QUADRATIC CONVERGENCE OF THE TANH-SINH ...

8 JONATHAN M. BORWEIN AND LINGYUN YEOur goal is to estimate the approximation error||E h,N || = ||Êh,N ||which will be computed by repeated use of Lemma 3.2 and (3.8).4. Evaluation of the error norm ||Êh,N ||Let ̂σ h,N := ̂T h − ̂T h,N . Then (3.8) implies that(4.1)||Êh,N || 2 = ||Êh + ̂σ h,N || 2= Êh,(z)Êh,(w) ̂K(z, w) +} {{ }Êh,(z)̂σ h,N,(w) ̂K(z, w)} {{ }e 1+ ̂σ h,N,(z) Ê h,(w) ̂K(z, w) + ̂σ h,N,(z)̂σ h,N,(w) ̂K(z, w)} {{ } } {{ }e 3e 2e 4.In (4.1), the error is divided into four parts denoted by e 1 , e 2 , e 3 , and e 4 . In thefollowing three subsections, we will evaluate these quantities term by term.4.1. Evaluation of e 2 and e 3 . The second and third terms of (4.1) are actuallyequal. Indeed:(4.2)e 3 = ̂σ h,N,(z) Ê h,(w) ̂K(z, w) = Ê h,(z)̂σ h,N,(w) ̂K(z, w)⎛⎞= Êh,(z) ⎝h ∑ ̂K(z, nh) ⎠= Êh,(z)= h ∑|n|>N⎛|n|>N⎝h ∑|n|>NÊ h F nh⎞F nh (z) ⎠whereF nh := P nh,(w) ̂K(z, w)(4.3)=π 2 cosh(z) cosh(nh)4 cosh ( π2 sinh(z) − π 2 sinh(nh)) cosh ( π2 sinh(z)) cosh ( π2 sinh(nh)).To calculate (4.2), first we compute ÎF nhÎF nh ===∫ ∞−∞∫ ∞−∞P nh,(w) ̂K(z, w)dzπ 2 cosh(z) cosh(nh)4 cosh ( π2 sinh(z) − π 2 sinh(nh)) cosh ( π2 sinh(z)) cosh ( πdz2sinh(nh))cosh(z)π 2 ∫cosh(nh) ∞4 cosh ( π2 sinh(nh)) −∞cosh ( π2 sinh(z) − π 2 sinh(nh)) cosh ( πdz.2sinh(z))