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ϑʔϦΤڃల։
ϕΫτϧͷղͰ͢ΑͶʂʁ
20173݄29@ϓϩάϥϚͷͨΊͷֶLTձ
horiem
@yellowshippo
Slide 2
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1. ϑʔϦΤڃల։ͱɺ
ؔΛෳͷࡾ֯ؔʹղ͢Δ͜ͱ
2. ؔϕΫτϧͰ͋Δ
3. ϑʔϦΤڃల։ϕΫτϧͷղͰ͢ΑͶʂʁ
Slide 3
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1. ϑʔϦΤڃల։ͱɺ
ؔΛෳͷࡾ֯ؔʹղ͢Δ͜ͱ
2. ؔϕΫτϧͰ͋Δ
3. ϑʔϦΤڃల։ϕΫτϧͷղͰ͢ΑͶʂʁ
Slide 4
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ϑʔϦΤڃల։ͱ
• 2 π पظؔΛ sin ͱ cos ʹղʢʹల։ʣ
• Իॲཧɺը૾ॲཧʹར༻
• ϑʔϦΤมɺϑʔϦΤڃల։ͷ֦ு
• पظؔͰͳͯ͘ల։Մೳ
f(x) =
a0
2
+
1
X
k=1
(ak cos kx + bk sin kx)
ak =
1
⇡
Z ⇡
⇡
f(x) cos kx dx
bk =
1
⇡
Z ⇡
⇡
f(x) sin kx dx
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f
(
x
) =a0
2
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f(x) =
a0
2
+ a1 cos 1x + b1 sin 1x
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f(x) =
a0
2
+ a1 cos 1x + b1 sin 1x + a2 cos 2x + b2 sin 2x
Slide 8
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f(x) =
a0
2
+ a1 cos 1x + b1 sin 1x + a2 cos 2x + b2 sin 2x
+ a3 cos 3x + b3 sin 3x
Slide 9
Slide 9 text
f(x) =
a0
2
+ a1 cos 1x + b1 sin 1x + a2 cos 2x + b2 sin 2x
+ a3 cos 3x + b3 sin 3x + a4 cos 4x + b4 sin 4x + a5 cos 5x + b5 sin 5x
Slide 10
Slide 10 text
f(x) =
a0
2
+ a1 cos 1x + b1 sin 1x + a2 cos 2x + b2 sin 2x
+ a3 cos 3x + b3 sin 3x + a4 cos 4x + b4 sin 4x + a5 cos 5x + b5 sin 5x
+ a6 cos 6x + b6 sin 6x + a7 cos 7x + b7 sin 7x + a7 cos 7x + b7 sin 7x
Slide 11
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ͷٻΊํෳࡶ͗͢
• ͳͥ͜Μͳܗͳͷ͔ʁ
➡ ϕΫτϧ͔ΒͷྨਪͰཧղՄೳʂ
f(x) =
a0
2
+
1
X
k=1
(ak cos kx + bk sin kx)
ak =
1
⇡
Z ⇡
⇡
f(x) cos kx dx
bk =
1
⇡
Z ⇡
⇡
f(x) sin kx dx
Slide 12
Slide 12 text
1. ϑʔϦΤڃల։ͱɺ
ؔΛෳͷࡾ֯ؔʹղ͢Δ͜ͱ
2. ؔϕΫτϧͰ͋Δ
3. ϑʔϦΤڃల։ϕΫτϧͷղͰ͢ΑͶʂʁ
Slide 13
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ؔϕΫτϧͰ͋Δ
• ؔແͷͷू·Γ
Slide 14
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ؔϕΫτϧͰ͋Δ
• ؔແͷͷू·Γ
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ؔϕΫτϧͰ͋Δ
൪߸
v =
0
B
B
B
@
v1
v2
.
.
.
vn
1
C
C
C
A
• ؔແͷͷू·Γ
• ϕΫτϧͷू·Γ
➡ ؔແݶݸͷΛͬͨϕΫτϧ
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ϕΫτϧͷੵ
U · V = U1V1 + U2V2 + · · · + UnVn =
X
k
UkVk
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ϕΫτϧͷੵ
U · V = U1V1 + U2V2 + · · · + UnVn =
X
k
UkVk
Slide 18
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ؔͷੵ
fk
fk 1
fk+1
fk+2
gk+2
gk+1
gk 1
gk
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ؔͷੵ
fk
fk 1
fk+1
fk+2
gk+2
gk+1
gk 1
gk
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ؔͷੵ
fk
fk 1
fk+1
fk+2
gk+2
gk+1
gk 1
gk
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ؔͷੵ
fk
fk 1
fk+1
fk+2
gk+2
gk+1
gk 1
gk
< f, g >
= lim
x
!0
[
f1g1 x
+
f2g2 x
+ · · · +
fngn x
]
=
Z
f
(
x
)
g
(
x
)
dx
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ؔͷੵ
fk
fk 1
fk+1
fk+2
gk+2
gk+1
gk 1
gk
͜ͷΜ
͍͍ײ͡ʹ
ܾΊΕOK
< f, g >
= lim
x
!0
[
f1g1 x
+
f2g2 x
+ · · · +
fngn x
]
=
Z
f
(
x
)
g
(
x
)
dx
Slide 23
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ϕΫτϧͷܭࢉ
• ϕΫτϧͷ͞ |U| =
p
U · U
Slide 24
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ϕΫτϧͷܭࢉ
• ϕΫτϧͷ͞
• ϕΫτϧͷަ
|U| =
p
U · U
U · V = 0
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ϕΫτϧͷܭࢉ
• ϕΫτϧͷ͞
• ϕΫτϧͷަ
• ਖ਼نަجఈ
|U| =
p
U · U
U · V = 0
ei
· ej = ij
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ϕΫτϧͷܭࢉ
• ϕΫτϧͷ͞
• ϕΫτϧͷަ
• ਖ਼نަجఈ
• ϕΫτϧͷղ
|U| =
p
U · U
U · V = 0
ei
· ej = ij
U = (U · e1)e1 + (U · e2)e2 + . . .
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ؔͷܭࢉ
• ؔͷ͞ʢϊϧϜʣ ||f|| =
p
< f, f >
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ؔͷܭࢉ
• ؔͷ͞ʢϊϧϜʣ
• ؔͷަ
||f|| =
p
< f, f >
< f, g >= 0
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ؔͷܭࢉ
• ؔͷ͞ʢϊϧϜʣ
• ؔͷަ
• ਖ਼نަجఈ
||f|| =
p
< f, f >
< f, g >= 0
< hi, hj >= ij
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ؔͷܭࢉ
• ؔͷ͞ʢϊϧϜʣ
• ؔͷަ
• ਖ਼نަجఈ
• ؔͷղ
||f|| =
p
< f, f >
< f, g >= 0
< hi, hj >= ij
f =< f, h1 > h1+ < f, h2 > h2 + . . .
Slide 31
Slide 31 text
1. ϑʔϦΤڃల։ͱɺ
ؔΛෳͷࡾ֯ؔʹղ͢Δ͜ͱ
2. ؔϕΫτϧͰ͋Δ
3. ϑʔϦΤڃల։ϕΫτϧͷղͰ͢ΑͶʂʁ
Slide 32
Slide 32 text
ؔΛࡾ֯ؔͰղ
• ͜Ε͕ϑʔϦΤڃల։
• ੵΛ͍͍ײ͡ʹܾΊ͍ͨ
➡ ࡾ͕֯ؔਖ਼نަجఈͱͳΔΑ͏ʹੵΛఆٛ
f =
a0
2
+ < f, cos 1x > cos 1x+ < f, cos 2x > cos 2x + . . .
+ < f, sin 1x > sin 1x+ < f, sin 2x > sin 2x + . . .
Slide 33
Slide 33 text
ϑʔϦΤڃల։ͷͨΊͷੵ
< f, g >
=
1
⇡
Z ⇡
⇡
fg dx
Slide 34
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ϑʔϦΤڃల։ͷͨΊͷੵ
• ੵΛ͜ͷΑ͏ʹఆٛ͢ΕɺҎԼ͕ຬͨ͞ΕΔɿ
< f, g >
=
1
⇡
Z ⇡
⇡
fg dx
< cos ix, cos jx > = ij
< sin ix, sin jx > = ij
< cos ix, sin jx > = 0
Slide 35
Slide 35 text
f(x) =
a0
2
+
1
X
k=1
(ak cos kx + bk sin kx)
ak =
1
⇡
Z ⇡
⇡
f(x) cos kx dx
bk =
1
⇡
Z ⇡
⇡
f(x) sin kx dx
Slide 36
Slide 36 text
f(x) =
a0
2
+
1
X
k=1
(ak cos kx + bk sin kx)
ak =
1
⇡
Z ⇡
⇡
f(x) cos kx dx
bk =
1
⇡
Z ⇡
⇡
f(x) sin kx dx
=< f, cos kx >
=< f, sin kx >
=< f, cos kx >
=< f, sin kx >
Slide 37
Slide 37 text
·ͱΊ
• ϑʔϦΤڃల։ͱɺ
ؔΛෳͷࡾ֯ؔʹղ͢Δ͜ͱ
• ؔϕΫτϧͰ͋Δ
• ϑʔϦΤڃల։ϕΫτϧͷղͰͨ͠
• ͬͱৄ͘͠Γ͍ͨਓ
ʮhoriem ϑʔϦΤʯͱ͔Ͱ̶̶ͬͯͶ