1. Introduction and Statement of the Problem

We consider the following quadrature formula

(1.1)

with the error functional given by

(1.2)

in the space , where and are coefficients and nodes of the formula (1.1), respectively, is the characteristic function of the interval , , is Dirac’s delta-function, function belongs to the space . is the Sobolev space of functions with square integrable m-th generalized derivative. In this space the norm of a function is defined by the formula

(1.3)

The difference

(1.4)

is called the error of the quadrature formula (1.1). The error (1.4) of the formula (1.1) is a linear functional in , where is the conjugate space to space.

For the error functional (1.4) to be defined on the space it is necessary to impose the following conditions (see ^[32])

(1.5)

By the Cauchy - Schwarz inequality

the error (1.4) of the formula (1.1) is estimated with the help of the norm

of the error functional (1.2). Consequently, estimation of the error of the quadrature formula (1.1) on functions of the space is reduced to finding the norm of the error functional in the conjugate space .

A minimization of the norm of the error functional with respect to the coefficients , when the nodes are fixed, is called as Sard’s problem. The obtained formula is called the optimal quadrature formula in the sense of Sard. This problem was first investigated by A.Sard ^[17].

There are several methods of construction of optimal quadrature formulas in the sense of Sard ^{[2, 34]}. In the space , based on these methods, Sard’s problem was investigated by many authors (see, for example, [1-10,13-16,19,20,22-29,31-36] and references therein).

In the present paper we give the sulution of Sard’s problem for the formula (1.1) in the space . Namely, we find the coefficients such that

(1.6)

Thus, in order to construct an optimal quadrature formula of the form (1.1) in the space , we need conse-quently to solve the following two problems:

Problem 1. Calculate the norm of the error functional for the given quadrature formula (1.1) in space.

Problem 2. Find the values of the coefficients such that the equality (1.6) be satisfied with fixed nodes .

In order to solve Problem 1, i.e., to calculate the norm of the error functional (1.2) in the space we use a concept of the extremal function for a given functional. The function is called the extremal for the functional (cf. ^[25]) if the following equality is fulfilled

(1.7)

For the extremal function of the error function the following result was proved by S.L.Sobolev ^[32].

Theorem 1. The extremal function of the error functional has the following form

where is a polynomial of degree , and

(1.8)

is the solution of the equation

The symbol * is the convolution of functions f and g is defined by the formula

Now, using Theorem 1, we immediately obtain a representation of the norm of the error functional

(1.9)

Thus, Problem 1 is solved.

The paper is organized as follows. In Section 2 we give some preliminaries. In Section 3 explicit formulas for the coefficients of the optimal quadrature formula of the form (1.1) are found; finally, In Section 4 some numerical results are presented.

2. Preliminaries

In the present section we give some definitions and known results that we need to prove the main results.

Below mainly we use the concept of discrete argument functions and operations on them. The theory of discrete argument functions is given in ^{[32, 33]}. For completeness we give some definitions about functions of discrete argument.

Assume that the nodes are equal spaced, i.e., , .

Definition 1. The function is a function of discrete argument if it is given on some set of integer values of .

Definition 2. The inner product of two discrete functions and is given by

if the series on the right hand side of the last equality converges absolutely.

Definition 3. The convolution of two functions and is the inner product

The Euler-Frobenius polynomials , are defined by the following formula ^[26]

(2.1)

.

The following theorem is true

Theorem 2. (Lemma 3 of ^[18]). Polynomial which is defined by the formula

is the Euler-Frobenius polynomial (2.1) of degree , i.e. , where

The following formula is valid ^[30]:

(2.2)

where is the finite difference of order of , is the ratio of a geometric progression. When from (2.2) we have

(2.3)

In our computations we need the discrete analogue of the differential operator which satisfies the following equality

(2.4)

where is the discrete argument function corresponding to the function defined by (1.8), is equal to 0 when and is equal to 1 when , i.e. is the discrete delta-function.

It should be noted that the operator was firstly introduced and investigated by S.L. Sobolev ^[32].

In ^[21] the discrete analogue of the differential operator , which satisfies equation (2.4), is constructed and the following theorem is proved.

Theorem 3. The discrete analogue of the differential operator has the form

(2.5)

where

(2.6)

is the Euler-Frobenius polynomial of degree , are the roots of the Euler-Frobenius polynomial , , is a small positive parameter.

Furthermore several properties of the discrete argument function were proved in ^[21]. Here we give the following property of which we need in our computations.

Theorem 4. The discrete argument function and the monomials are related to each other as follows

where is the Bernoulli number.

3. The Optimal Coefficients of the Quadrature Formula (1.1)

Let the nodes of the quadrature formula (1.1) be fixed. The error functional (1.2) satisfies the conditions (1.5). Norm of the error functional is a multidimensional function of the coefficients . For finding its minimum under the conditions (1.5), we apply the Lagrange method.

Namely, we consider the function

where and , it’s partial derivatives by and equating to zero, so that we obtain the following system of linear equations

(3.1)

(3.2)

where is a polynomial of degree , and is defined by (1.8),

(3.3)

Further we investigate Problem 2. Instead of we introduce the following functions

(3.4)

In such statement it is necessary to express the coefficients by the function .

Then taking into account (2.4), (3.4) and Theorems 3, 4, for the coefficients we have

(3.5)

Thus, if we find the function , then the coefficients will be found from equality (3.5).

To calculate the convolution (3.5) it is required to find the representation of the function for all integer values of . From equality (3.1) we get that when . Now we need to find the representation of the function when and .

Since when then

Now we calculate the convolution when and .

Suppose then taking into account equalities (3.2) we have

Thus, when , we get

(3.6)

where

is the polynomial of degree and

is a unknown polynomial of degree of .

Similarly, in the case for the convolution we obtain

(3.7)

We denote

(3.8)

(3.9)

where

Taking into account (3.4), (3.6) and (3.7) we get the following problem

Problem 3. Find the solution of the equation

(3.10)

having the form:

(3.11)

Here and are unknown polynomials of degree with respect to .

If we find , then from (3.8), (3.9) we have

(3.12)

Unknowns , can be found from equation (3.10), using the function defined by (2.5). Then we obtain explicit form of the function and from (3.5) we find the coefficients . Furthermore from (3.12) we get .

Thus Problem 3 and respectively Problem 2 will be solved.

The main result of the present is the following theorem.

Theorem 5. Coefficients of the optimal quadrature formula (1.1), with equally spaced nodes in the space , have the following form

(3.13)

(3.14)

(3.15)

where

(3.16)

(3.17)

and , , are defined by (2.6), are the roots of the Euler-Frobenius polynomial , , and are defined from the system (3.18)-(3.19), (3.21)-(3.22).

Proof. First we find the expressions for and . When and from (3.11) for and we get

(3.18)

(3.19)

Now we have unknowns , , .

From equation (3.10), by choosing and , we are able to find and .

Taking into account (3.11), from (3.10) we get the following system

(3.20)

where and .

First we consider the cases . From (3.20) replacing by and using (2.5) and (2.3), after some calculations for , we get the following system of linear equations

(3.21)

where

Here and .

Now we consider the cases

From (3.20) replacing by and using (2.5) and (2.3), after some calculations for we get the following system of linear equations

(3.22)

where

Here and .

Thus for the unknowns , , we have obtaint the system (3.21), (3.22) of linear equations. Since our problem has a unique solution, the main matrix of this system is non singular. Unknowns , can be found from system (3.21), (3.22). Then taking into account (3.12), using (3.18) and (3.19) we have

Now we find the coefficients , .

From (3.5), taking into account (2.5), we deduce

From here, using (2.5) and formula (2.3), taking into account (3.16) and (3.17), after some calculations we arrive at the expressions of the coefficients , which are given in the assertion of the theorem.

Theorem 5 is proved.

To point out the applicability of the formulas obtained above, we will focus on the particular cases and .

The case .

In this case Problem 2 is as follows.

Problem 4. Find coefficients of the optimal quadrature formula (1.1) in the space

The solution of Problem 4 is the coefficients , and . They satisfy thefollowing system

where

##

So, as a direct consequence of Theorem 5, we arrive at the following result

Corollary 1. The coefficients of the optimal quadrature formula (1.1) in the space have the following form

The case . Then from Problem 2 we have

Problem 5. Find coefficients of the optimal quadrature formula (1.1) in the space

The solution of Problem 5, the coefficients , , and which satisfy the system

where

In this case we have the following result as an immediate corollary of Theorem 5.

Corollary 2. The coefficients of the optimal quadrature formula (1.1) in the space have the form ^[11]

where

The coefficients of the optimal quadrature formula (1.1) in the space have given in ^[12].

4. Numerical Results

It should be noted that constructed optimal quadrature formulas of the form (1.1) with the error functional (1.2), the coefficients which are determined by formulas (3.13)–(3.15) are exact for monomials . This statement is also checked numerically.

Clearly, the optimal coefficients (3.13)–(3.15) depend only on the roots (where ) of the Euler–Frobenius polynomial , which is defined by formula (2.1). Therefore to obtain numerical values of the coefficients it is sufficient to calculate the roots of the Euler–Frobenius polynomial , whose absolute values are less than 1.

It should be noted that for the Euler–Frobenius polynomials and their roots are given in ^[30].

Below we consider some particular cases.

We consider the case . In this case we obtain the optimal quadrature formulas of the form (1.1) in the space . Here we need the roots of the Euler–Frobenius polynomial , in which . From (2.1) we get

and the roots of this polynomial, whose absolute values less than 1 are

(4.1)

Now we give tables of values of the coefficients of optimal quadrature formulas of the form (1.1) for the cases and . For and solving the system (3.21), (3.22) and using (4.1) from (3.13)–(3.15) we get the following optimal quadrature formula of the form (1.1) in the space

(4.2)

The coefficients of the optimal formula (4.2) are presented in Table 1.

Table 1. The coefficients of the optimal quadrature formula (4.2)

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Using (1.9) we get the following estimation of the formula (4.2)

The case . In this case we obtain the optimal quadrature formulas of the form (1.1) in the space . We need the roots of the Euler–Frobenius polynomial , in which . From (2.1) we get

and the roots of this polynomial, whose absolute values less than 1 are

(4.3)

For and solving the system (3.21), (3.22) and using (4.3) from (3.13)–(3.15) we get the following optimal quadrature formula of the form (1.1) in the space

(4.4)

The coefficients of the optimal formula (4.4) are presented in Table 2.

Table 2. The coefficients of the optimal quadrature formula (4.4)

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Using (1.9) we get the following estimation of the formula (4.4)

Remark. It should be noted that when from Theorem 5 we get Theorem 2.1. of ^[13].

5. Conclusion

In the present work we constructed the optimal quadrature formulas with polynomial weight by Sobolev method in the space .

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