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
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