Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration

Abstract

Error estimates with explicit constants are given for approximations of functions, definite integrals and indefinite integrals by means of the Sinc approximation. Although in the literature various error estimates have already been given for these approximations, those estimates were basically for examining the rates of convergence, and several constants were left unevaluated. Giving more explicit estimates, i.e., evaluating these constants, is of great practical importance, since by this means we can reinforce the useful formulas with the concept of “verified numerical computations.” In this paper we reveal the explicit form of all constants in a computable form under the same assumptions of the existing theorems: the function to be approximated is analytic in a suitable region. We also improve some formulas themselves to decrease their computational costs. Numerical examples that confirm the theory are also given.

Appendix A: On the differences between the function spaces

Here we explain how Definition 2.1 ($\mathbf{L}_{K,\alpha ,\beta }({\fancyscript{D}})$) differs from the definition in Stenger [18, Definition 4.1.1] ($\tilde{\mathbf{L}}_{\alpha ,\beta }({\fancyscript{D}})$), which is described below.

Definition 6.1

Let ${\fancyscript{D}}$ be a simply connected domain which satisfies $(a,\,b)\subset {\fancyscript{D}}$, and let $\alpha ,\,\beta$ be positive constants. Let $\phi$ be a conformal map of ${\fancyscript{D}}$ onto ${\fancyscript{D}}_{ d}$, for a constant ${ d}>0$. Then $\tilde{\mathbf{L}}_{\alpha ,\beta }({\fancyscript{D}})$ denotes the family of all functions $f$ that are analytic on ${\fancyscript{D}}$ and satisfy for all $z$ in ${\fancyscript{D}}$ the condition that

$$\begin{aligned} |f(z)|\le C\frac{1}{(1+|\,\mathrm{e}^{-\phi (z)}|)^{\alpha }(1+|\,\mathrm{e}^{\phi (z)}|)^{\beta }}. \end{aligned}$$

(6.1)

1.1 Appendix A.1: The difference in the SE-Sinc case

If $\phi (z)$ is the SE transformation $\psi _{\text{ SE }}^{-1}(z)$, the condition (2.7) is equivalent to

$$\begin{aligned} |f(z)| \le K \frac{(b-a)^{\alpha +\beta }}{|1 + \mathrm{e}^{-\phi (z)}|^{\alpha }|1 + \mathrm{e}^{\phi (z)}|^{\beta }}. \end{aligned}$$

(6.2)

The substantive difference between (6.1) and (6.2) is the place of the absolute value sign $|\,\cdot \,|$. And, as written in Stenger [18, Example 4.2.8], the conditions (6.1) and (6.2) are essentially equivalent if we do not care about the concrete forms of the constants $C$ and $K$. Theorems 2.3, 2.5 and 2.8 hold even if we replace $\tilde{\mathbf{L}}_{\alpha ,\beta }({\fancyscript{D}})$ with $\mathbf{L}_{K,\alpha ,\beta }({\fancyscript{D}})$, and vice versa.

However, the goal of this paper is to reveal all the constants in the error estimates. From this point of view clearly (6.1) and (6.2) are not exactly the same conditions. Therefore we have to determine which function spaces to employ in order to give the desired estimates. In this paper, Definition 2.1 is employed for the two reasons described below.

1.2 Appendix A.2: The difference in the DE-Sinc case

If $\phi (z)$ is the DE transformation $\psi _{\text{ DE }}^{-1}(z)$, we can rewrite the condition (6.1) as

$$\begin{aligned} |f(\psi _{\text{ DE }}(\zeta ))| \le C \frac{1}{(1 + |\,\mathrm{e}^{-\zeta }|)^{\alpha }(1 + |\,\mathrm{e}^{\zeta }|)^{\beta }}, \end{aligned}$$

(6.3)

substituting the DE transformation $z=\psi _{\text{ DE }}(\zeta )$. On the other hand, the condition (2.7) is equivalent to

$$\begin{aligned} |f(\psi _{\text{ DE }}(\zeta ))| \le K \frac{(b-a)^{\alpha +\beta }}{|1 + \mathrm{e}^{-\uppi \sinh (\zeta )}|^{\alpha }|1 + \mathrm{e}^{\uppi \sinh (\zeta )}|^{\beta }}. \end{aligned}$$

(6.4)

Compare (6.3) with (6.4). The decay rate (as $|\mathrm Re \zeta |\rightarrow \infty$) of (6.3) is single-exponential, whereas the rate of (6.4) is double-exponential. This means that the function space $\tilde{\mathbf{L}}_{\alpha ,\beta }({\fancyscript{D}})$ does not include the DE case. Actually, It is well known that the condition (6.1) is not suited for the DE transformation, and for this reason some authors [12, 25, 26] have already adopted the condition (2.7).

Furthermore, as described above, the condition (2.7) is also meaningful in the SE-Sinc case, and thus can be used throughout the present paper. This is convenient from a theoretical viewpoint.

1.3 Appendix A.3: Practicality of Definition 2.1

The second reason is that Definition 2.1 is more natural from a practical viewpoint, since the condition (2.7) is directly related to popular conditions such as Lipschitz/Hölder continuity. In fact, Stenger [18, Example 4.2.8] has explained a way to construct a function $G$ in $\mathbf{L}_{K,\alpha ,\alpha }({\fancyscript{D}})$ from a function $F$ analytic in ${\fancyscript{D}}$ and $\alpha$-order Hölder continuous on $\overline{{\fancyscript{D}}}$. This means the constant $K$ in (2.7) is more easily found in practice, and the constant $C$ in (6.1) should be estimated from the condition (2.7) (or (6.2)). This is why we believe that the condition (2.7) is more natural than (6.1) from a practical viewpoint.