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直交多項式と表現論

 直交多項式と表現論

数理空間“τόπος” (トポス)新歓イベントhttps://peatix.com/event/643037 での発表資料です。

Naoya Umezaki

May 11, 2019
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  1. ௚ަଟ߲ࣜͱදݱ࿦
    ക࡚௚໵@unaoya
    2019/5/11 ਺ཧۭؒ τ ´
    oπoζʢτϙεʣ৽׻Πϕϯτ
    1

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  2. ࡾ֯ؔ਺ͷഒ֯ެࣜ
    sin 2θ = 2 sin θ cos θ
    sin 3θ = sin 2θ cos θ + cos 2θ sin θ
    = 2 sin θ cos2 θ + (2 cos2 θ − 1) sin θ
    = sin θ(4 cos2 θ − 1)
    sin 4θ = sin 3θ cos θ + cos 3θ sin θ
    = sin θ(4 cos2 θ − 1) cos θ + (4 cos3 θ − 3 cos θ) sin θ
    = sin θ(8 cos3 θ − 4 cos θ)
    2

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  3. Chebyshev ଟ߲ࣜ
    sin θ = sin θ × 1 U0(x) = 1
    sin 2θ = sin θ × 2 cos θ U1(x) = 2x
    sin 3θ = sin θ × (4 cos2 θ − 1) U2(x) = 4x2 − 1
    sin 4θ = sin θ × (8 cos3 θ − 4 cos θ) U3(x) = 8x3 − 4x
    ͱ͢Δͱ
    sin(n + 1)θ = sin θ × Un(cos θ)
    ͱͳΔɻ͜ͷ Un(x) Λୈ 2 छ Chebyshev ଟ߲ࣜͱ͍͏ɻ
    3

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  4. ઴Խࣜ
    Ճ๏ఆཧ͔Β
    sin(n + 1)θ = cos θ sin nθ + cos nθ sin θ
    sin(n − 1)θ = cos θ sin nθ − cos nθ sin θ
    sin(n + 1)θ + sin(n − 1)θ = 2 sin nθ cos θ
    ͜͜Ͱ sin(n + 1)θ = sin θ × Un(cos θ) Λࢥ͍ग़͢ɻ྆ลΛ sin θ Ͱ
    ׂͬͯ x = cos θ ͱ͢Δͱɺ
    Un+1(x) + Un−1(x) = 2xUn(x)
    ͕੒Γཱͭɻ
    4

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  5. ฼ؔ਺
    ઴Խࣜ Un(x) − 2xUn−1(x) + Un−2(x) = 0, U0(x) = 1, U1(x) = 2x
    Λ࢖ͬͯ

    n=0
    Un(x)tn = U0(x) + U1(x)t + U2(x)t2 + U3(x)t3 + · · ·
    Λมܗ͢Δɻt ഒ͢Δͱ antn ͸ antn+1 ͱͳΔ͔Βɺtn ͷ܎਺͸
    an−1
    ʹͳΔ͜ͱʹ஫ҙ͢Δͱɺ
    −2xt

    n=0
    Un(x)tn = −2xU0(x)t − 2xU1(x)t2 − 2xU2(x)t3 − 2xU3(x)t4 −
    t2

    n=0
    Un(x)tn = U0(x)t2 + U1(x)t3 + U2(x)t4 + U3(x)t5 + · · ·
    (1 − 2tx + t2)

    n=0
    Un(x) = 1
    5

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  6. ฼ؔ਺
    1
    1 − 2tx + t2
    = U0(x) + U1(x)t + U2(x)t2 + U3(x)t3 + · · ·
    ٯʹࠨลΛల։͢Δ͜ͱͰ
    1 + (2tx − t2) + (2tx − t2)2 + (2tx − t2)3 + · · ·
    = 1 + 2tx − t2 + 4t2x2 − 4t3x + t4 + 8t3x3 − 12t4x2 + · · ·
    = 1 + (2x)t + (−1 + 4x2)t2 + (−4x + 8x3)t3 + · · ·
    = U0(x) + U1(x)t + U2(x)t2 + U3(x)t3 + · · ·
    ͱఆٛ͢Δ͜ͱ΋Ͱ͖Δɻ
    6

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  7. ঢ߱ԋࢉࢠ
    1
    1 − 2tx + t2
    Λ t, x ͰͦΕͧΕඍ෼͢Δͱɺ
    2x − 2t
    (1 − 2tx + t2)2
    =
    2(x − t)
    (1 − 2tx + t2)2
    ,
    2t
    (1 − 2tx + t2)2
    ͱͳΔɻ
    t(x − t)
    2(x − t)
    (1 − 2tx + t2)2
    = 2t
    (x − t)2
    (1 − 2tx + t2)2
    = 2t
    x2 − 2xt + t2
    (1 − 2tx + t2)2
    (1 − x2)
    2t
    (1 − 2tx + t2)2
    = 2t
    1 − x2
    (1 − 2tx + t2)2
    (t(x − t)
    d
    dt
    + (1 − x2)
    d
    dx
    )
    1
    1 − 2tx + t2
    = 2t
    1 − 2xt + t2
    (1 − 2tx + t2)2
    = 2t
    1
    1 − 2tx + t2
    ͱͳΔɻ
    7

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  8. ঢ߱ԋࢉࢠ
    ͭ·Γ
    (t(x − t)
    d
    dt
    + (1 − x2)
    d
    dx
    − 2t)
    1
    1 − 2tx + t2
    = 0
    ͕੒Γཱͭɻ
    1
    1 − 2tx + t2
    = U0(x) + U1(x)t + U2(x)t2 + U3(x)t3 + · · ·
    ͷӈลʹ΋ಉ͡ܭࢉΛ͢Δɻantn Λ t ഒͱ t Ͱͷඍ෼ʹΑͬͯ tn
    ͷ܎਺͕ an−1, (n + 1)an+1
    ͱͳΔ͜ͱʹ஫ҙ͢Δͱɺ
    nxUn(x) − (n − 1)Un−1(x) + (1 − x2)
    d
    dx
    Un(x) − 2Un−1(x) = 0
    {(1 − x2)
    d
    dx
    + nx}Un(x) = (n + 1)Un−1
    8

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  9. ঢ߱ԋࢉࢠ
    {(1 − x2)
    d
    dx
    + nx}Un(x) = (n + 1)Un−1(x)
    (n + 1)2xUn(x) = (n + 1)Un+1(x) + (n + 1)Un−1(x)
    ྆ลҾ͘ͱ
    {(1 − x2)
    d
    dx
    − (n + 2)x}Un(x) = −(n + 1)Un+1(x)
    ͜ͷೋ͕ͭঢ߱ԋࢉࢠͱݺ͹ΕΔɻ
    {(1 − x2)
    d
    dx
    + nx}Un(x) = (n + 1)Un−1
    {(1 − x2)
    d
    dx
    − (n + 2)x}Un(x) = −(n + 1)Un+1(x)
    9

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  10. ඍ෼ํఔࣜ
    ঢ߱ԋࢉࢠΛଓ͚ͯ࡞༻ͤ͞Δͱɺ
    {(1 − x2)
    d
    dx
    − (n + 1)x}{(1 − x2)
    d
    dx
    + nx}Un(x) = −n(n + 1)Un−1
    ({(1 − x2)
    d
    dx
    − (n + 1)x}{(1 − x2)
    d
    dx
    + nx} + n(n + 1)})Un(x) = 0
    ࠨลΛܭࢉ͢Δͱɺ
    {(1 − x2)2
    d2
    dx2
    + (1 − x2)(−2x)
    d
    dx
    − (n + 1)x(1 − x2)
    d
    dx
    + nx(1 − x2)
    d
    dx
    + n(1 − x2) − n(n + 1)x2 + n(n + 1)}Un(x) = 0
    (1 − x2)
    d2
    dx
    Un(x) − 3x
    d
    dx
    Un(x) + n(n + 2)Un(x) = 0
    ͕͑ΒΕΔɻ
    10

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  11. ඍ෼ํఔࣜ
    d2
    dθ2
    (sin θUn(cos θ))
    = sin θ(−Un(cos θ) − 3 cos θ
    d
    dx
    Un(cos θ) + (1 − cos2 θ)
    d2
    dx2
    Un(cos θ))
    ͱͳΔͷͰɺ
    (1 − x2)
    d2
    dx
    Un(x) − 3x
    d
    dx
    Un(x) + n(n + 2)Un(x) = 0
    ͱ߹ΘͤΔͱɺ
    d2
    dθ2
    sin θUn(cos θ) = −(n + 1)2 sin θUn(cos θ)
    ͜Ε͸྆୺͕ݻఆ͞Εͨ೾ͷํఔࣜʢແݶͷߴ͞ͷҪށܕϙςϯ
    γϟϧʣ
    11

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  12. ௚ަؔ܎
    π
    0
    sin(n + 1)θ sin(m + 1)θdθ =



    π (n = m)
    0 (n ̸= m)
    ࠨลΛ
    π
    0
    sin(n + 1)θ
    sin θ
    sin(m + 1)θ
    sin θ
    sin θ sin θdθ
    ͱมܗ͔ͯ͠Β x = cos θ ͱஔ׵ੵ෼͢Δͱ
    1
    −1
    Un(x)Um(x) 1 − x2dx =



    π (n = m)
    0 (n ̸= m)
    ͜ΕΛ࢖͏ͱؔ਺Λ Chebyshev ଟ߲ࣜͰల։Ͱ͖ɺ਺஋ܭࢉͳͲ
    ʹԠ༻͞ΕΔɻ
    12

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  13. SU(2) ͷදݱ
    SU(2) =
    a b
    −¯
    b ¯
    a
    , |a|2 + |b|2 = 1, a, b ∈ C ͷ 2 ม਺ଟ߲ࣜ
    ΁ͷ࡞༻Λ
    a b
    −¯
    b ¯
    a
    f (x, y) = f (¯
    ax − by, ¯
    bx + ay) ͰఆΊΔɻ
    eiθ 0
    0 e−iθ
    f (x, y) = f (e−iθx, eiθy) Λܭࢉ͢Δͱ
    eiθ 0
    0 e−iθ
    x = e−iθx,
    eiθ 0
    0 e−iθ
    y = eiθy,
    eiθ 0
    0 e−iθ
    x2 = e−2iθx2,
    eiθ 0
    0 e−iθ
    xy = xy,
    eiθ 0
    0 e−iθ
    y2 = e2iθy2
    13

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  14. දݱͷࢦඪ
    eiθ 0
    0 e−iθ
    x3 = e−3iθx3,
    eiθ 0
    0 e−iθ
    x2y = e−iθx2y,
    eiθ 0
    0 e−iθ
    xy2 = eiθxy2,
    eiθ 0
    0 e−iθ
    y3 = e3iθy3
    ಉ࣍͡਺ͷ෦෼ʹ͍ͭͯ܎਺͚ͩ଍͢ɻ౳ൺ਺ྻͷ࿨ͷެࣜͱ
    Euler ͷެࣜͰܭࢉ͢Δͱ
    e−iθ + eiθ =
    e−2iθ − e2iθ
    e−iθ − eiθ
    =
    sin 2θ
    sin θ
    e−2iθ + 1 + e2iθ =
    e−3iθ − e3iθ
    e−iθ − eiθ
    =
    sin 3θ
    sin θ
    e−3iθ + e−iθ + eiθ + e3iθ =
    e−4iθ − e4iθ
    e−iθ − eiθ
    =
    sin 4θ
    sin θ
    14

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  15. ࢦඪͷ௚ަੑ
    ܈ͷදݱͷࢦඪʹ͸௚ަੑ͕͋Δɻྫ͑͹ n ̸= m ͳΒ
    sin(n + m)

    N
    + sin 2(n + m)

    N
    + · · · + sin(N − 1)(n + m)

    N
    = 0
    ͕੒Γཱͭͱ͍͏͜ͱɻ
    ಉ͡Α͏ʹ n ̸= m ͷͱ͖ɺ
    π
    0
    sin(n + 1)θ
    sin θ
    sin(m + 1)θ
    sin θ
    sin θ sin θdθ = 0
    0
    −1
    Un(x)Um(x) 1 − x2dx = 0
    ͕੒Γཱͭɻ
    15

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  16. ௚ަଟ߲ࣜ
    ্ͰݟͨΑ͏ͳ௚ަؔ܎ɺ઴Խࣜɺඍ෼ํఔࣜͳͲͷੑ࣭Λຬͨ
    ͢ଟ߲ࣜͷܥྻ͕͍͔ͭ͋͘Δɻ
    ྫ͑͹ྔࢠྗֶʹग़ͯ͘Δ΋ͷͱͯ͠
    • Hermite ଟ߲ࣜ͸ௐ࿨ৼಈࢠ
    • Legendre ଟ߲ࣜ͸֯ӡಈྔͷྔࢠԽ
    • Laguerre ଟ߲ࣜ͸ਫૉݪࢠͷಈܘํఔࣜ
    ͕͋Δɻ
    ௚ަଟ߲ࣜͷੑ࣭Λදݱ࿦ͰཧղͰ͖Δʢͱࢥ͍ͬͯΔͷͰ͜Ε
    ͔Βษڧ͢Δʣ
    ɻ
    16

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