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
(1) 
is efficient if is an analytic function on the entire infinite interval , and
(2) 
is efficient if is an analytic periodic function of period . 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]
(3) 
It is an interpolation formula with sampling points , that is,
it works very well especially if is an analytic function over the entire infinite interval , and we obtain the trapezoidal formula over the entire infinite interval by integrating (3) formally term by term, that is,
As the trapezoidal formula for numerical integration over the entire infinite interval , 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 for periodic functions with equal mesh which generates the trapezoidal formula for periodic functions, that is,
(4)  
where is the period of , and  
(5) 
This paper gives the answer “yes” to this question, that is, there exists an interpolation formula 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 be a periodic function of period . We propose the following interpolation formula for .
(6)  
We can easily check that

is an interpolation of with sampling points , that is,

is a periodic function of period , that is,

The trapezoidal formula for over with an equal mesh is obtained by integrating , that is,

The limit as and with fixed gives the Sinc approximation
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
(7)  
where is the characteristic function of the interval  
We compare (7) with the Fourier transform of
(8) 
We find that is nonzero only in the interval and, in (7), the integral on the righthand side of (8) is replaced with its approximation by the trapezoidal formula of equal mesh .
Let be a periodic function of period and expand it to the Fourier series
(9)  
where  
(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 by truncating the infinite sum on the righthand side of (8) and replace the integral on the righthand 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 righthand side of (9) as
(11) 
Then, we approximate the integral on the righthand side of (10) by its approximation by the point trapezoidal formula
(12)  
Substituting (12) into (11), we have
Using the formula
we obtain the Sinc approximation (6) for .
Second, we derive the Sinc approximation for the case that the number of sampling points is even. We truncate the Fourier series (9) as
(13)  
Then, we approximate the integral on the righthand side of (10) by its approximation by the point trapezoidal formula
(14)  
Substituting (14) into (13), we have
(15)  
Using the formula
we obtain the Sinc approximation (6) for .
3 Convergence theorem
The following theorem shows that the Sinc approximation for periodic functions converges exponentially as the number of sampling points increases if the function is an analytic periodic function.
Theorem 1
We suppose that is a periodic function of period and analytic in a complex domain containing the strip . Then we have the inequality
(16) 
where
(17) 
and is a positive constant depending and 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 be the following complex domain defined for by
We suppose that a function is analytic in the strip domain
and satisfies
Then, we have
This theorem shows that the error of the Sinc approximation is of order as the mesh goes to zero. On the other hand, the error of the approximation (6) for periodic analytic functions is of order where is the mesh size . 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 . We can prove the theorem for similarly. Since
we have
(18) 
Each term of the first term on the righthand side of (18) is the error of the approximation of the integral
by the point trapezoidal formula of the equal mesh , and, from the theorem in §4.6.5 of [1], we have
Then, we have
As for the second term on the righthand side of (18), since
we have
Therefore, we have (16) for .
4 Numerical examples
We evaluated the errors of the point Sinc approximation and the point Chebyshev interpolation
(19) 
where is the Sinc approximation or the Chebyshev interpolation, of the periodic functions with period
(20) 
The maximum on the righthand side of (19) was actually evaluated on equidistant points in the interval . All the computations were performed using programs coded in C++ and multiprecision arithmetics with 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 increases.
We evaluated the decay rate of the error 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
by using Theorem 1 for the functions (2) and (3) in (20). Since these functions have poles respectively atthe value in Theorem 1 is estimated as for the function (2) and for the function (3). Using them, we can estimate theoretically the decay rates of the error of the Sinc approximation 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).
N e  N e 
(1)  (2) 
N e  
(3) 
function  (1)  (2)  (3)  

numerical  
theoretical 
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.
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