arXiv subjects

Tomoaki Okayama

Publications and source records attributed to Tomoaki Okayama.

At least 19 recordsLinked to original sources

arXiv2026

Relation between two Sinc-collocation methods for Volterra integral equations of the second kind and further improvement

Stenger and Rashidinia--Zarebnia independently proposed two Sinc-collocation methods for Volterra integral equations of the second kind, but the relation between the methods has not been clarified. This study reformulates Stenger's method for general two-variable kernels and rigorously establishes its applicability and convergence. We prove that the approximate functions produced by the two methods are not generally identical, although they coincide at every collocation point. We also provide a rigorous convergence proof for the Rashidinia--Zarebnia method and show that both methods attain the same root-exponential convergence rate. Because Stenger's method is simpler to implement than the Rashidinia--Zarebnia method, we adopt it as the basis for further improvement. By replacing the tanh transformation with the double-exponential transformation, we develop a new Sinc-collocation method and prove its almost exponential convergence. Numerical examples support the theoretical results and demonstrate the favorable balance between accuracy and computational cost of the proposed method.

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arXiv2026

DE-Sinc approximation for unilateral rapidly decreasing functions and its computable error bound

The Sinc approximation is highly effective for functions that decay rapidly at both ends of the real axis. For unilateral rapidly decreasing functions, which decay algebraically as $t\to-\infty$ and exponentially as $t\to\infty$, an appropriate variable transformation is required. Existing single-exponential transformations for this class of functions yield only root-exponential convergence, even when improved transformations are employed. This paper develops a double-exponential (DE)-Sinc approximation based on the transformation $t=2\sinh(\log(\log(1+\exp(π\sinh x))))$, which was previously introduced for numerical integration. The main contribution is a rigorous and computable error bound of order $\operatorname{O}(\exp(-cn/\log n))$ with a constant explicitly expressed in terms of the problem parameters. Under the stated assumptions, the resulting approximation achieves almost exponential convergence and is suitable for computation with guaranteed accuracy. Numerical examples satisfying these assumptions confirm the predicted convergence behavior and the validity of the derived error bound.

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arXiv2026

Improvement of conformal maps combined with the Sinc approximation for derivatives over infinite intervals

F. Stenger proposed efficient approximation formulas for derivatives over infinite intervals. These formulas were derived by combining the Sinc approximation with appropriate conformal maps. It has been demonstrated that these formulas can attain root-exponential convergence. In this study, we enhance the convergence rate by improving the conformal maps employed in those formulas. We provide a theoretical error analysis and numerical experiments that confirm the effectiveness of our new formulas.

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arXiv2026

Refinement of the theory and convergence of the Sinc convolution -- beyond Stenger's conjecture

The Sinc convolution is an approximate formula for indefinite convolutions proposed by Stenger. The formula was derived based on the Sinc indefinite integration formula combined with the single-exponential transformation. Although its efficiency has been confirmed in various fields, several theoretical issues remain unresolved. The first contribution of this study is to resolve those issues by refining the underlying theory of the Sinc convolution. This contribution includes an essential resolution of Stenger's conjecture. The second contribution of this study is to improve the convergence rate by replacing the single-exponential transformation with the double-exponential transformation. Theoretical analysis and numerical experiments confirm that the modified formula achieves superior convergence compared to Stenger's original formula.

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arXiv2025

Explicit error bounds of the SE and DE formulas for integrals with logarithmic and algebraic singularity

The single exponential (SE) and double exponential (DE) formulas are widely recognized as efficient quadrature formulas for evaluating integrals with endpoint singularity. For integrals exhibiting algebraic singularity, explicit error bounds in a computable form have been provided, enabling computations with guaranteed accuracy. Such explicit error bounds have also been provided for integrals exhibiting logarithmic singularity. However, these error bounds have two points to be discussed. The first point is on overestimation of divergence speed of logarithmic singularity. The second point is on the case where there exist both logarithmic and algebraic singularity. To address these issues, this study provides new error bounds for integrals with logarithmic and algebraic singularity. Although existing and new error bounds described above pertain to integrals over the finite interval, the SE and DE formulas are also applicable to integrals over the semi-infinite interval. On the basis of the new results, this study provides new error bounds for integrals over the semi-infinite interval with logarithmic and algebraic singularity at the origin.

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arXiv2025

Improvement of Sinc-collocation methods for Volterra-Fredholm integral equations of the second kind and their theoretical analysis

Sinc-collocation methods for Volterra-Fredholm integral equations of the second kind were proposed independently by multiple authors: by Shamloo et al. in 2012 and by Mesgarani and Mollapourasl in 2013. Their theoretical analyses and numerical experiments suggest that the presented methods can attain root-exponential convergence. However, their convergence has not been strictly proved. This study improves these methods to facilitate implementation, and provides a convergence theorem for the improved method. For the same equations, another Sinc-collocation method was proposed in 2016 by John and Ogbonna, which is regarded as an improvement to the variable transformation employed by Shamloo et al. It may attain a higher rate than the previous methods, but its convergence has not yet been proved. Therefore, this study improves it to facilitate implementation, and provides a convergence theorem for the improved method.

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arXiv2024

Error analyses of Sinc-collocation methods for exponential decay initial value problems

Nurmuhammad et al. developed the Sinc-Nyström methods for initial value problems in which the solutions exhibit exponential decay end behavior. In these methods, the Single-Exponential (SE) transformation or the Double-Exponential (DE) transformation is combined with the Sinc approximation. Hara and Okayama improved on these transformations to attain a better convergence rate, which was later supported by theoretical error analyses. However, these methods have a computational drawback owing to the inclusion of a special function in the basis functions. To address this issue, Okayama and Hara proposed Sinc-collocation methods, which do not include any special function in the basis functions. This study conducts error analyses of these methods.

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arXiv2023

Improvement of selection formulas of mesh size and truncation numbers for the DE-Sinc approximation and its theoretical error bound

The Sinc approximation applied to double-exponentially decaying functions is referred to as the DE-Sinc approximation. Because of its high efficiency, this method has been used in various applications. In the Sinc approximation, the mesh size and truncation numbers should be optimally selected to achieve its best performance. However, the standard selection formula has only been "near-optimally" selected because the optimal formula of the mesh size cannot be expressed in terms of elementary functions of truncation numbers. In this study, we propose two improved selection formulas. The first one is based on the concept by an earlier research that resulted in a better selection formula for the double-exponential formula. The formula performs slightly better than the standard one, but is still not optimal. As a second selection formula, we introduce a new parameter to propose truly optimal selection formula. We provide explicit error bounds for both selection formulas. Numerical comparisons show that the first formula gives a better error bound than the standard formula, and the second formula gives a much better error bound than the standard and first formulas.

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arXiv2023

Sinc-collocation methods with consistent collocation points for Fredholm integral equations of the second kind

Sinc-collocation methods are known to be efficient for Fredholm integral equations of the second kind, even if functions in the equations have endpoint singularity. However, existing methods have the disadvantage of inconsistent collocation points. This inconsistency complicates the implementation of such methods, particularly for large-scale problems. To overcome this drawback, this study proposes another Sinc-collocation methods with consistent collocation points. The results of a theoretical error analysis show that the proposed methods have the same convergence property as existing methods. Numerical experiments suggest the superiority of the proposed methods in terms of implementation and computational cost.

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arXiv2022

Yet another DE-Sinc indefinite integration formula

Based on the Sinc approximation combined with the tanh transformation, Haber derived an approximation formula for numerical indefinite integration over the finite interval (-1, 1). The formula uses a special function for the basis functions. In contrast, Stenger derived another formula, which does not use any special function but does include a double sum. Subsequently, Muhammad and Mori proposed a formula, which replaces the tanh transformation with the double-exponential transformation in Haber's formula. Almost simultaneously, Tanaka et al. proposed another formula, which was based on the same replacement in Stenger's formula. As they reported, the replacement drastically improves the convergence rate of Haber's and Stenger's formula. In addition to the formulas above, Stenger derived yet another indefinite integration formula based on the Sinc approximation combined with the tanh transformation, which has an elegant matrix-vector form. In this paper, we propose the replacement of the tanh transformation with the double-exponential transformation in Stenger's second formula. We provide a theoretical analysis as well as a numerical comparison.

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arXiv2020

New conformal map for the trapezoidal formula for infinite integrals of unilateral rapidly decreasing functions

While the trapezoidal formula can attain exponential convergence when applied to infinite integrals of bilateral rapidly decreasing functions, it is not capable of this in the case of unilateral rapidly decreasing functions. To address this issue, Stenger proposed the application of a conformal map to the integrand such that it transforms into bilateral rapidly decreasing functions. Okayama and Hanada modified the conformal map and provided a rigorous error bound for the modified formula. This paper proposes a further improved conformal map, with two rigorous error bounds provided for the improved formula. Numerical examples comparing the proposed and existing formulas are also given.

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arXiv2019

New conformal map for the Sinc approximation for exponentially decaying functions over the semi-infinite interval

The Sinc approximation has shown high efficiency for numerical methods in many fields. Conformal maps play an important role in the success, i.e., appropriate conformal map must be employed to elicit high performance of the Sinc approximation. Appropriate conformal maps have been proposed for typical cases; however, such maps may not be optimal. Thus, the performance of the Sinc approximation may be improved by using another conformal map rather than an existing map. In this paper, we propose a new conformal map for the case where functions are defined over the semi-infinite interval and decay exponentially. Then, we demonstrate in both theoretical and numerical ways that the convergence rate is improved by replacing the existing conformal map with the proposed map.

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arXiv2017

Theoretical analysis of a Sinc-Nyström method for Volterra integro-differential equations and its improvement

A Sinc-Nyström method for Volterra integro-differential equations was developed by Zarebnia in 2010. The method is quite efficient in the sense that exponential convergence can be obtained even if the given problem has endpoint singularity. However, its exponential convergence has not been proved theoretically. In addition, to implement the method, the regularity of the solution is required, although the solution is an unknown function in practice. This paper reinforces the method by presenting two theoretical results: 1) the regularity of the solution is analyzed, and 2) its convergence rate is rigorously analyzed. Moreover, this paper improves the method so that a much higher convergence rate can be attained, and theoretical results similar to those listed above are provided. Numerical comparisons are also provided.

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arXiv2017

Error estimates with explicit constants for the Sinc approximation over infinite intervals

The Sinc approximation is a function approximation formula that attains exponential convergence for rapidly decaying functions defined on the whole real axis. Even for other functions, the Sinc approximation works accurately when combined with a proper variable transformation. The convergence rate has been analyzed for typical cases including finite, semi-infinite, and infinite intervals. Recently, for verified numerical computations, a more explicit, "computable" error bound has been given in the case of a finite interval. In this paper, such explicit error bounds are derived for other cases.

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arXiv2016

An optimal approximation formula for functions with singularities

We propose an optimal approximation formula for analytic functions that are defined on a complex region containing the real interval $(-1,1)$ and possibly have algebraic singularities at the endpoints of the interval. As a space of such functions,we consider a Hardy space with the weight given by $w_μ(z) = (1-z^{2})^{μ/2}$ for $μ> 0$, and formulate the optimality of an approximation formula for the functions in the space. Then, we propose an optimal approximation formula for the space for any $μ> 0$ as opposed to existing results with the restriction $0 < μ< μ_{\ast}$ for a certain constant $μ_{\ast}$. We also provide the results of numerical experiments to show the performance of the proposed formula.

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arXiv2016

Potential theoretic approach to design of accurate formulas for function approximation in symmetric weighted Hardy spaces

We propose a method for designing accurate interpolation formulas on the real axis for the purpose of function approximation in weighted Hardy spaces. In particular, we consider the Hardy space of functions that are analytic in a strip region around the real axis, being characterized by a weight function $w$ that determines the decay rate of its elements in the neighborhood of infinity. Such a space is considered as a set of functions that are transformed by variable transformations that realize a certain decay rate at infinity. Popular examples of such transformations are given by the single exponential (SE) and double exponential (DE) transformations for the SE-Sinc and DE-Sinc formulas, which are very accurate owing to the accuracy of sinc interpolation in the weighted Hardy spaces with single and double exponential weights $w$, respectively. However, it is not guaranteed that the sinc formulas are optimal in weighted Hardy spaces, although Sugihara has demonstrated that they are near optimal. An explicit form for an optimal approximation formula has only been given in weighted Hardy spaces with SE weights of a certain type. In general cases, explicit forms for optimal formulas have not been provided so far. We adopt a potential theoretic approach to obtain almost optimal formulas in weighted Hardy spaces in the case of general weight functions $w$. We formulate the problem of designing an optimal formula in each space as an optimization problem written in terms of a Green potential with an external field. By solving the optimization problem numerically, we obtain an almost optimal formula in each space. Furthermore, some numerical results demonstrate the validity of this method. In particular, for the case of a DE weight, the formula designed by our method outperforms the DE-Sinc formula.

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arXiv2015

Explicit error bound for modified numerical iterated integration by means of Sinc methods

This paper reinforces numerical iterated integration developed by Muhammad--Mori in the following two points: 1) the approximation formula is modified so that it can achieve a better convergence rate in more general cases, and 2) explicit error bound is given in a computable form for the modified formula. The formula works quite efficiently, especially if the integrand is of a product type. Numerical examples that confirm it are also presented.

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arXiv2013

Error Estimates with Explicit Constants for Sinc Quadrature and Sinc Indefinite Integration over Infinite Intervals

The Sinc quadrature and the Sinc indefinite integration are approximation formulas for definite integration and indefinite integration, respectively, which can be applied on any interval by using an appropriate variable transformation. Their convergence rates have been analyzed for typical cases including finite, semi-infinite, and infinite intervals. In addition, for verified automatic integration, more explicit error bounds that are computable have been recently given on a finite interval. In this paper, such explicit error bounds are given in the remaining cases on semi-infinite and infinite intervals.

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