On an optimal quadrature formula for approximation of Fourier integrals in the space L_2^(1)

This paper deals with the construction of an optimal quadrature formula for the approximation of Fourier integrals in the Sobolev space L_2^(1)[a,b] of non-periodic, complex valued functions which are square integrable with first order derivative. Here the quadrature sum consists of linear combination of the given function values in a uniform grid. The difference between the integral and the quadrature sum is estimated by the norm of the error functional. The optimal quadrature formula is obtained by minimizing the norm of the error functional with respect to coefficients. Analytic formulas for optimal coefficients can also be obtained using discrete analogue of the differential operator d^2/d x^2. In addition, the convergence order of the optimal quadrature formula is studied. It is proved that the obtained formula is exact for all linear polynomials. Thus, it is shown that the convergence order of the optimal quadrature formula for functions of the space C^2[a,b] is O(h^2). Moreover, several numerical results are presented and the obtained optimal quadrature formula is applied to reconstruct the X-ray Computed Tomography image by approximating Fourier transforms.

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1 Introduction

In practice, since we have discrete values of an integrand, the Fourier transforms are reduced to an approximation of the integral of type

(1.1)

with . For example, the problem of X-ray Computed Tomography (CT) is to reconstruct the function from its Radon transform. One of the widely used analytic methods in CT image reconstruction is the filtered back-projection method in which the Fourier transforms are used (see [14, Chapter 3] or formulas (4.9)-(4.11) of section 4.2).

It should be recalled that integrals of type (1.1) with strongly oscillating integrands are used in applications of mathematics and other sciences. They are mainly calculated using special effective methods of numerical integration (for review see, for example, [1, 3, 4, 11, 13, 16, 17, 18, 19, 22, 28], and references therein).

Based on Sobolev’s method, the problem of the construction of optimal quadrature formulas for numerical calculation of Fourier coefficients (1.1) with in Hilbert spaces and was studied in [6] and [7], respectively. In these works, explicit formulas of optimal coefficients were obtained for . In particular, for , the convergence order of optimal quadrature formulas was studied.

Recently, in [29] the optimal quadrature formulas were studied for integrals with arbitrary weights in Sobolev space . General formulas were obtained for the worst-case error depending on nodes. Especially, when calculating Fourier coefficients of the form (1.1) with real , it was proved that equidistant nodes are optimal if , where is the number of nodes in the quadrature formula.

It should be noted that for numerical calculation of the integral (1.1) with real , a quadrature formula with explicit coefficients is needed. Therefore, in this paper, we study the construction of optimal quadrature formulas in the sense of Sard for the approximation of Fourier integrals of the form (1.1) with in the Sobolev space of non-periodic square integrable functions with the first order derivative. We obtain explicit formulas for optimal coefficients and calculate the norm of the error functional of the optimal quadrature formula. We note that the obtained optimal quadrature formula can be used to approximate Fourier integrals and reconstruct a function from its discrete Radon transform.

The rest of the paper is organized as follows. In Section 2, an optimal quadrature formula in the sense of Sard is constructed to approximate Fourier integrals in the space . In Section 3, the results of Section 2 are extended to the case of arbitrary interval

by linear transformation. That is, an optimal quadrature formula is obtained for approximate Fourier integrals in the space

. Finally, in Section 4 the obtained quadrature formula is applied to the approximation of Fourier transforms of a function using the given values of the function and to the reconstruction of the X-ray CT image.

2 Construction of optimal quadrature formula for the interval

Consider the quadrature formula

(2.1)

with the error

(2.2)

where

and the corresponding error functional

(2.3)

Here, are coefficients of the formula (2.1), , , , with ,

is the characteristic function of the interval

, and is the Dirac’s delta-function. The function belongs to the Sobolev space of complex valued functions which are defined in the interval and square integrable with the first order derivative. In this space, the inner product is defined as

(2.4)

where is the complex conjugate function for the function and the norm of the function is denoted by

We note that the coefficients in the formula (2.1) vary by and , that is .

The error (2.2) in the quadrature formula (2.1) is a linear functional in , where is the conjugate space for the space .

The absolute value of the error (2.2) is estimated by Cauchy-Schwarz inequality as

where

(2.5)

is the norm of the error functional (2.3).

In the sense of Sard [20], the problem of construction of the optimal quadrature formula (2.1) is to find the minimum of the norm (2.5) of the error functional by coefficients when nodes are fixed. Here, we note that distances between adjacent nodes in the formula (2.1) are the same. For the quadrature formulas of the form (2.1) with , this problem was first studied by Sard in space for some , where is the space of real-valued functions which are square integrable with th generalized derivative. Also this problem for the case has been investigated by many authors using splines, function and Sobolev methods. For example, see [2, 9, 12, 15, 23, 25, 26, 27] and references therein.

Therefore, in order to construct optimal quadrature formulas of the form (2.1) in the sense of Sard in the space , the following problem needs to be solved.

Problem 1

Find the coefficients that satisfy the equality

(2.6)

In this section we solve Problem 1 for the case with by finding the norm (2.5) and minimizing it by coefficients .

2.1 The norm of the error functional (2.3)

To find the norm (2.5), we use the extremal function for the error functional (see [26, 27]) that satisfies the following equality:

(2.7)

Since is a Hilbert space, we obtain

(2.8)

using the Riesz theorem for , where is the inner product of the functions and defined by (2.4) and , respectively. In addition, the equality is achieved. Then we obtain

(2.9)

from (2.7). In order for the error functional (2.3) to be defined in the space , the condition

(2.10)

must be imposed which means that the quadrature formula (2.1) is exact for any constant term.

For in (2.8) we have

(2.11)
(2.12)

where is the complex conjugate to . Then the following theorem holds.

Theorem 1

The solution of the boundary value problem (2.11)-(2.12) is the extremal function of the error functional , expressed as

(2.13)

where

(2.14)

, a complex number, and is the convolution operation.

From Sobolev’s result (see [26, 27]) on the extremal function of quadrature formulas in the space , we can get the statement of Theorem 1, especially when .

Next, we assume that

(2.15)

where and are real numbers. Then, using (2.10) and (2.13) for the norm of the error functional with (2.9), we get

Therefore, by direct calculation with (2.15), we get

(2.16)

Then from (2.10) with (2.15), we obtain the following equalities:

(2.17)
(2.18)

Thus, we get the expression (2.16) for the norm of the error functional (2.3).

Further, in the next section we will solve Problem 1.

2.2 Minimization of the expression (2.16) by coefficients

Problem 1 is equivalent to the problem minimizing (2.16) in and using Lagrange method under the conditions (2.17) and (2.18).

Now we consider the function

By making the partial derivatives of with respect to , , , and equal to zero, we get the following system of linear equations:

(2.19)
(2.20)
(2.21)
(2.22)

We multiple both sides of (2.21) and (2.22) by and add these to (2.19) and (2.20), respectively, to obtain a system of linear equations with unknowns , , and :

(2.23)
(2.24)

where is defined in (2.14). The system (2.23)-(2.24) has a unique solution. The uniqueness of the solution of this system can be proved by the uniqueness of the solution of the system (3.1)-(3.2) in [24]. The solution of the system (2.23)-(2.24) provides the minimum of at . The quadrature formula of the form (2.1) with coefficients is called the optimal quadrature formula in the sense of Sard, and are said to be the optimal coefficients. For convenience, the optimal coefficients will be denoted as .

The purpose of this section is to obtain an analytic solution for the system (2.23)-(2.24). To do this, we use the concept of discrete argument functions and operations. The theory of discrete argument functions is given in [26, 27]. We give the definition for the function of discrete argument. Suppose that nodes has uniform spacing (i.e., is a small positive parameter), and functions and are complex-valued and defined on the real line or on an interval of .

The function is a function of discrete argument if it is given on some set of integer values of . The inner product of two discrete argument functions and is given by

if the series on the right hand side of the last equality converges absolutely. The convolution of two functions and is the inner product

We also use the discrete analogue for the operator , that satisfies

(2.25)

where , is equal to 0 when , and 1 when .

It should be noted that the discrete analogue of the differential operator was first introduced and investigated by Sobolev [26, 27] and it was constructed in [21]. In particular, from the results of [21] for , the following are obtained.

Theorem 2

The discrete analogue to the operator satisfying (2.25) has the form

(2.26)

and satisfies

(2.27)

Now we return to our problem.

We regard the coefficients as a discrete argument function and assume for and . Then, considering the above definitions, we rewrite the system (2.23)-(2.24) in the convolution form as

(2.28)
(2.29)

where

(2.30)
(2.31)

and is defined by (2.14).

Now we have the following problem.

Problem 2

Find , , and satisfying the system (2.28)-(2.29) for given and .

Note that Problem 2 is equivalent to Problem 1. The main result of this section is as follows.

Theorem 3

For with , coefficients of the optimal quadrature formulas of the form (2.1) in the sense of Sard in the space have the form

(2.32)

In addition, for the square of the norm of the error functional (2.3) of the optimal quadrature formula (2.1) in the space , the following holds:

(2.33)

Proof. We consider a discrete argument function

(2.34)

Then, considering (2.25) and (2.27), we have

(2.35)

Calculating the convolution (2.35) requires the representation of the function for all integer values of . From (2.28) we have

(2.36)

Now we need to find the representation of for and . Using (2.14) and (2.29) for and , respectively, we get

(2.37)

where is defined as (2.31), and and are unknowns. Then from the last two equalities when and , we get the following system of two linear equations for these unknowns:

Therefore, solving this system using (2.30) and (2.31), we get

(2.38)
(2.39)

With (2.38) and (2.39) in mind, the combination of (2.36) and (2.37) results in

The analytic formulas (2.32) is now obtained from (2.35) by taking into account (2.26) and (2.27), using the last representation of , and by direct calculation of the optimal coefficients .

Now we are going to get (2.33). We rewrite (2.16) in the following form:

(2.40)

Since , considering (2.38), we have

Therefore, these two last equalities are used in (2.19) and (2.21) to obtain

and

Then the expression (2.40) for takes the form

Therefore calculating the definite integrals, keeping (2.15) in mind and using (2.32), we get (2.33) after some simplifications. Theorem 3 has been proved. 

We note that in Theorem 3, the formulas for the optimal coefficients are decomposed into two parts: real and imaginary parts. Therefore from the formulas (2.32) of Theorem 3, we get the following results.

Corollary 1

For with , coefficients of the optimal quadrature formula of the form

in the sense of Sard in have the form

Corollary 2

For with , coefficients of the optimal quadrature formula of the form

in the sense of Sard in have the form

It is easy to see that for Sard’s following result [20] on the optimality of the trapezoidal quadrature formula in is obtained from Theorem 3.

Corollary 3

Coefficients of the optimal quadrature formula of the form

(2.41)

in the space have the form

and for the norm of the error functional of the optimal quadrature formula (2.41) in the space , the following holds

In addition, for with we obtain the following corollary from (2.32) and (2.33).

Corollary 4

For with , coefficients of the optimal quadrature formula of the form (2.1) in the sense of Sard in the space have the form

and for the norm of the error functional (2.3) of the optimal quadrature formula (2.1) in the space , the following holds:

i.e., the convergence order of the optimal quadrature formula of the form (2.1) is for with .

Remark 1 It should be noted that for a fixed , we obtain

from (2.33), i.e., the convergence order of the optimal quadrature formula of the form (2.1) is .

Remark 2 In particular, in the case with , the results of [5] and of Section 6 of [6] are obtained from Theorem 3.

Remark 3 The equality (2.39) means that the optimal quadrature formula of the form (2.1) with coefficients (2.32) is exact to because

The equality (2.39) together with (2.29) provides the exactness of our optimal quadrature formula for all linear functions. Therefore, for functions with a continuous second derivative, the convergence order of the optimal quadrature formula (2.1) with coefficients (2.32) is concluded as .

3 Optimal quadrature formula for the interval [a,b]

Here, optimal quadrature formulas for the interval are obtained by a linear transform from the results of the previous section.

We consider the construction of the optimal quadrature formula of the form

(3.1)

in the Sobolev space . Here are coefficients, are the nodes of the formula (3.1), , , and for .

Now, by a linear transformation , where , we obtain

Finally, by applying Theorem 3 and Corollary 3 to the integral on the right-hand side of the last equality, we have the following main result of the present work.

Theorem 4

For with , coefficients of the optimal quadrature formula of the form

(3.2)

in the sense of Sard in the space have the form

(3.3)

and for , the coefficients take the form

(3.4)

where .

The monomials for are now considered as a function in the integral in the left-hand side of (3.2). Then we get

(3.5)

where .

Remark 4 We find from Remark 3 that the optimal quadrature formula (3.2) is exact for all linear functions, that is the following equalities hold: