31 / 43
Perittec epekt�seic sunart sewn kai hmitonoeideÐc seirèc
Ac upojèsoume oti mia sunarthsh orÐzetai mìno sto [0, L] kai ìqi seìlo to [−L, L], dhlad f : [0, L]→ R.
Orismìc
H peritt epèktash thc f : [0, L]→ R eÐnai mia sun�rthsh
fo : [−L, L]→ R h opoÐa orÐzetai wc
fo(x) =
{−f (−x) gia x ∈ [L, 0),f (x) gia x ∈ [0, L],
H sunarthsh aut epekteinetai periodika gia x 6∈ [−L, L] sÔmfwna me
ton kanìna f (x + 2L) = f (x) gia k�je x ∈ R.
H fe eÐnai mia peritt sunarthsh h opoÐa tautÐzetai me thn arqik
sunarthsh f sto di�sthma [0, L]
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 32 / 43
H seira Fourier gia thn fo ja eÐnai mia seir� pou ja perièqei mìno
ìrouc sin(nπL x)kai kai onomazetai seir� hmitìnwn ( half range sine
series)H seir� aut apoteleÐ mia prosèggish thc arqikhc sunarthsh f sto
di�sthma [0, L], mìno apo hmÐtona,
Fs =∞∑n=1
bn sin(nπ
Lx),
bn =2
L
∫ L
−Lf (x) sin
(nπLx)dx .
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 33 / 43
Par�deigma
DÐnetai h sun�rthsh f : [0, π]→ R me tÔpo f (x) = x + x2.
BreÐte thn �rtia kai thn peritt epèktash thc sun�rthshc aut c sto
di�sthma [−π, π].
UpologÐste thn seir� sunhmitìnwn kai thn seir� hmitìnwn (half rangeseries) gia thn sun�rthsh f .
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 34 / 43
H �rtia epèktash fe orÐzetai wc
fe(x) =
{f (−x) gia x ∈ [−π, 0),f (x) gia x ∈ [0, π],
Epeid f (−x) = (−x) + (−x)2 = −x + x2, èqoume ìti
fe(x) =
{−x + x2 gia x ∈ [−π, 0),x + x2 gia x ∈ [0, π],
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 35 / 43
H peritt epèktash fo orÐzetai wc
fo(x) =
{−f (−x) gia x ∈ [−π, 0),f (x) gia x ∈ [0, π],
Epeid −f (−x) = −(−x)− (−x)2 = x − x2, èqoume ìti
fo(x) =
{x − x2 gia x ∈ [−π, 0),x + x2 gia x ∈ [0, π],
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 36 / 43
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 37 / 43
H seir� sunhmitìnwn gia thn f eÐnai
Fc(x) =a02
+∞∑n=1
an cos(nx), x ∈ (0, π),
a0 =2
π
∫ π
0f (x)dx =
2
π
∫ π
0(x + x2)dx = 2
(π2
3+π
2
),
an =2
π
∫ π
0f (x) cos(nx)dx =
2
π
∫ π
0(x + x2) cos(nx)dx
=2
π
[2π + 1
n2(−1)n − 1
n2
]'Ara
Fc(x) =
(π2
3+π
2
)+
2
π
∞∑n=1
[2π + 1
n2(−1)n − 1
n2
]cos(nx), x ∈ (0, π).
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 38 / 43
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 39 / 43
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 40 / 43
H seir� hmitìnwn gia thn f eÐnai
Fs(x) =∞∑n=1
bn sin(nx), x ∈ (0, π),
bn =2
π
∫ π
0f (x) sin(nx)dx =
2
π
∫ π
0(x + x2) sin(nx)dx
=2
π
[π2 + π
n(−1)n+1 +
2
n3((−1)n − 1)
]'Ara
Fs(x) =2
π
∞∑n=1
[π2 + π
n(−1)n+1 +
2
n3((−1)n − 1)
]sin(nx), x ∈ (0, π).
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 41 / 43
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 42 / 43
A. N. Giannakìpouloc (O.P.A) Logismoc II Earinì Ex�mhno 2017 43 / 43