Design of highly accurate formulas for numerical integration in weighted Hardy spaces with the aid of potential theory 1 Ken ichiro Tanaka 1 Ω R m F I = F (t) dt (1.1) Ω m m 1 m = 1 1 Newton-Cotes Gauss [3, 9] F F (t) n F (a i ) l i (t) = i=1 Ω F (t) dt n c i F (a i ) i=1 ( ) c i = l i (t) dt Ω (1.2) 1 / 36

( ) {a i } {l i (t)} I Ω F t = ψ(x) I I = ψ 1 (Ω) F (ψ(x))ψ (x) dx (1.3) DE [13] Ω = ( 1, 1) DE DE ( π ) ψ : (, ) ( 1, 1), ψ(x) = tanh 2 sinh x (1.3) R = (, ) 2 I h = h k= M k= M (1.4) F (ψ(kh))ψ (kh) (1.5) F ( ( π )) tanh 2 sinh(kh) π cosh(kh) 2 cosh 2 ((π/2) sinh(kh)) (1.6) M N h R M + N + 1 h 1 DE f(t) = F (ψ(t))ψ (t) [7, 14] (1.5) R f R 3 4 (1.6) F x = ±1 ψ f DE F [7] DE f 2 [16] 3 [7] 4 1970 [4] 2.3 [4] DE 37

No.1 (2016) 1 Newton-Cotes Simpson Gauss DE (4.1) R (1.7) (1.8) DE-Sinc Gauss 4 (3.12) B { } H (, w) := f : C f f := sup f(z) z w(z) < d := {z C Im z < d} w 2.1 2.1 (2.3) f w [12] f H (, w) f w R 5 DE 6 (1.2) R Stenger [10, 11] sinc sinc sinc sinc sinc(x) = (sin πx)/(πx) 5 [7] w 1.1 SE 6 w(x) = O(exp( x )) (x ± ) Andersson [1], Andersson & Bojanov [2] 38

( ) f(x) f(kh) sinc(x kh) (1.7) k= M (1.7) (1.2) (, ) f(x) dx h f(kh) (1.8) k= M (1.5) 7 [8] sinc (1.7) DE DE-Sinc DE-Sinc [15] w H (, w) [6] (1.2) (1.8) 1 2 3 [15] 4 5 6 1.1. DE f f(x) = O(exp( (β x ) ρ )) (x ± ) Stenger [10, 11] SE SE F DE DE SE F DE [14] [5, 14] 7 h (1.7) (1.8) 39

No.1 (2016) 2 2.1 [7] w d := {z C Im z < d} B( ) ζ d lim ζ(x + iy) dy = 0 (2.1) x ± d lim ( ζ(x + iy) + ζ(x iy) ) dx < (2.2) y d 0 w 1 w 1. w B( ) 1 w { } H (, w) := f : C f f := sup f(z) z w(z) <. (2.3) w 2. w R 3. log w R 2.2 1 3 w H (, w) [8] f H (, w) R (2N + 1) N R f f e N (f) f f 1 f H (, w) e N (f) e N (f) f EN min (H (, w)) 40

:= inf 1 l N inf m l,...,m l m l + +m l =2N+1 ( ) m l j 1 inf inf sup sup a j ϕ jk f 1 x R f(x) f (k) (a j ) ϕ jk (x), distinct j= l k=0 (2.4) ϕ jk 2 {a j } EN min(h (, w)) ϕ jk a j 2.3 E. B. Saff V. Totik [6] [6] [15, 2.3] Green g Dd (x, z) = log tanh((π/(4d))(x z)) supp µ R Borel µ Green ( U µ π ) (x) = log tanh 4d (x z) dµ(z) (2.5) R U µ log w(x) 1 I w (µ) = = R R R R ( g Dd (x, z) log(w(x)w(z)) 1/M ) dµ(z)dµ(x) (2.6) ( ( π ) ) log tanh 4d (x z) log(w(x)w(z)) 1/M dµ(z)dµ(x) (2.7) µ [6, II.5.10] w 1 3 µ 2.1. w 1 3 M V w := inf µ M(R,M) I w (µ) (2.8) M(R, M) R Borel M 1. V w 2. µ w M(R, M) I w (µ w ) = V w. (2.9) µ w. 41

No.1 (2016) 3 [15] 3 4 2.2 EN min(h (, w)) [8] 4.3 w 1 ([8, 4.3]). w 1 2 E min N (H (, w)) [ N sup sup f 1 x R f(x) k= N [ sup B N (x; {a l }, ) w(x) x R = inf a l R = inf a l R ] f(a k ) B N;k(x; {a l }, )w(x) 4d B N;k (a k ; {a l }, )w(a k ) π T (a k x) ] (3.1) B N (x; {a l }, ) = B N;k (x; {a l }, ) = ( π ) T (x) = tanh 4d x, (3.2) N k= N N m N, m k ( π ) tanh 4d (x a k), (3.3) ( π ) tanh 4d (x a m) (3.4) (3.1) {a l } f N (x) = f(a k ) B N;k(x; {a l }, )w(x) B N;k (a k ; {a l }, )w(a k ) k= N 4d π T (a k x) (3.1) {a l } log B N (x; {a l }, ) w(x) = V σ a (x) + log w(x) (3.5) V σ a (x) = k= N ( π log tanh k)) 4d (x a (3.6) 42

( ) (2.5) µ µ(z) = σ a (z) := δ(z a k ) k= N Green ( 1) δ Dirac µ V µ = U µ (3.5) (3.1) 1. [ ( inf sup V σ a D a l R d (x) + log w(x) )] (3.7) x R {a l } 1 R Borel µ N M(R, 2(N + 1)) µ N (R) = 2(N + 1) 1 2N + 1 8 2. [ ( inf sup V µ N D µ N M(R,2(N+1)) d (x) + log w(x) )]. (3.8) x R supp µ N =[ α N,α N ] α N µ N M(R, 2(N + 1)) 2 2.1 (2.8) V µ N Green U µ N 2 V µ N µ N µ N µ N M(R, 2(N + 1)) C 2 ( α N, α N ) M(R, 2(N + 1)) R 2 3 3. V µ N (x) + log w(x) = K N for any x [ α N, α N ], (3.9) V µ N (x) + log w(x) K N for any x R \ [ α N, α N ] (3.10) α N, K N µ N M(R, 2(N + 1)) C 2 ( α N, α N ) 8 2N + 1 43

No.1 (2016) 3 V µ N (x) + log w(x) [ α N, α N ] 3 R V µ N (x, y) R 2 V µ N (x + i y) \ [ α N, α N ] Laplace V µ N 9 R V µ N v [15] Laplace v α N K N V µ N = v R 4. SP1 SP2 µ N M(R, 2(N + 1)) C 2 ( α N, α N ) α N, K N R V µ N v SP1 α N, K N, V µ N αn, K N, v R V µ N = v α N α N ( π ) log tanh 4d (x z) dµn (z) = v (x) x R, (3.11) µ N 4 SP1 (3.11) µ N (R) = 2(N + 1) v α N, K N (3.11) Fourier Fourier (FFT) 1 {a l } 2 µ N µ N 3 µ N 4 Green SP1 3 V µ N SP2 µ N 9 R 2 V µ N D = 0 d [ α N, α N ] V µ N D = log w K d N Green 44

( ) 1. d w, N 4 SP1 α N K N 2. 4 SP2 x [ α N, α N ] ν N := µ N ν N 10 3. x [ α N, α N ] I[ ν N ](x) := 4. I[ ν N ] I[ ν N ] 1 x 0 ν N (t) dt 5. a i a i = I[ ν N ] 1 (i) (i = N,..., N) 6. f N f N (x) := j= N f(a j ) B N:j(x; {a i }, ) w(x) B N:j (a j ; {a i }, ) w(a j ) 4d π T (a j x) (3.12) 1 4 (3.12) EN min(h (, w)) (3.12) w Ganelius [15] [15] w (3.12) w DE-Sinc 4 (3.12) x (, ) f(x) dx f N (x) dx = c N:j ({a i },, w) f(a j ) (4.1) j= N c N:j ({a i },, w) = B N:j (x; {a i }, ) w(x) B N:j (a j ; {a i }, ) w(a j ) 4d π T (a j x) dx (4.2) 10 F[ ν N ] Fourier FFT 45

No.1 (2016) (4.1) (3.12) H (, w) (4.1) d w N (4.2) c N:j (4.2) c N:j c N:j (4.1) (4.1) (4.2) (4.2) c N:j ({a i },, w) h n k= n B N:j (k h; {a i }, ) w(k h) B N:j (a j ; {a i }, ) w(a j ) 4d π T (a j k h) (4.3) 11 5 5.1 5.2 5.1 ( ). f(x) = sech(2x), f(x) dx = π 2. (5.1) 5.2 (Gaussian ). f(x) = x 2 (π/4) 2 + x 2 exp( x2 ), f(x) dx = π π2 exp((π/4) 2 ) 4 ( π ) erfc. (5.2) 4 5.3 ( DE ). f(x) = π cosh(2x) cosh((π/2) sinh(2x)), f(x) dx = π. (5.3) (4.1) H (, w) 11 (4.2) w T (x) = (π/(4d)) sech 2 ((π/(4d))x) 46

( ) 2 d d = π/4 1 (1.8) M = N 12 5.1 5.2 w SE 5.3 w DE h N w A MATLAB 2 H (D π/4, w) w w(x) (4.1) N (4.3) n h 5.1 sech(2x) N = 5,..., 100 (5 ) (n, h) = (500, 0.04) 5.2 exp( x 2 ) N = 5,..., 50 (5 ) (n, h) = (500, 0.02) 5.3 sech((π/2) sinh(2x)) N = 5,..., 25 (5 ) (n, h) = (500, 0.005) 1 3 (4.1) (1.8) 5.1 5.2 5.3 (4.1) 13 B 12 w 2 13 Gaussian (1.8) (4.1) N 10 10 15 10 16 N (4.2) 47

No.1 (2016) -2-4 Errors for the SE weighted function f SE trapezoid formula (4.1) -6 log 10 (error) -8-10 -12-14 -16 10 20 30 40 50 60 70 80 90 100 N 1 5.1 trapezoid formula (4.1) (4.1) -2-4 Errors for the Gauss weighted function f Gauss trapezoid formula (4.1) -6 log 10 (error) -8-10 -12-14 -16 5 10 15 20 25 30 35 40 45 50 N 2 5.2 trapezoid formula (4.1) (4.1) 48

( ) -2-4 Errors for the DE weighted function f DE trapezoid formula (4.1) -6 log 10 (error) -8-10 -12-14 -16 5 10 15 20 25 N 3 5.3 trapezoid formula (4.1) (4.1) 6 [15] (3.12) (4.1) SE DE (4.2) [1] J.-E. Andersson, Optimal quadrature of H p functions, Math. Z. 172 (1980), pp. 55 62. [2] J.-E. Andersson and B. D. Bojanov, A note on the optimal quadrature in H p, Numer. Math. 44 (1984), pp. 301 308. [3] P. J. Davis and P. Rabinowitz, Methods of numerical integration, second edition, Dover, New York, 1984. [4] M. Mori, Discovery of the double exponential transformation and its developments, Publ. RIMS Kyoto Univ. 41 (2005), pp. 897 935. [5] T. Okayama, K. Tanaka, T. Matsuo, and M. Sugihara, DE-Sinc methods have almost 49

No.1 (2016) the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods, Part I: Definite integration and function approximation, Numer. Math. 125 (2013), pp. 511 543. [6] E. B. Saff and V. Totik, Logarithmic potentials with external fields, Springer, Berlin Heidelberg, 1997. [7] M. Sugihara, Optimality of the double exponential formula functional analysis approach, Numer. Math. 75 (1997), pp. 379 395. [8] M. Sugihara, Near optimality of the sinc approximation, Math. Comp. 72 (2003), pp. 767 786. [9] 1994. [10] F. Stenger, Numerical methods based on sinc and analytic functions, Springer, New York, 1993. [11] F. Stenger, Handbook of sinc numerical methods, CRC Press, Boca Raton, 2011. [12] 253 (1975), pp. 24 37. [13] H. Takahasi and M. Mori, Double exponential formulas for numerical integration, Publ. RIMS Kyoto Univ. 9 (1974), pp. 721 741. [14] K. Tanaka, M. Sugihara, K. Murota, and M. Mori, Function classes for double exponential integration formulas, Numer. Math. 111 (2009), pp. 631 655. [15] K. Tanaka, T. Okayama, and M. Sugihara, Potential theoretic approach to design of highly accurate formulas for function approximation in weighted Hardy spaces, arxiv:1511.04530, 14 Nov. 2015. (http://arxiv.org/abs/1511.04530) [16] L. N. Trefethen and J. A. C. Weideman, The exponentially convergent trapezoidal rule, SIAM Review 56 (2014), pp. 385 458. A (1.8) M = N N w h 5.1 5.3 (1.8) f(x) dx h k= N f(kh) = f(x) dx h f(kh) + h k= k >N f(kh) E D (f, h) + E T (f, h, N) (A.1) 50

( ) E D (f, h) E T (f, h, N) E D (f, h) = f(x) dx h E T (f, h, N) = h f(kh) k >N k= f(kh), (A.2) (A.3) E D (f, h) f H (, w) ( E D (f, h) C exp 2πd ) h (A.4) C h E T (f, h, N) f H (, w) E T (f, h, N) f h w(kh) (A.5) k N w (A.5) 5.1 E T (f, h, N) C 1 exp( 2Nh), 5.2 E T (f, h, N) C 2 exp( (Nh) 2 ), 5.3 E T (f, h, N) C 3 exp( (π/4) exp(2nh)) (A.6) (A.7) (A.8) C 1, C 2, C 3 N h (A.4) (A.6) (A.8) h h h 5.1 (A.4) (A.6) d = π/4 2πd 2πd h = 2Nh h = 2N = π 2 N (A.9) 5.2 5.3 2πd h = (Nh)2 h = 2πd h = π 4 ( ) 1/3 ( ) 2πd π 2 1/3 N 2 = 2N 2, (A.10) exp(2nh) h = W (16dN) 2N = W (4πN) 2N log(4πn) 2N (A.11) W g(x) = x e x Lambert W 5 5.1 5.3 (A.9) (A.11) h (A.11) log 51

No.1 (2016) B 3 4 w w 1/2 3 ( ) (B.1) 1/2 w 1/2 1 1/2 1 w ( ) w w 1/2 4 6-2 -4 Errors for the SE weighted function f SE trapezoid formula (4.1) modified formula (4.1) -6 log 10 (error) -8-10 -12-14 -16 10 20 30 40 50 60 70 80 90 100 N 4 5.1 trapezoid formula (4.1) (4.1) modified formula (4.1) (4.1) (B.1) 52

( ) -2-4 Errors for the Gauss weighted function f Gauss trapezoid formula (4.1) modified formula (4.1) -6 log 10 (error) -8-10 -12-14 -16 5 10 15 20 25 30 35 40 45 50 N 5 5.2 trapezoid formula (4.1) (4.1) modified formula (4.1) (4.1) (B.1) N = 35, 45 0-2 -4 Errors for the DE weighted function f DE trapezoid formula (4.1) modified formula (4.1) -6 log 10 (error) -8-10 -12-14 -16 5 10 15 20 25 N 6 5.3 trapezoid formula (4.1) (4.1) modified formula (4.1) (4.1) (B.1) ( : 2015 12 18 ; : 2016 1 25 ) 53