フーリエ級数展開は ベクトルの分解ですよね！？

2017年3月29日@プログラマのための数学LT会
詳細: http://blog.physips.com/entry/fourier_orthogonal

Transcript

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

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2. 1. ϑʔϦΤڃ਺ల։ͱ͸ɺ

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

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3. 1. ϑʔϦΤڃ਺ల։ͱ͸ɺ

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

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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
   

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5. f
   (
   x
   ) =a0
   2
   

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

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

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

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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
   

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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
    

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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
    

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12. 1. ϑʔϦΤڃ਺ల։ͱ͸ɺ

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

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13. ؔ਺͸ϕΫτϧͰ͋Δ
    • ؔ਺͸ແ਺ͷ఺ͷू·Γ
    

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14. ؔ਺͸ϕΫτϧͰ͋Δ
    • ؔ਺͸ແ਺ͷ఺ͷू·Γ
    

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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
    

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

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

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

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

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

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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
    

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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
    

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23. ϕΫτϧͷܭࢉ
    • ϕΫτϧͷ௕͞ |U| =
    p
    U · U
    

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

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

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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 + . . .
    

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

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

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

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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 + . . .
    

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31. 1. ϑʔϦΤڃ਺ల։ͱ͸ɺ

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

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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 + . . .
    

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33. ϑʔϦΤڃ਺ల։ͷͨΊͷ಺ੵ
    < f, g >
    =
    1
    ⇡
    Z ⇡
    ⇡
    fg dx
    

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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
    

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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
    

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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 >
    

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37. ·ͱΊ
    • ϑʔϦΤڃ਺ల։ͱ͸ɺ

    ؔ਺Λෳ਺ͷࡾ֯ؔ਺ʹ෼ղ͢Δ͜ͱ
    • ؔ਺͸ϕΫτϧͰ͋Δ
    • ϑʔϦΤڃ਺ల։͸ϕΫτϧͷ෼ղͰͨ͠
    • ΋ͬͱৄ͘͠஌Γ͍ͨਓ͸

    ʮhoriem ϑʔϦΤʯͱ͔Ͱ̶̶ͬͯͶ
    

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