QUADRATIC CONVERGENCE OF THE TANH-SINH ...

6 JONATHAN M. BORWEIN AND LINGYUN YEThen G 2 is a Hilbert space and the mapping f ↦→ ̂f is an isomorphism of H 2onto G 2 . Also, G 2 has an orthonormal basis with elements(3.2) φ n (z) := ψ ′ (z) (ψ(z)) nandLemma 3.1. Assume the reproducing kernel of H 2 is K(z, w), then the reproducingkernel of G 2 is(3.3) ̂K(z, w) = K (ψ(z), ψ(w)) · ψ ′ (z) · ψ ′ (w)given the transformation ψ(z) from H 2 to G 2 .Proof. Given the reproducing property (2.1) and the definition of the inner product(3.1), if ̂K(z, w) denotes the reproducing kernel of G 2 then〈 ̂f(z), ̂K(z, w)〉 z = 1 ∫̂f(z)2π̂K(z, w)| dz/ψ ′ (z)|= ̂f(w)∂A= ψ ′ (w) · 〈f(z), K(z, ψ(w))〉∫z= ψ ′ 1(w) · ̂f(z)K(ψ(z), ψ(w))ψ2π′ (z)| dz/ψ ′ (z)|∂A= 1 ∫̂f(z)K(ψ(z), ψ(w))ψ2π′ (z)ψ ′ (w) |dz/ψ ′ (z)|∂AComparing the last integral with the one on the first line, we get̂K(z, w) = K (ψ(z), ψ(w)) · ψ ′ (z) · ψ ′ (w).Therefore for the tanh-sinh transformation, by (2.2), G 2 has a reproducing kernel()̂K(z, w) = 1/ 1 − ψ(z)ψ(w) · ψ ′ (z) · ψ ′ (w)(3.4)=×π 24 cosh ( π2 sinh(z) − π 2 sinh(w))cosh(z) cosh(w)cosh ( π2 sinh(z)) cosh ( π2 sinh(w)).Consequently, the point functionals on G 2 , ̂P z , defined for z ∈ A bŷP z ̂f = ̂f(z)are bounded linear functionals (see Thm. 12.6.1 of [4]) and satisfy(3.5) | ̂P z f| 2 ≤ ̂K(z, z) ‖ ̂f‖ 2 .3.1. Further working notation. We use similar notation in G 2 as in H 2 :Î ̂f :=∫ 1−1f(x)dx =̂T h,N ̂f := Th,N f = h∫ ∞−∞N∑n=−NÊ h,N ̂f :=(Î − ̂Th,N)̂f,̂f(u)du,̂f(nh),□