1997
DOI: 10.1007/s002110050244
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Optimality of the double exponential formula - functional analysis approach -

Abstract: In the light of the functional analysis theory we establish the optimality of the double exponential formula. The argument consists of the following three ingredients: (1) introduction of a number of spaces of functions analytic in a strip region about the real axis, each space being characterized by the decay rate of their elements (functions) in the neighborhood of the infinity; (2) proof of the (near-) optimality of the trapezoidal formula in each space introduced in (1) by showing the (near-) equality betw… Show more

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Cited by 86 publications
(80 citation statements)
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“…Here, the optimality means the superiority of the formula in its accuracy over any other formula applicable to the elements of the function space. For example, such a problem has been addressed for a trapezoidal formula [8] and a sinc interpolation [9].…”
Section: Discussionmentioning
confidence: 99%
See 1 more Smart Citation
“…Here, the optimality means the superiority of the formula in its accuracy over any other formula applicable to the elements of the function space. For example, such a problem has been addressed for a trapezoidal formula [8] and a sinc interpolation [9].…”
Section: Discussionmentioning
confidence: 99%
“…A variety of numerical methods based on sinc approximations has been studied during the last three decades [6,7,8,9]. The methods cover function approximation, approximation of derivatives, approximate definite and indefinite integration, approximate solution of initial and boundary value ODE problems, and so on.…”
Section: Introductionmentioning
confidence: 99%
“…In the introduction we have remarked that for small values of n, the approach we have proposed gives slightly better results than the DE-rule. This is due to the fact that the stepsize h of the DE-rule is chosen to optimize the rate of convergence, that is, the behavior of the remainder term as n → ∞ (see [15]). Indeed, as n → ∞ the accuracy given by the DE-rule is superior to that given by our numerical integration approach.…”
Section: One-dimensional Integration Rulesmentioning
confidence: 99%
“…It is the well-known double exponential (DE) transformation (see [16], [10], [15]). In this paper we examine the use of nonlinear changes of variable for the numerical evaluation of some 4-dimensional integrals arising in the numerical solution of hypersingular boundary integral equations.…”
Section: §1 Introductionmentioning
confidence: 99%
“…By means of the DE-formula of Takahashi and Mori (1974) (see also Mori and Sugihara 2001), whose optimality is mathematically proven by Sugihara (1997), Kawabata (2016) repeated his foregoing calculations (Kawabata and Limaye 2011) of the isotropic scattering H-function, and demonstrated that it is possible to get numerical values with accuracy of 15 digits in double precision arithmetic. In fact, the results are in perfect agreement within one unit difference in the 15-th decimal place with those of Jablonski (2015).…”
Section: Introductionmentioning
confidence: 99%