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1 RIMS Kokyuroku, vol.084, (999), Euler Fourier Euler Fourier S = ( ) n f(n) = e in f(n) (.) I = 0 e ix f(x) dx (.2) Euler Fourier Fourier Euler Euler Fourier Euler Euler Fourier Fourier [5], [6] Euler Fourier 2 Euler Euler S = ( ) n a n (2.) S Euler = ( 2 2 Δ)n a 0, (Δa n a n+ a n ) (2.2) Euler N [3] S (N) Euler = N ( 2 2 Δ)n a 0 = N m=0 w (N) m ( ) m a m (2.3)

2 w m (N) Euler w (N) m = N n=m+ 2 N ( ) N n N w m (N) w m (N) w (N) m m+ 2 N/4 e (x N/2)2 /(2 N/4) dx, m/ N/2 N/2 N w (N) m N (2.4) e t2 dt (2.5) w m (N) N/2 2 erfc(m/ N/2), N (2.6) erfc(x) 2 e t2 dt p, q N x w(x; p, q) = erfc(x/p q) (2.7) 2 weight m : w (N) m w(x; p, q) ( N =6 p = q =(N/2) /2 ) Euler w m (N) w(x; p, q) S (N) = N ( ) n f(n) (2.8) N S w (N) = w(n; p, q)( ) n f(n) (2.9) 2

3 w(x; p, q) = 2erfc(x/p q) (p, q ) f(z) arg(z +/2) δ ( δ δ</2 ) f(z) M lim max f(r /2+iR tan θ) =0 R θ δ α tan δ, 0 <α< S ( ) S ( ) w < M +α 2 e α/2 ( p α 2 e(q αp/2) 2 /( α 2) + 2 α e(q αp)q ) e q 2 (2.0) q = q +/(2p) S (N) S w (N) S (N) S w (N) = 2i ( w(z; p, q))f(z) dz sin z (2.) C 2 C Im z N 2 + iαn C + C N N N Re z C N 2 iαn S (N) S (N) w 2 C 2: C sin z w(z; p, q) f(z) dz (2.2) erfc(z) = e (t+z)2 dt 2 0 = 0 e (t+z)2 dt (2.3) + e t2 z 2 dt =+ 2 erfc(z) 0 2 e z2, Re z 0 0 e t2 z 2 dt = 2 e z2, Re z<0 (2.4) 3

4 (2.2) C N 2 C N sin z w(z; p, q) f(z) dz < ( 2 cosh y dy + ) +(Nα/p) 2 2 e ((N /2)/p q)2 max f(n /2+iNy) y α 0, N (2.5) (2.2) C + ( z = t /2+iαt) / sin z sin z = w(z; p, q) w(z; p, q) = < 2 e 2iz eiz 2 +e 2αt cos 2t e αt 2 e α/2 e αt (2.6) 2 erfc((t /2)/p q + iαt/p) + ) 2 +(αt/p) 2 2 e (t/p q, t pq (2.7) ) 2 +(αt/p) 2 2 e (t/p q, t < pq q = q +/(2p) C + 2 C + sin z w(z; p, q) f(z) dz < ( 2 2 e α/2 e αt ) 2 +(αt/p) e (t/p q M +α 2 dt + e αt M ) +α 2 dt pq ) < M ( +α 2 p e α/2 2 α 2 e(q αp/2)2 /( α2) + αp)q α e(q e q 2 (2.8) (2.2) C C + S ( ) S ( ) w = lim w N S(N) S (N) < M +α 2 e α/2 ( p α 2 e(q αp/2) 2 /( α 2) + 2 α e(q αp)q ) e q 2 (2.9) [] p, q α q = q + 2p = α 2 p (2.20) 4

5 S w ( ) S ( ) S ( ) w < M +α 2 e α/2 ( p + 2 ) e q2 (2.2) α 2 α (2.20) α q S w ( ) S w ( ) N S w (N) w(n; p, q) n w(n; p, q) 2 exp( (n/p q)2 ) N ( ) (2.20) (2.22) S w (N) S ( ) S (N) w S ( ) S ( ) w + N = 2pq + = 4 α q 2 (2.22) S ( ) S ( ) w + M 2 w(n; p, q)( ) n f(n) n=n n=n exp( (n/p q) 2 ) = O(α 2 ( α) /2 qe q2 ) = O(α 3/2 ( α) /2 Ne αn/4 ), N (2.23) Euler S w (N) N S = ( ) n n + = log 2 (2.24). Euler 2. Euler S (N) Euler = N ( 2 2 Δ)n a 0 N S w (N) = w(n; p, q)( ) n a n 2 p, q N =2pq, q = p/2 Euler Euler 5

6 N Euler Euler Fourier Euler Fourier I = ( ω>0 ) Euler 0 f(x)e iωx dx (3.) L I w (L) = 0 w(x; p, q)f(x)e iωx dx (3.2) w(x; p, q) φ(t) dt =, lim φ(t) = 0 (3.3) t ± φ(t) w(x; p, q) = x/p q φ(t) dt (3.4) p, q L ω f(x) Euler I w (L) L I φ(t) Euler 2 w(x; p, q) = 2erfc(x/p q) ( p, q ) f(z), E(z; ω) 0 arg(z r) δ ( r ω, δ δ</2 ) f(z) M, E(z; ω) M 2 e iωz lim max f(r + r + ir tan θ) =0 R 0 θ δ α tan δ, 0 <α< f(x)e(x; ω) dx w(x; p, q)f(x)e(x; ω) dx r r ( M M 2 +α 2 p 2 α 2 e(q ωαp/2)2 /( α2) + ) ωαp)q ωα e(q e q 2 (3.5) q = q r/p ΔI (R) w = = r+r r r+r r r+r f(x)e(x; ω) dx w(x; p, q)f(x)e(x; ω) dx r ( w(x; p, q))f(x)e(x; ω) dx (3.6) (r, r + R) 3 C +, C R 6

7 Im z C + R + r + iαr C R Re z r R 3: C + + C R ΔI (R) w = ( w(z; p, q))f(z)e(z; ω) dz C + +C R w(z; p, q) f(z) E(z; ω) dz (3.7) C + +C R [ ] r =0, E(z; ω) = exp(iωz) Fourier (3.) w(x; p, q) I w = w(x; p, q)f(x)e iωx dx (3.8) p, q α I I w <M +α 2 0 q = ωα 2 p (3.9) ( p 2 α + ) e q2 (3.0) 2 ωα w(x; p, q) I I w Euler (3.2) (3.8) L I w I (L) w L =2pq = 4 ωα q2 (3.) L w(x; p, q)f(x)e iωx dx M exp( (x/p q) 2 ) dx 2 2pq M p e q2 (3.2) 4 7

8 (3.0) 2( Euler ) (3.9) (3.) w(x; p, q) = 2erfc(x/p q) Euler I I w (L) I I w + I w I w (L) ( +α < M 2 ) p p +α α 2 4 ωα e q2 = O((ωα) /2 ( α) /2 Le ωαl/4 ), L (3.3) p, q e q2 q p p α ω L 2 E(z; ω) Hankel Fourier Bessel e iωz 4 Euler Euler Fourier Euler f(x) Fourier F (ω) = f(x)e iωx dx (4.) 2 ŵ(x; p, q) = x/p+q x/p q φ(t) dt (4.2) Fŵ(ω) = ŵ(x; p, q)f(x)e iωx dx (4.3) 2 φ(t) (3.3) (4.3) Fŵ(ω) = Ŵ (ω ω )F (ω ) dω (4.4) Ŵ (ω) ŵ(x; p, q) Fourier Ŵ (ω) = ŵ(x; p, q)e iωx dx 2 = sin(pqω) 2Φ(pω) (4.5) ω Φ(ω) = φ(x)e iωx dx (4.6) 2 8

9 (4.4) Fŵ(ω) = F (ω ) sin(pq(ω ω )) (ω ω 2Φ(p(ω ω )) dω (4.7) ) sinc F (ω), Φ(ω) sinc q O(exp( Cq)) O( x m ) Fourier ω m f(z) f( z) z 2 F (ω) Re ω = ±0 sinc (4.7) F (ω) Φ(ω) 3 F (ζ) F ( ζ) Φ(p(ω ζ)) Φ(p(ω + ζ)) arg(ζ) <γ (γ >0 ω 0 ) γ lim F (Re iθ )Φ(p(ω Re iθ )) exp( pq Im Re iθ ) dθ =0 R ± γ lim ε ±0 γ 0 <θ<γ γ F (εe iθ )Φ(p(ω εe iθ )) εdθ =0 exp( pq Im ζ ) F (ω) Fŵ(ω) F (ζ) Φ(p(ω ζ)) dζ (4.8) C θ+ +C θ ω ζ C θ+ C θ 4 Im ζ C θ C θ+ θ θ C θ C θ+ Re ζ 4: C θ+ C θ sin(pqω) ω = exp(+ipqω) exp( ipqω) 2iω 2iω = exp(+ipq(ω + i0)) exp( ipq(ω i0)) 2i(ω + i0) 2i(ω i0) = exp(+ipq(ω + i0)) exp( ipq(ω i0)) + δ(ω) 2i(ω + i0) 2i(ω i0) (4.9) 9

10 (4.7) C θ+ C θ [] 3 ω Φ(ω ) p Φ(p(ω ζ)) φ(t) exp( pq Im ζ ) sinc p, q Euler Fŵ(ω) L ± f(±z) ŵ(±z; p, q) 2 φ(t) t t φ(±t)dt. ω Φ(ω ) 2. t t φ(±t)dt Euler I 0 -sinh Φ(ω) = I 0(β ω 2 ), ω (4.0) 2I 0 (β) Φ(ω) = 0, ω > (4.) [2] w(x; p, q) = sinh β 2 t 2 dt (4.2) I 0 (β) x/p q β 2 t 2 Euler f(x) 5 Fourier Fourier Euler Fourier I = 0 cos x +x 2 dx = K 0() (5.) Euler Legendre-Gauss (5.) (j+) cos x I = I (j), I (j) = dx (5.2) j=0 j +x 2 I (j) I [5] 0

11 . Euler : Euler (φ(t) = /2 e t2 q = p/2 =4 L =2pq) 60 Gauss 2. : Euler (24 ) 2 Gauss 0 7 : I Euler Euler /5 I sin x I 2 = dx = (5.3) +x2 0 2 Euler φ(t) = /2 e t2 q = p/2 =4 L =2pq. 2. Legendre-Gauss 3. Clenshaw-Curtis (Chebyshev ) 2: I I 2 Euler I I Legendre-Gauss Clenshaw-Curtis Clenshaw-Curtis Legendre-Gauss 2 2 x = ±i I [], [4] Bessel xj 0 (x) I 3 = 0 +x 2 dx = K 0() (5.4) J 0 (x) I 4 = dx = I +x 2 0(/2)K 0 (/2) (5.5) 0 Fourier Euler ( φ(t) = /2 e t2 q = p/2 =4 L =2pq)

12 . Legendre-Gauss 2. Clenshaw-Curtis 3 3: Bessel I 3 I 4 I 3 I 4 Legendre-Gauss Clenshaw-Curtis Bessel Euler Euler I I 2. Euler ( ) q = p/2 =4 2. Euler 2(I 0 -sinh ) p =/ω = β = q =20 w(x; p, q) = Legendre-Gauss 4 w(x; p, q) = erfc(x/p q) (5.6) 2 sinh β 2 t 2 dt (5.7) I 0 (β) x/p q β 2 t 2 4: I I I 0 -sinh I 0 -sinh Euler 2/3 6 Fourier f(x) Fourier ( ω ) F (ω) = f(x)e iωx dx (6.) 2 f(x) x Euler F w (L) (ω) = L+ w( x ; p, q)f(x)e iωx dx (6.2) 2 L 2

13 F w (L) (ω) h F w (N,h) (ω) = h 2 N + n= N w( nh ; p, q)f(nh)e iωnh ( N ± = L ± /h ) Euler p, q L ± ω p, q L + = N/2, L = N/2 ω =2k/(Nh) F w (N,h) ( 2 Nh k)= h 2 N/2 n= N/2 (6.3) w( nh ; p, q)f(nh)e 2ink/N (6.4) FFT w(x; p, q) = 2erfc(x/p q) (6.4) 5, 6 Fourier f (x) = +x 2, (F (ω) = K 0( ω )) f 2 (x) = x 3 4+x 4, (F 2(ω) =i sgn ω e ω cos ω) 2 F w (N,h) 0 0 F w (N,h) k k (a) f(x) =f (x) (b) f(x) =f 2 (x) 5: Fourier : F w (N,h) ( 2 Nhk), N = 52, h =0.25 N = 52, h = ( 6 ) Euler 0 2 p, q exp( q 2 )=0 2, L = Nh/2=2pq w(x; p, q) 6 k ω = 2 2 Nh N 2 Nh k ω ω max = =8 ω ω ω =8 2 Euler ω ω =2q/p =3.45 ( k =35.) h ω </h=8 I 0 -sinh w(x; p, q) = sinh β 2 t 2 dt (6.5) I 0 (β) x/p q β 2 t 2 3

14 F F w (N,h) F F w (N,h) no weight no weight k k (a) f(x) =f (x) (b) f(x) =f 2 (x) 6: Fourier : F w (N,h) ( 2 Nhk), N = 52, h =0.25 (6.4) 7 F F w (N,h) F F w (N,h) k k (a) f(x) =f (x) (b) f(x) =f 2 (x) 7: Fourier - I 0 -sinh : F w (N,h) ( 2 Nhk), N = 52, h =0.25 N = 52, h =0.25 Euler 0 5 p, q, β β = q = log(0 5 ), L = Nh/2=2pq w(x; p, q) 7 k < 2 ω =/p =2.6 ( k =22.0) Euler ω 2/3 ω 4

15 7 Fourier Euler Euler Fourier Fourier FFT Euler Bessel [] H. Takahasi and M. Mori, Error Estimation in the Numerical Integration of Analytic Functions, Rep. Comput. Centre Univ. Kyoto 3, 970, [2] J.F.Kaiser, Nonrecursive digital filter design using the I 0 sinh window function, Proc. IEEE International Symposium on Circuits and Systems, 974. [3] Jet Wimp, Sequence Transformations and their Applications, Academic Press, 98. [4],,, 975. [5] Philip J. Davis and Philip Rabinowitz, Methods of Numerical Integration (Second Edition), Academic Press, 984. [6] T. Ooura and M. Mori, The Double Exponential Formula for Oscillatory Functions Over the Half Infinite Interval, Journal of Computational and Applied Mathematics 38, 99, [7],,,

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