The Hardy-Littlewood Function

Transcription

1 Hardy-Littlewood p. 1/2 The Hardy-Littlewood Function An Exercise in Slowly Convergent Series Walter Gautschi Purdue University

2 Hardy-Littlewood p. 2/2 OUTLINE The Hardy-Littlewood (HL) function

3 Hardy-Littlewood p. 2/2 OUTLINE The Hardy-Littlewood (HL) function Summation by integration

4 Hardy-Littlewood p. 2/2 OUTLINE The Hardy-Littlewood (HL) function Summation by integration A quadrature problem and its solution

5 Hardy-Littlewood p. 2/2 OUTLINE The Hardy-Littlewood (HL) function Summation by integration A quadrature problem and its solution Gauss quadrature rules

6 Hardy-Littlewood p. 2/2 OUTLINE The Hardy-Littlewood (HL) function Summation by integration A quadrature problem and its solution Gauss quadrature rules Orthogonal polynomials: 3-term recurrence

7 Hardy-Littlewood p. 2/2 OUTLINE The Hardy-Littlewood (HL) function Summation by integration A quadrature problem and its solution Gauss quadrature rules Orthogonal polynomials: 3-term recurrence Quadrature approximation of the HL-function

8 Hardy-Littlewood p. 2/2 OUTLINE The Hardy-Littlewood (HL) function Summation by integration A quadrature problem and its solution Gauss quadrature rules Orthogonal polynomials: 3-term recurrence Quadrature approximation of the HL-function Direct summation with acceleration

9 Hardy-Littlewood p. 3/2 The function of Hardy and Littlewood H(x) = k=1 1 k sin x k, x > 0 background summation procedure of Lambert Hardy and Littlewood (1936): there exist infinitely many x with x such that H(x) > C(log log x) 1/2

10 Hardy-Littlewood p. 3/2 The function of Hardy and Littlewood H(x) = k=1 1 k sin x k, x > 0 background summation procedure of Lambert Hardy and Littlewood (1936): there exist infinitely many x with x such that H(x) > C(log log x) 1/2 complete monotonicity C. Berg and H. Alzer (work in progress): complete monotonicity for all m of [x m ψ (m) (x)] (m) is equivalent to H(x) π 2 Ismail and Clark (2003): true if m = 2, 3,..., 16

11 Hardy-Littlewood p. 4/2 Summation by integration (G. and Milovanović, 1985) where S = k=1 a k, a k = (Lf)(k) (Lf)(s) = 0 e st f(t)dt

12 Hardy-Littlewood p. 4/2 Summation by integration (G. and Milovanović, 1985) where summation procedure S = k=1 a k, a k = (Lf)(k) (Lf)(s) = 0 e st f(t)dt S = k=1 (Lf)(k) = k=1 = 0 = 0 k=1 e (k 1)t e t f(t)dt 1 e t f(t)dt = 1 e t 0 0 e kt f(t)dt t 1 e t f(t) t e t dt poles at ±2µiπ, µ = 1, 2, 3,...

13 Hardy-Littlewood p. 5/2 Progress report # H(x) x The HL-function for 0 x 100

14 Hardy-Littlewood p. 6/2 A quadrature problem R g(t)dλ(t) = n ν=1 λ νg(τ ν ) + R n (g) determine λ ν, τ ν such that R n (g) = 0 if g S 2n, where S 2n = Q m P 2n m 1, 0 m 2n P 2n m 1 = polynomials of degree 2n m 1 Q m = rational functions with prescribed poles

15 Hardy-Littlewood p. 6/2 A quadrature problem R g(t)dλ(t) = n ν=1 λ νg(τ ν ) + R n (g) determine λ ν, τ ν such that R n (g) = 0 if g S 2n, where S 2n = Q m P 2n m 1, 0 m 2n P 2n m 1 = polynomials of degree 2n m 1 Q m = rational functions with prescribed poles specifically: Q m = span { r(t) = 1 1+ζ µ t, µ = 1, 2,..., m } ζ µ C, ζ µ 0, 1 + ζ µ t 0 on supp(dλ)

16 Hardy-Littlewood p. 7/2 Theorem (G., 2000; Vanherwegen et al., 2000) Let ω m (t) = m µ=1 (1 + ζ µt). Assume the existence of a (polynomial) Gauss formula R g(t) dλ(t) ω m (t) = n ν=1 λg ν g(τ G ν ), g P 2n 1. Then τ ν = τ G ν, λ ν = ω m (τ G ν )λ G ν, ν = 1, 2,..., n, yields the desired formula.

17 Hardy-Littlewood p. 7/2 Theorem (G., 2000; Vanherwegen et al., 2000) Let ω m (t) = m µ=1 (1 + ζ µt). Assume the existence of a (polynomial) Gauss formula R g(t) dλ(t) ω m (t) = n ν=1 λg ν g(τ G ν ), g P 2n 1. Then τ ν = τ G ν, λ ν = ω m (τ G ν )λ G ν, ν = 1, 2,..., n, yields the desired formula. Example m even, ζ µ = ± i 2µπ, µ = 1, 2,..., m/2 ω m (t) = m/2 µ=1 ( 1 + t2 4µ 2 π 2 ) > 0 on R

18 Hardy-Littlewood p. 8/2 Progress report # H(x) x The HL-function for 900 x 1000

19 Hardy-Littlewood p. 9/2 Construction of Gauss quadrature rules R g(t)dσ(t) = n ν=1 σ νg(τ ν ), g P 2n 1

20 Hardy-Littlewood p. 9/2 Construction of Gauss quadrature rules R g(t)dσ(t) = n ν=1 σ νg(τ ν ), g P 2n 1 orthogonal polynomials π k (t) = π k (t; dσ) : R π k(t)π l (t)dσ(t) = 0 if k l τ ν = zeros of π n ( ; dσ)

21 Construction of Gauss quadrature rules R g(t)dσ(t) = n ν=1 σ νg(τ ν ), g P 2n 1 orthogonal polynomials π k (t) = π k (t; dσ) : R π k(t)π l (t)dσ(t) = 0 if k l τ ν = zeros of π n ( ; dσ) three-term recurrence relation π k+1 (t) = (t α k )π k (t) β k π k 1 (t), π 1 (t) = 0, π 0 (t) = 1 where α k = α k (dσ), β k = β k (dσ), and by convention β 0 = R dσ(t) Hardy-Littlewood p. 9/2

22 Hardy-Littlewood p. 10/2 Jacobi matrix J n (dσ) = α 0 β1 0 β1 α 1 β βn 1 βn 1 α n 1

23 Hardy-Littlewood p. 10/2 Jacobi matrix J n (dσ) = α 0 β1 0 β1 α 1 β βn 1 βn 1 α n 1 Theorem (Golub & Welsch, 1969) The Gauss nodes τ ν are the eigenvalues of J n, J n (dσ)v ν = τ ν v ν, v T ν v ν = 1, and the Gauss weights σ ν given by σ ν = β 0 v 2 ν,1, v ν = [v ν,1,... ] T.

24 Hardy-Littlewood p. 11/2 Progress report # H(x) x The HL-function for 9, 900 x 10, 000

25 Hardy-Littlewood p. 12/2 Computation of recurrence coefficients discretization method approximation of dσ by a discrete N-point measure dσ N R p(t)dσ(t) N k=1 w kp(t k ) =: R p(t)dσ N(t) then α k (dσ) α k (dσ N ), β k (dσ) β k (dσ N ) (discrete) inner product (u, v) N = R u(t)v(t)dσ N(t) = N k=1 w ku(t k )v(t k )

26 Hardy-Littlewood p. 13/2 Darboux s formulae (I) α k = (tπ k,π k ) N (π k,π k ) N, k = 0, 1,..., n 1, β 0 = (π 0, π 0 ) N, β k = (π k,π k ) N (π k 1,π k 1 ) N, k = 1, 2,..., n 1

27 Hardy-Littlewood p. 13/2 Darboux s formulae (I) α k = (tπ k,π k ) N (π k,π k ) N, k = 0, 1,..., n 1, β 0 = (π 0, π 0 ) N, β k = (π k,π k ) N (π k 1,π k 1 ) N, k = 1, 2,..., n 1 recurrence relation (II) π k+1 (t) = (t α k )π k (t) β k π k 1 (t)

28 Hardy-Littlewood p. 13/2 Darboux s formulae (I) α k = (tπ k,π k ) N (π k,π k ) N, k = 0, 1,..., n 1, β 0 = (π 0, π 0 ) N, β k = (π k,π k ) N (π k 1,π k 1 ) N, k = 1, 2,..., n 1 recurrence relation (II) π k+1 (t) = (t α k )π k (t) β k π k 1 (t) Stieltjes s procedure π 0 = 1 (I) = α 0, β 0 (II) = π 1 (I) = α 1, β 1 (II) = (I) = α n 1, β n 1

29 Hardy-Littlewood p. 14/2 Progress report # H(x) x x 10 4 The HL-function for 99, 900 x 100, 000

30 Hardy-Littlewood p. 15/2 Back to Hardy-Littlewood H(x) = k=1 a k(x), a k (x) = 1 k sin x k general term as a Laplace transform 1 s ex/s = ( L (t) I 0 (2 xt) ) (s), I 0 = modified Bessel 1 s sin x s = ( 1 1 s 2i e ix/s e ix/s) = a k (x) = (Lf)(k), f(t) = f(t; x) [ I0 (2 ixt) I 0 (2 ixt) ] = 1 2i

31 Hardy-Littlewood p. 16/2 The function f power series f(t; x) = k=0 ( 1) k u 2k+1 (2k + 1)! 2, u = xt f(t; x) lim t 0 t = x

32 Hardy-Littlewood p. 16/2 The function f power series f(t; x) = k=0 integral representation f(t; x) = 1 π ( 1) k u 2k+1 (2k + 1)! 2, u = xt f(t; x) lim t 0 t π composite trapezoidal rule 0 = x e 2u cos θ sin( 2u cos θ)dθ

33 Hardy-Littlewood p. 17/2 HL-function (cont ) H(x) = 0 quadrature approximation t 1 e t f(t;x) t e t dt H(x) n ν=1 λ ν τ ν 1 e τ ν f(τ ν ;x) τ ν

34 Hardy-Littlewood p. 17/2 HL-function (cont ) H(x) = 0 quadrature approximation t 1 e t f(t;x) t e t dt H(x) n ν=1 λ ν τ ν 1 e τ ν f(τ ν ;x) τ ν polynomial/rational Gauss with Q m : m = 2 (n + 1)/2

35 Hardy-Littlewood p. 17/2 HL-function (cont ) H(x) = 0 quadrature approximation t 1 e t f(t;x) t e t dt H(x) n ν=1 λ ν τ ν 1 e τ ν f(τ ν ;x) τ ν polynomial/rational Gauss with Q m : m = 2 (n + 1)/2 performance with error tolerance x n # of digits lost

36 Hardy-Littlewood p. 18/2 Progress report #5 (smoking gun?) H(x) x x 10 5 The HL-function for 999, 900 x 1, 000, 000

37 Hardy-Littlewood p. 19/2 Direct summation with acceleration Assume x 1 and let n = x H(x) = = n k=1 n 1 k sin x k + k=1 ( π 2 n +x 6 1 k sin x k + k=1 k=n+1 1 k 2 ) k=n+1 1 k x3 6 1 k sin x k ( sin x k x k ( π 4 n 90 k=1 ( x k) 3 ) ) 1 k 4

38 Hardy-Littlewood p. 20/2 Euler-Maclaurin summation π 2 6 n k=1 1 k 2 B 0 n+1 B 1 (n+1) 2 + B 2 (n+1) 3 + B 4 (n+1) B 10 (n+1) 11 π 4 90 n k=1 1 k B 4 0 3(n+1) B 3 1 (n+1) + 2B 4 2 (n+1) + 5B 5 4 (n+1) 7 where + 28B 6 3(n+1) 9 + 3B 8 (n+1) B 10 (n+1) 13 B 0 = 1, B 1 = 1 2, B 2 = 1 6,..., B 10 = 5 66 are the Bernoulli numbers

39 Hardy-Littlewood p. 21/2 Progress report # H(x) x x 10 6 The HL-function for 9, 999, 900 x 10, 000, 000

40 Hardy-Littlewood p. 22/2 Progress report # H(x) x x 10 7 The HL-function for 99, 999, 900 x 100, 000, 000

41 Kommt Zeit, kommt Rat... Hardy-Littlewood p. 23/2