An analog of the Sinc approximation for periodic functions

1 Introduction

The trapezoidal formula with equal mesh is an elementary numerical integration formula, and it works very well for the integral of an analytic function over the entire infinite interval and the integral of an analytic periodic function over an interval of its period length, that is, the formula

\int_{- \infty}^{\infty} f (x) d x ≃ h \infty \sum k = - \infty f (k h)

(1)

is efficient if $f (x)$ is an analytic function on the entire infinite interval $(- \infty, + \infty)$ , and

\int_{0}^{a} f (x) d x ≃ h {\frac{1}{2} f (0) + N - 1 \sum k = 1 f (k h) + \frac{1}{2} f (a)} (h = \frac{a}{N})

(2)

is efficient if $f (x)$ is an analytic periodic function of period $a$ . Especially, the formula (1) over the entire infinite interval is widely used in science and engineering combined with the technique of a variable transform [9].

By the way, there exists an interpolation formula with equal mesh over the entire infinite interval. It is the Sinc approximation [7]

f (x) ≃ L_{h} [f] (x) = \infty \sum k = - \infty f (k h) \frac{sin [(π / h) (x - k h)]}{(π / h) (x - k h)} .

(3)

It is an interpolation formula with sampling points $k h$ $(k \in Z)$ , that is,

L_{h} [f] (k h) = f (k h) (k \in Z) .

it works very well especially if $f (x)$ is an analytic function over the entire infinite interval $(- \infty, + \infty)$ , and we obtain the trapezoidal formula over the entire infinite interval by integrating (3) formally term by term, that is,

	$\int_{- \infty}^{\infty} f (x) d x ≃$	$\int_{- \infty}^{\infty} L_{h} [f] (x) d x$
	$=$	$\infty \sum k = - \infty f (k h) \int_{- \infty}^{\infty} \frac{sin [(π / h) (x - k h)]}{(π / h) (x - k h)} d x$
	$=$	$h \infty \sum k = - \infty f (k h) .$

As the trapezoidal formula for numerical integration over the entire infinite interval $(- \infty, + \infty)$ , the Sinc approximation is applied to various numerical computations combined with the technique of variable transforms such as indefinite integrals [3]

, initial value problems of ordinary differential equations

[5], boundary value problems of ordinary differential equations [8] and integral equations [4, 6].

Taking into account that

the trapezoidal formula works very well for analytic functions over the entire infinite interval and analytic periodic functions, and
the Sinc approximation is an interpolation formula over the entire infinite interval which generates the trapezoidal formula over the entire infinite interval,

it is a natural question whether there exists an interpolation formula $f (x) ≃ L [f] (x)$ for periodic functions $f (x)$ with equal mesh which generates the trapezoidal formula for periodic functions, that is,

	$L [f] (k h) = f (k h) (k = 0, 1, 2, \dots, N - 1; h = \frac{}{a} N),$	(4)
where $a$ is the period of $f (x)$ , and
	$\int_{0}^{a} L [f] (x) d x = h {\frac{1}{2} f (0) + N - 1 \sum k = 1 f (k h) + \frac{1}{2} f (a)} .$	(5)

This paper gives the answer “yes” to this question, that is, there exists an interpolation formula $f (x) ≃ L [f] (x)$ for periodic functions which satisfies the properties (4) and (5).

The remainder of this paper are structured as follows. In Section 2, we propose an interpolation formula for periodic functions which we desire. In Section 3, we show a theorem on the exponential convergence of the proposed method for periodic analytic functions. In Section 4, we show the effectiveness of the proposed method by some numerical examples. In Section 5, we show concluding remarks.

2 Sinc approximation for periodic functions

Let $f (x)$ be a periodic function of period $a$ . We propose the following interpolation formula for $f (x)$ .

	$f (x) ≃$	$L_{N} [f] (x)$		(6)
	$=$	$⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \begin{matrix} \frac{1}{2 N_{0} + 1} 2 N_{0} \sum k = 0 f (\frac{k a}{2 N_{0} + 1}) \frac{sin [(2 N_{0} + 1) \frac{π}{a} (x - \frac{k a}{2 N_{0} + 1})]}{sin [\frac{π}{a} (x - \frac{k a}{2 N_{0} + 1})]} if N = 2 N_{0} + 1 (odd), \end{matrix} \begin{matrix} \frac{1}{2 N_{0}} 2 N_{0} - 1 \sum k = 0 f (\frac{k a}{2 N_{0}}) cot [\frac{π}{a} (x - \frac{k a}{2 N_{0}})] sin [2 N_{0} \frac{π}{a} (x - \frac{k a}{2 N_{0}})] if N = 2 N_{0} (even) . \end{matrix} \end{matrix}$		(6)

We can easily check that

$L_{N} [f] (x)$ is an interpolation of $f (x)$ with sampling points $k a / N$ $(k = 0, 1, 2, \dots, N - 1)$ , that is,

$L_{N} [f] (\frac{k a}{N}) = f (\frac{k a}{N}) (k = 0, 1, 2, \dots, N - 1) .$
$L [f] (x)$ is a periodic function of period $a$ , that is,

$L_{N} [f] (x + a) = L_{N} [f] (x) .$
The trapezoidal formula for $f (x)$ over $(0, a)$ with an equal mesh $h = a / N$ is obtained by integrating $L_{N} [f] (x)$ , that is,

$\frac{a}{N} {\frac{1}{2} f (0) + N - 1 \sum k = 1 f (\frac{k a}{N}) + \frac{1}{2} f (a)} = \int_{0}^{a} L_{N} [f] (x) d x .$
The limit as $N \to \infty$ and $a \to \infty$ with $h = a / N$ fixed gives the Sinc approximation

$L_{N} [f] (x) \to L_{h} [f] (x) = \infty \sum k = - \infty f (k h) \frac{sin [(π / h) (x - k h)]}{(π / h) (x - k h)} .$

The first and second properties are satisfied also by the Sinc approximation. We call the approximation (6) the Sinc approximation for periodic functions.

Derivation

We show here how to find the Sinc approximation for periodic functions (6).

We begin with the Fourier transform of the Sinc approximation over the entire infinite interval

	$\begin{matrix} F [L_{h} [f]] (ξ) = & \int_{- \infty}^{\infty} L_{h} [f] (x) e^{- 2 π i ξ x} d x = & h \infty \sum k = - \infty f (k h) e^{- 2 π i k h} χ_{(- π / h, π / h)} (ξ), \end{matrix}$	(7)
where $χ_{(- π / h, π / h)} (ξ)$ is the characteristic function of the interval $(- π / h, π / h)$
	$χ_{(- π / h, π / h)} (ξ) = {\begin{matrix} 1 & ξ \in (- π / h, π / h) 0 & ξ \notin [- π / h, π / h] \end{matrix} .$

We compare (7) with the Fourier transform of $f (x)$

F [f] (ξ) = \int_{- \infty}^{\infty} f (x) e^{- 2 π i ξ x} d x .

(8)

We find that $F [L_{h} [f]] (ξ)$ is non-zero only in the interval $(- π / h, π / h)$ and, in (7), the integral on the right-hand side of (8) is replaced with its approximation by the trapezoidal formula of equal mesh $h$ .

Let $f (x)$ be a periodic function of period $a$ and expand it to the Fourier series

	$f (x) = \sum n \in Z {ˆ f}_{n} exp (i \frac{2 n π}{a} x),$	(9)
where
	${ˆ f}_{n} = \frac{1}{a} \int_{- \infty}^{\infty} f (x) exp (- i \frac{2 n π}{a} x) d x$	(10)

From the above observation of the Sinc approximation over the entire infinite interval, we expect that we can obtain the Sinc approximation of the periodic function $f (x)$ by truncating the infinite sum on the right-hand side of (8) and replace the integral on the right-hand side of (10) by its approximation by the trapezoidal formula.

First, we derive the Sinc approximation of the case that the number of sampling points is odd. We truncate the right-hand side of (9) as

f (x) ≃ N_{0} \sum n = - N_{0} {ˆ f}_{n} exp (i \frac{2 n π}{a} x) .

(11)

Then, we approximate the integral on the right-hand side of (10) by its approximation by the $(2 N_{0} + 1)$ -point trapezoidal formula

${ˆ f}_{n} ≃$	${ˆ f}_{n}^{(2 N_{0} + 1)}$	(12)
$=$	$\frac{1}{a} \frac{a}{2 N_{0} + 1} 2 N_{0} \sum k = 0 f (\frac{k a}{2 N_{0} + 1}) exp (- i \frac{2 n π}{a} \frac{k a}{2 N_{0} + 1})$
$=$	$\frac{1}{2 N_{0} + 1} 2 N_{0} \sum k = 0 f (\frac{k a}{2 N_{0} + 1}) exp (- i \frac{2 n k π}{2 N_{0} + 1}) .$

Substituting (12) into (11), we have

	$f (x) ≃$	$N_{0} \sum n = - N_{0} {ˆ f}_{n}^{(2 N_{0} + 1)} exp (i \frac{2 n π}{a} x)$
	$=$	$\frac{1}{2 N_{0} + 1} 2 N_{0} \sum k = 0 f (\frac{k a}{2 N_{0} + 1}) N_{0} \sum n = - N_{0} exp [i \frac{2 n π}{a} (x - \frac{}{k a} 2 N_{0} + 1)] .$

Using the formula

N_{0} \sum n = - N_{0} e^{i κ n} = \frac{sin [(N_{0} + 1 / 2) κ]}{sin (κ / 2)},

we obtain the Sinc approximation (6) for $N = 2 N_{0} + 1$ .

Second, we derive the Sinc approximation for the case that the number of sampling points is even. We truncate the Fourier series (9) as

	$f (x) ≃$	$N_{0} - 1 \sum n = - N_{0} - 1 {ˆ f}_{n} exp (i \frac{2 n π}{a} x) + \frac{1}{2} {ˆ f}_{2 N_{0}} exp (i \frac{2 (2 N_{0}) π}{a} x)$		(13)
		$+ \frac{1}{2} {ˆ f}_{- 2 N_{0}} exp (i \frac{2 \cdot (- 2 N_{0}) π}{a} x) .$		(13)

Then, we approximate the integral on the right-hand side of (10) by its approximation by the $(2 N_{0})$ -point trapezoidal formula

${ˆ f}_{n} ≃$	${ˆ f}_{n}^{(2 N_{0})}$	(14)
$=$	$\frac{1}{a} \frac{a}{2 N_{0}} 2 N_{0} - 1 \sum k = 0 f (\frac{k a}{2 N_{0}}) exp (- i \frac{2 n π}{a} \frac{k a}{2 N_{0}})$
$=$	$\frac{1}{2 N_{0}} 2 N_{0} - 1 \sum k = 0 f (\frac{k a}{2 N_{0}}) exp (- i \frac{2 n k π}{2 N_{0}}) .$

Substituting (14) into (13), we have

$f (x) ≃$	$N_{0} - 1 \sum n = - N_{0} + 1 {ˆ f}_{n}^{(2 N_{0})} exp (i \frac{2 n π}{a} x) + \frac{1}{2} {ˆ f}_{N_{0}}^{(2 N_{0})} exp (i \frac{2 N_{0} π}{a} x)$	(15)
	$+ \frac{1}{2} {ˆ f}_{- N_{0}}^{(2 N_{0})} exp (- i \frac{2 N_{0} π}{a} x)$
$=$	$\frac{1}{2 N_{0}} 2 N_{0} - 1 \sum k = 0 f (\frac{k a}{2 N_{0}}) ⎧ ⎨ ⎩ N_{0} - 1 \sum n = - N_{0} + 1 exp [i \frac{2 n π}{a} (x - \frac{k a}{2 N_{0}})]$
	$+ \frac{1}{2} exp [i \frac{2 N_{0} π}{a} (x - \frac{k a}{2 N_{0}})] + \frac{1}{2} exp [- i \frac{2 N_{0} π}{a} (x - \frac{k a}{2 N_{0}})]} .$

Using the formula

N_{0} - 1 \sum n = - N_{0} + 1 e^{i κ n} + \frac{1}{2} e^{i κ N_{0}} + \frac{1}{2} e^{- i κ N_{0}} = cot (\frac{κ}{2}) sin (N_{0} κ),

we obtain the Sinc approximation (6) for $N = 2 N_{0}$ .

3 Convergence theorem

The following theorem shows that the Sinc approximation for periodic functions converges exponentially as the number of sampling points $N$ increases if the function $f (x)$ is an analytic periodic function.

Theorem 1

We suppose that $f (z)$ is a periodic function of period $a$ and analytic in a complex domain containing the strip $| Im z | ≦ d$ . Then we have the inequality

| f (x) - L_{N} [f] (x) | ≦ C_{d, a} ∥ f ∥_{d} exp (- \frac{π d}{a} N) (\forall x \in R),

(16)

where

∥ f ∥_{d} = max | Im z | = d | f (z) |

(17)

and $C_{d, a}$ is a positive constant depending $d$ and $a$ only.

We compare this theorem with the following theorem for the Sinc approximation over the entire infinite interval (Theorem 3.1.3 in [7]).

Theorem 2

Let $D_{d} (ϵ)$ be the following complex domain defined for $0 < ϵ < 1$ by

D_{d} (ϵ) = {z \in C | | Re z | < 1 / ϵ, | Im z | < d (1 - ϵ)} .

We suppose that a function $f (z)$ is analytic in the strip domain

D_{d} = {z \in C | | Im z | < d}

and satisfies

N (f, D_{d}) \equiv lim ϵ \to 0 \int_{\partial D_{d} (ϵ)} | f (z) | | d z | < \infty .

Then, we have

| f (x) - L_{h} [f] (x) | ≦ \frac{1}{π d} N (f, D_{d}) \frac{exp (- π d / h)}{1 - exp (- 2 π d / h)} (\forall x \in R) .

This theorem shows that the error of the Sinc approximation is of order $O [exp (- π d / h)]$ as the mesh $h$ goes to zero. On the other hand, the error of the approximation (6) for periodic analytic functions is of order $O [exp (- π d / h)]$ where $h$ is the mesh size $h = a / N$ . Therefore, the error of the Sinc approximation for periodic functions is of the same order as that of the Sinc approximation over the entire infinite interval.

Proof of Theorem 1

We prove Theorem 1 only for $N = 2 N_{0} + 1$ . We can prove the theorem for $N = 2 N_{0}$ similarly. Since

f (x) - L_{2 N_{0} + 1} [f] (x) = N_{0} \sum n = - N_{0} ({ˆ f}_{n} - {ˆ f}_{n}^{(2 N_{0} + 1)}) exp (i \frac{2 n π}{a} x) - \sum | n | > N_{0} {ˆ f}_{n} exp (i \frac{2 n π}{a} x),

we have

| f (x) - L_{2 N_{0} + 1} [f] (x) | ≦ N_{0} \sum n = - N_{0} | {ˆ f}_{n} - {ˆ f}_{n}^{(2 N_{0} + 1)} | + \sum | n | > N_{0} | {ˆ f}_{n} | .

(18)

Each term of the first term on the right-hand side of (18) is the error of the approximation of the integral

by the $(2 N_{0} + 1)$ -point trapezoidal formula of the equal mesh $h = a / (2 N_{0} + 1)$ , and, from the theorem in §4.6.5 of [1], we have

| {ˆ f}_{n} - {ˆ f}_{n}^{(2 N_{0} + 1)} | ≦ 2 ∥ f ∥_{d} exp (\frac{2 π d}{a} | n |) \frac{exp (- (2 π d / a) (2 N_{0} + 1))}{1 - exp (- 2 π d / a)} .

Then, we have

	$\| {ˆ f}_{n} - {ˆ f}_{n}^{(2 N_{0} + 1)} \| ≦$	$2 ∥ f ∥_{d} \frac{exp (- (2 π d / a) (2 N_{0} + 1))}{1 - exp (- 2 π d / a)} N_{0} \sum n = - N_{0} exp (\frac{2 π d}{a} \| n \|)$
	$≦$	$2 ∥ f ∥_{d} \frac{exp (- (2 π d / a) N_{0})}{[1 - exp (- 2 π d / a)]^{2}} .$

As for the second term on the right-hand side of (18), since

| f_{n} | ≦ ∥ f ∥_{d} exp (- \frac{2 | n | π d}{a}) (n \neq 0),

we have

	$\sum \| n \| > N_{0} \| {ˆ f}_{n} \| ≦$	$∥ f ∥_{d} \sum \| n \| > N_{0} exp (- \frac{2 \| n \| π d}{a})$
	$≦$	$2 ∥ f ∥_{d} \frac{exp (- (2 π d / a) (N_{0} + 1))}{1 - exp (- 2 π d / a)} .$

Therefore, we have (16) for $N = 2 N_{0} + 1$ .

4 Numerical examples

We evaluated the errors of the $N$ -point Sinc approximation and the $N$ -point Chebyshev interpolation

ϵ_{N} = max 0 ≦ x ≦ a | f (x) - L_{N} [f] (x) |,

(19)

where $L_{N} [f] (x)$ is the Sinc approximation or the Chebyshev interpolation, of the periodic functions with period $a = 2 π$

(1) exp (sin x), (2) \frac{1}{2 + cos x}, (3) \frac{1}{5 + cos x} .

(20)

The maximum on the right-hand side of (19) was actually evaluated on $1024$ equidistant points in the interval $[0, 2 π)$ . All the computations were performed using programs coded in C++ and multi-precision arithmetics with $100$ decimal digit precision by the software exflib [2]. Figure 1 shows the errors of the two methods applied to the functions (20). From these figures, the Sinc approximation is superior to the Chebyshev interpolation and converges exponentially as the number of sampling points $N$ increases.

We evaluated the decay rate of the error $ϵ_{N}$ of the Sinc approximation using the fit command of the software gnuplot. Table 1

shows the results. We can also estimate the decay rate of the error

ϵ_{N}

by using Theorem 1 for the functions (2) and (3) in (20). Since these functions have poles respectively at

	$(2)$	$z = - i log (- 2 \pm \sqrt{3}) = \mp 1.3170 \dots i + (2 n + 1) π (n \in Z),$
	$(3)$	$z = - i log (\frac{- 5 \pm \sqrt{21}}{2}) = \mp 1.566 8 \dots i + (2 n + 1) π (n \in Z),$

the value $d$ in Theorem 1 is estimated as $d \approx 1.3170 \dots$ for the function (2) and $d \approx 1.5669 \dots$ for the function (3). Using them, we can estimate theoretically the decay rates of the error of the Sinc approximation $ϵ_{N} = O [exp (- (π d / a) N)] = O [exp ((- d / 2) N)],$ which are shown in Table 1. From this table, the theoretical error estimate make a good agreement with the numerical result especially for the function (2).

Figure 1: The errors of the proposed method and the Chebyshev interpolation applied to the functions (20).

function		(1)	(2)	(3)
$ϵ_{N}$	numerical	$O ({0.22}^{N})$	$O ({0.52}^{N})$	$O ({0.32}^{N})$
	theoretical		$O ({0.52}^{N})$	$O ({0.46}^{N})$

Table 1: The decay rate of the error of the proposed method applied to the functions (20).

5 Summary

In this paper, we proposed an approximation of periodic functions, which can be regarded as an analog of the Sinc approximation for functions over the entire infinite interval in the sense that it has equidistant sampling points and it gives the trapezoidal formula with equal mesh by integrating it. From theoretical error estimate and numerical examples, it converges exponentially as the number of sampling points increases if the approximated function is analytic.

As an application of the proposed approximation, it is expected that it can be applied to the boundary element method for boundary value problems of partial differential equations.

References

[1] P. J. Davis and P. Rabinowitz (1984) Methods of numerical integration, second ed.. Academic Press, San Diego. Cited by: §3.
[2] H. Fujiwara Exflib information. Note: http://www-an.acs.i.kyoto-u.ac.jp/~fujiwara/exflib/ Cited by: §4.
[3] M. Muhammad and M. Mori (2003) Double exponential formula for numerical indefinite integration. J. Comput. Appl. Math. 161, pp. 431–448. Cited by: §1.
[4] M. Muhammad, A. Nurmuhammad, M. Mori, and M. Sugihara (2005) Numerical solution of integral equations by means of the sinc collocation method based on the double exponential transformation. J. Comput. Appl. Math. 177, pp. 269–286. Cited by: §1.
[5] A. Nurmuhammad, M. Muhammad, and M. Mori (2005) Numerical solution of initial value problems based on the double exponential transformation. Publ. RIMS, Kyoto Univ. 41, pp. 937–948. Cited by: §1.
[6] T. Okayama, T. Matsuo, and M. Sugihara (2011) Improvement of a sinc-collocation method for fredholm integral equations of the second kind. BIT Numer. Math. 51, pp. 339–366. Cited by: §1.
[7] F. Stenger (1993) Numerical methods based on sinc and analytic functions. Springer-Verlag, New York. Cited by: §1, §3.
[8] M. Sugihara (2002) Double exponential transformation in the sinc-collocation method for two-point boundary value problems. J. Comput. Appl. Maht. 149, pp. 239–250. Cited by: §1.
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4 Numerical examples

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An analog of the Sinc approximation for periodic functions

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