horiem
March 29, 2017
20k

# フーリエ級数展開は ベクトルの分解ですよね！？

2017年3月29日@プログラマのための数学LT会

March 29, 2017

## Transcript

1. ϑʔϦΤڃ਺ల։͸
ϕΫτϧͷ෼ղͰ͢ΑͶʂʁ
2017೥3݄29೔@ϓϩάϥϚͷͨΊͷ਺ֶLTձ
horiem
@yellowshippo

2. 1. ϑʔϦΤڃ਺ల։ͱ͸ɺ
ؔ਺Λෳ਺ͷࡾ֯ؔ਺ʹ෼ղ͢Δ͜ͱ
2. ؔ਺͸ϕΫτϧͰ͋Δ
3. ϑʔϦΤڃ਺ల։͸ϕΫτϧͷ෼ղͰ͢ΑͶʂʁ

3. 1. ϑʔϦΤڃ਺ల։ͱ͸ɺ
ؔ਺Λෳ਺ͷࡾ֯ؔ਺ʹ෼ղ͢Δ͜ͱ
2. ؔ਺͸ϕΫτϧͰ͋Δ
3. ϑʔϦΤڃ਺ల։͸ϕΫτϧͷ෼ղͰ͢ΑͶʂʁ

4. ϑʔϦΤڃ਺ల։ͱ͸
• 2 π पظؔ਺Λ sin ͱ cos ʹ෼ղʢʹల։ʣ
• Ի੠ॲཧ΍ɺը૾ॲཧʹར༻
• ϑʔϦΤม׵͸ɺϑʔϦΤڃ਺ల։ͷ֦ு
• पظؔ਺Ͱͳͯ͘΋ల։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

5. f
(
x
) =a0
2

6. f(x) =
a0
2
+ a1 cos 1x + b1 sin 1x

7. f(x) =
a0
2
+ a1 cos 1x + b1 sin 1x + a2 cos 2x + b2 sin 2x

8. f(x) =
a0
2
+ a1 cos 1x + b1 sin 1x + a2 cos 2x + b2 sin 2x
+ a3 cos 3x + b3 sin 3x

9. 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

10. 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

11. ܎਺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

12. 1. ϑʔϦΤڃ਺ల։ͱ͸ɺ
ؔ਺Λෳ਺ͷࡾ֯ؔ਺ʹ෼ղ͢Δ͜ͱ
2. ؔ਺͸ϕΫτϧͰ͋Δ
3. ϑʔϦΤڃ਺ల։͸ϕΫτϧͷ෼ղͰ͢ΑͶʂʁ

13. ؔ਺͸ϕΫτϧͰ͋Δ
• ؔ਺͸ແ਺ͷ఺ͷू·Γ

14. ؔ਺͸ϕΫτϧͰ͋Δ
• ؔ਺͸ແ਺ͷ఺ͷू·Γ

15. ؔ਺͸ϕΫτϧͰ͋Δ
੒෼൪߸
੒෼
v =
0
B
B
B
@
v1
v2
.
.
.
vn
1
C
C
C
A
• ؔ਺͸ແ਺ͷ఺ͷू·Γ
• ϕΫτϧ΋੒෼ͷू·Γ
➡ ؔ਺͸ແݶݸͷ੒෼Λ΋ͬͨϕΫτϧ

16. ϕΫτϧͷ಺ੵ
U · V = U1V1 + U2V2 + · · · + UnVn =
X
k
UkVk

17. ϕΫτϧͷ಺ੵ
U · V = U1V1 + U2V2 + · · · + UnVn =
X
k
UkVk

18. ؔ਺ͷ಺ੵ
fk
fk 1
fk+1
fk+2
gk+2
gk+1
gk 1
gk

19. ؔ਺ͷ಺ੵ
fk
fk 1
fk+1
fk+2
gk+2
gk+1
gk 1
gk

20. ؔ਺ͷ಺ੵ
fk
fk 1
fk+1
fk+2
gk+2
gk+1
gk 1
gk

21. ؔ਺ͷ಺ੵ
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

22. ؔ਺ͷ಺ੵ
fk
fk 1
fk+1
fk+2
gk+2
gk+1
gk 1
gk
͜ͷ΁Μ
͍͍ײ͡ʹ
ܾΊΕ͹OK
< f, g >
= lim
x
!0
[
f1g1 x
+
f2g2 x
+ · · · +
fngn x
]
=
Z
f
(
x
)
g
(
x
)
dx

23. ϕΫτϧͷܭࢉ
• ϕΫτϧͷ௕͞ |U| =
p
U · U

24. ϕΫτϧͷܭࢉ
• ϕΫτϧͷ௕͞
• ϕΫτϧͷ௚ަ
|U| =
p
U · U
U · V = 0

25. ϕΫτϧͷܭࢉ
• ϕΫτϧͷ௕͞
• ϕΫτϧͷ௚ަ
• ਖ਼ن௚ަجఈ
|U| =
p
U · U
U · V = 0
ei
· ej = ij

26. ϕΫτϧͷܭࢉ
• ϕΫτϧͷ௕͞
• ϕΫτϧͷ௚ަ
• ਖ਼ن௚ަجఈ
• ϕΫτϧͷ෼ղ
|U| =
p
U · U
U · V = 0
ei
· ej = ij
U = (U · e1)e1 + (U · e2)e2 + . . .

27. ؔ਺ͷܭࢉ
• ؔ਺ͷ௕͞ʢϊϧϜʣ ||f|| =
p
< f, f >

28. ؔ਺ͷܭࢉ
• ؔ਺ͷ௕͞ʢϊϧϜʣ
• ؔ਺ͷ௚ަ
||f|| =
p
< f, f >
< f, g >= 0

29. ؔ਺ͷܭࢉ
• ؔ਺ͷ௕͞ʢϊϧϜʣ
• ؔ਺ͷ௚ަ
• ਖ਼ن௚ަجఈ
||f|| =
p
< f, f >
< f, g >= 0
< hi, hj >= ij

30. ؔ਺ͷܭࢉ
• ؔ਺ͷ௕͞ʢϊϧϜʣ
• ؔ਺ͷ௚ަ
• ਖ਼ن௚ަجఈ
• ؔ਺ͷ෼ղ
||f|| =
p
< f, f >
< f, g >= 0
< hi, hj >= ij
f =< f, h1 > h1+ < f, h2 > h2 + . . .

31. 1. ϑʔϦΤڃ਺ల։ͱ͸ɺ
ؔ਺Λෳ਺ͷࡾ֯ؔ਺ʹ෼ղ͢Δ͜ͱ
2. ؔ਺͸ϕΫτϧͰ͋Δ
3. ϑʔϦΤڃ਺ల։͸ϕΫτϧͷ෼ղͰ͢ΑͶʂʁ

32. ؔ਺Λࡾ֯ؔ਺Ͱ෼ղ
• ͜Ε͕ϑʔϦΤڃ਺ల։
• ಺ੵΛ͍͍ײ͡ʹܾΊ͍ͨ
➡ ࡾ֯ؔ਺͕ਖ਼ن௚ަجఈͱͳΔΑ͏ʹ಺ੵΛఆٛ
f =
a0
2
+ < f, cos 1x > cos 1x+ < f, cos 2x > cos 2x + . . .
+ < f, sin 1x > sin 1x+ < f, sin 2x > sin 2x + . . .

33. ϑʔϦΤڃ਺ల։ͷͨΊͷ಺ੵ
< f, g >
=
1

Z ⇡

fg dx

34. ϑʔϦΤڃ਺ల։ͷͨΊͷ಺ੵ
• ಺ੵΛ͜ͷΑ͏ʹఆٛ͢Ε͹ɺҎԼ͕ຬͨ͞ΕΔɿ
< f, g >
=
1

Z ⇡

fg dx
< cos ix, cos jx > = ij
< sin ix, sin jx > = ij
< cos ix, sin jx > = 0

35. 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

36. 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 >

37. ·ͱΊ
• ϑʔϦΤڃ਺ల։ͱ͸ɺ
ؔ਺Λෳ਺ͷࡾ֯ؔ਺ʹ෼ղ͢Δ͜ͱ
• ؔ਺͸ϕΫτϧͰ͋Δ
• ϑʔϦΤڃ਺ల։͸ϕΫτϧͷ෼ղͰͨ͠
• ΋ͬͱৄ͘͠஌Γ͍ͨਓ͸
ʮhoriem ϑʔϦΤʯͱ͔Ͱ̶̶ͬͯͶ