1. Introduction

As is generally known, numerical integration formulae, or quadrature formulae, are methods for the approximate evaluation of definite integrals. They are needed for the computation of those integrals for which either the antiderivative of the integrand cannot be expressed in terms of elementary functions or for which the integrand is available only at discrete points, for example from experimental data. In addition and even more important, quadrature formulae provide a basic and important tool for the numerical solution of differential and integral equations.

Consider the following general quadrature formula

(1.1)

with the error functional

(1.2)

in a Banach space . Here are the coefficients and are the nodes of the formula (1.1), , , is a weight function, is the indicator of the interval [0,1], is the Dirac delta-function, is an element of the space .

The difference

(1.3)

is called the error of the quadrature formula (1.1).

By the Cauchy-Schwarz inequality

the error (1.3) of the formula (1.1) is estimated with the help of the norm of the error functional (1.2) in the conjugate space , i.e. by

Thus estimation of the error (1.3) 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 .

Obviously the norm of the error functional depends on the coefficients and the nodes of the quadrature formula (1.1). The problem of finding the minimum of the norm of the error functional by coefficients and by nodes is called S.M. Nikol’skii problem, and obtained formula is called optimal quadrature formula in the sense of Nikol’skii. This problem was first considered by S.M. Nikol’skii ^[17], and continued by many authors, see e.g. ^{[3, 4, 5, 6, 18, 38]} and references therein. Minimization of the norm of the error functional by coefficients when the nodes are fixed is called Sard’s problem. And obtained formula is called optimal quadrature formula in the sense of Sard. First this problem was investigated by A.Sard ^[19].

The results of this paper are related to Sard’s problem. So here we discuss some of the previous results about optimal quadrature formulas in the sense of Sard which are closely connected to our results.

There are several methods of construction of optimal quadrature formulas in the sense of Sard such as spline method, function method (see e.g. ^{[3, 21]}) and Sobolev’s method which is based on construction of discrete analogue of a linear differential operator (see e.g. ^{[34, 35]}). In the different spaces, based on these methods, the Sard’s problem was investigated by many authors, see, for example, [2,3,5,7,8,10,12-16,20-23,25,28-37] and references therein.

In the paper ^[21], using spline method, optimality of the classical Euler-Maclaurin formula was proved and the error of this quadrature formula is calculated in , where is the space of functions which are square integrable with -th generalized derivative.

Let ( ) be a class of functions , having on the [0,1] - absolute continuoues derivative and , where . In ^[38] it is proved, that among quadrature formulas (1.1) when the Euler-Maclaurin quadrature formula is optimal in the space . And in ^[27] optimality of the lattice cubature formulas of Euler-Maclaurin type is proved in the space .

Using -function method optimality of the Euler-Maclaurin quadrature formula is proved and the error of this formula is calculated by T. Catinas and Gh. Coman ^[5] in the space . Also using this method in ^[14] a procedure of construction of quadrature formulas of the form (1.1), which are exact for solutions of linear differential equations and are optimal in the sense of Sard is discussed.

It should be noted, that in applications the formula (1.1) is interesting for small values of . Optimal quadrature formulas in the sense of Sard for the case has already been discussed by many authors, mainly in the space (see [2,3,5,7,8,10,12-16,20-23,25,28,29,32-37] and references therein).

The main aim of this paper is to construct optimal quadrature formulas of the form (1.1) in the sense of Sard for the case when in the space equipped with the norm

(1.4)

and . The equality (1.4) is the semi-norm and if and only if , where is a polynomial of degree , .

It should be noted that is a Hilbert space if we identify functions that differ by a solution of .

Here we use the Sobolev’s method ^{[34, 35]} which is based on the discrete analogue of the differential operator .

We consider the following quadrature formula

(1.5)

with the error functional

(1.6)

in the space for . Here are known coefficients and , are unknown coefficients of the formula (1.5), is a natural number.

For the error functional (1.6) to be defined on the space it is necessary to impose the following conditions (see ^[33])

(1.7)

Hence it is clear that for existence of the quadrature formulas of the form (1.5) the condition has to be met.

Note that here in after means the functional (1.6).

As was noted above by the Cauchy-Schwarz inequality, the error of the formula (1.5) is estimated by the norm of the error functional (1.6). Furthermore the norm of the error functional (1.6) depends on the coefficients . We minimize the norm of the error functional (1.6) by the coefficients , i.e., we find

(1.8)

The coefficients which satisfy the equality (1.8) are called the optimal coefficients and denoted by and the corresponding quadrature formula is called the optimal quadrature formula in the sense of Sard. In the sequel, for the purposes of convenience the optimal coefficients will be denoted as .

Thus to construct optimal quadrature formulas in the form (1.5) in the sense of Sard we have to consequently solve following problems.

Problem 1. Find the norm of the error functional (1.6) of the quadrature formula of the form (1.5) in the space .

Problem 2. Find coefficients which satisfy the equality (1.8).

The paper is organized as follows. In section 2 we give some definitions and known formulas. In section 3 we determine the extremal function which corresponds to the error functional and give a representation of the norm of the error functional (1.6). Section 4 is devoted to a minimization of with respect to the coefficients We obtain a system of linear equations for the coefficients of the optimal quadrature formula of the form (1.5) in the sense of Sard in the space . Explicit formulas for coefficients of the optimal quadrature formula of the form (1.5) are found in section 5. Moreover we calculate the norm of the error functional (1.6) of the optimal quadrature formula of the form (1.5). In section 6 some numerical results are presented.

2. Definitions and Known Formulas

In this section we give some definitions and formulas that we need to prove the main results.

Here the main concept used is that of functions of discrete argument and operations on them (see. ^{[33, 35]}). For the purposes of completeness we give some definitions about functions of discrete argument.

Assume that and are real-valued functions of real variable and are defined in real line .

Definition 2.1. Function is called function of discrete argument, if it is given on some set of integer values of .

Definition 2.2. The inner product of two discrete functions and is called the number

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

Definition 2.3. The convolution of two discrete functions and is called the inner product

The Euler-Frobenius polynomials , are defined by the following formula ^[35]

(2.1)

.

For the Euler-Frobenius polynomials the following identity holds

(2.2)

and also the following theorem is true

Theorem 2.1 (Lemma 3 of ^[24]). Polynomial which is defined by the formula

(2.3)

is the Euler-Frobenius polynomial (1) of degree , i.e. , where

The following formula is valid ^[11]:

(2.4)

where is the finite difference of order of , is ratio of a geometric progression.

At last we give the following well known formulas from ^[9]

(2.5)

where are Bernoulli numbers,

(2.6)

3. The Extremal Function and the Representation of the Error Functional Norm

To solve Problem 1, i.e., for finding the norm of the error functional (1.6) on the space a concept of the extremal function is used ^[33]. The function is said to be the extremal function of the error functional (1.6) if the following equality holds

(3.1)

In the space the extremal function of a functional is found by S.L. Sobolev ^{[33, 35]}. This extremal function has the form

(3.2)

where

(3.3)

is a solution of the equation

(3.4)

is a polynomial of degree , * - is operation of convolution, i.e.

It is well known that for any functional in the equality

holds ^[33].

Applying this equality to the error functional (1.6) we obtain the following

(3.5)

Thus Problem 1 is solved for quadrature formulas of the form (1.5) in the space .

4. The System for Optimal Coefficients

Now we investigate Problem 2. For finding the minimum of the under the conditions (1.7) the Lagrange method is used. For this we consider the following function

where are unknown multipliers. The function is the multidimensional function with respect to the coefficients and . Equating to zero partial derivatives of the by coefficients together with conditions (1.7) we get the following system of linear equations

(4.1)

(4.2)

where is defined by (3), is unknown polynomial of degree and

(4.3)

here is the Bernoulli number.

It is clear that

(4.4)

and the following holds

(4.5)

The system (4.1)-(4.2) is called the discrete system of Wiener-Hopf type for the optimal coefficients ^{[33, 35]}. In the system (4.1)-(4.2) the coefficients and polynomial are unknowns. The system (4.1)-(4.2) has unique solution and this solution gives the minimum to the . Here we omitted the proof of the existence and uniqueness of the solution of the system (4.1)-(4.2). The proof of the existence and uniqueness of the solution of this system is as the proof of the existence and uniqueness of the solution of discrete Wiener-Hopf type system of the optimal coefficients in the space for quadrature formulas of the form (1.1) for the case (see ^{[33, 35]}). It should be noted, that in ^[14] the uniqueness of the optimal quadrature formulas in the Sard’s sense of the form (1.1) is discussed.

5. The Coefficients and the Norm for the Error Functional of the Optimal Quadrature Formulas (1.5)

In the present section we study the solution of the system (4.1)-(4.2). To solve this system we use the approach which was suggested by S.L. Sobolev in ^[34]. Furthermore we investigate order of convergence of optimal quadrature formulas of the form (1.5).

5.1. The Coefficients of the Optimal Quadrature Formulas (1.5)

Suppose that for and . Using Definition 2.3 and keeping in mind (4.4) we rewrite the equation (4.1) in the convolution form:

(5.1)

We consider the following problem

Problem A. Find the discrete function and unknown polynomial of degree , which satisfy the system (4.1)-(4.2).

Further, instead of we introduce the functions

(5.2)

(5.3)

In this statement it is necessary to express by the function . For this we need such operator , which satisfies the equation

(5.4)

where is the discrete argument function corresponding to the function defined by (4.4), is equal to 0 when and is equal to 1 when , i.e. is the discrete delta-function. The equation (5.4) is the discrete analogue of the equation (4.5). So the discrete function is called the discrete analogue of the differential operator ^[33].

It should be noted that the operator , which is the discrete analogue of the operator , was firstly introduced and investigated by S.L. Sobolev ^[33].

In ^[26] the discrete analogue of the differential operator , which satisfies equation is constructed and its properties are investigated.

Following Theorem 1 and Property 2 of the work ^[26] for the discrete analogue of the operator we respectively have the following theorems.

Theorem 5.1. The discrete analogue of the differential operator has the form

(5.5)

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

Theorem 5.2. The discrete argument function and the monomials are related to each other as follows

(5.6)

Then, taking into account (5.4) and Theorems 5.1, 5.2, for the optimal coefficients we have

(5.7)

Thus, if we find the function , then the optimal coefficients will be found from the equality (5.7).

To calculate the convolution (5.7) it is required to find the representation of the function for all integer values of . From the equality (5.1) we get, that when , where is defined by equality (4.3). Now we need to find the representation of the function when and .

Since when , then

Now we calculate the convolution when .

Suppose , then taking into account (4.2), we have

Hence, denoting by

(5.8)

for the case of we get

(5.9)

Now suppose then for we get

(5.10)

Denoting

(5.11)

and taking into account (5.1), (5.3), (5.9)-(5.11) we have the following problem

Problem B. Find the solution of the equation

(5.12)

having the form:

(5.13)

where and are unknown polynomials of degree .

If we find and , then from (5.11) we obtain

Unknowns and can be found from equation (5.12), using the discrete argument function . Then we can obtain the explicit form of the function and respectively we can find the optimal coefficients (). Thus Problem B and respectively Problem A can be solved.

But here we will not find , . Instead, using and the form (5.13) of the discrete argument function , taking into account (5.7), we find the expressions for the optimal coefficients when .

We introduce the following notations

(5.14)

where , is the Euler-Frobenius polynomial of degree , are given in Theorem 5.1. Note that because of the series in (5.14) are convergent.

The following holds

Theorem 5.3. The coefficients , of the optimal quadrature formulas of the form (1.5) in the space have the following form

(5.15)

where are defined by (5.14), are given in Theorem 5.1.

Proof. Since . Then from (5.7), using Definition 2.3, equalities (5.5) and (5.13), we have

Now, adding and subtracting the expressions and to and from the last expression and taking into account Definition 2.3 we get

Since , defined by equality (4.3), is the polynomial of degree with respect to then keeping in mind (5.6) we have .

Therefore taking into account the denotations (5.14) for we get (5.15).

Theorem 5.3 is proved.

For the coefficients of the optimal quadrature formulas of the form (1.5) the following holds.

Theorem 5.4. Among quadrature formulas of the form (1.5) with the error functional (1.6) in the space there exists unique optimal formula which coefficients are determined by the following formulas

(5.16)

(5.17)

(5.18)

(5.19)

(5.20)

(5.21)

where satisfy the following system of linear equations

(5.22)

here are Bernoulli numbers, is the finite difference of order of , is given in Theorem 2.1, are given in Theorem 5.1.

Proof. The coefficients , are given in quadrature formula (1.5) and have the forms (5.16)-(5.18). Thus we need to find only the coefficients , . First we consider the cases and . From (4.2) when for the coefficients and we obtain

(5.23)

(5.24)

It is clear from equalities (5.23)-(5.24) that the coefficients and are expressed by the coefficients , . From (5.15) it is obviously that the coefficients are expressed by and . Therefore it is sufficient to find unknowns and , .

Further we find unknowns and , from system (4.1)-(4.2).

Now we consider the first sum of equation (4.1). For this sum taking into account (4.4) we have

Hence using binomial formula, taking into account formulas (4.2), (5.15) and (2.4), keeping in mind that is the root of the Euler-Frobenius polynomial of degree , for we have

Putting the last expression of and the expression (4.3) for into (4.1) we get the following identity with respect to

(5.25)

From here, equating the corresponding coefficients of when of both sides of (5.25), for unknowns and we obtain the following system of equations

(5.26)

(5.27)

Further equating the corresponding coefficients of when of both sides of (5.25) we find evident form of the polynomial , i.e.

(5.28)

Thus, from (5.25) (i.e. from (4.1)) for unknowns and () we obtained the system (5.26)-(5.27) of equations. To find others equations for unknowns and we use (4.2). For this, replacing by , we reduce equations (4.2) to the following form

(5.29)

For the left hand side of (5.29), using (5.15) and (2.4), we get

(5.30)

And using formula (2.5) for the sum of the right hand side of (5.29) we have

(5.31)

Further, taking into account equalities (5.30), (5.31) and getting the difference the left and the right hand sides of (5.29) we get the polynomial of degree with respect to which is identically 0. i.e. we have the following

Hence, equating the coefficients of the same powers of to zero, we get the following system

(5.32)

(5.33)

Using (5.32) and Theorem 2.1 one can show that system (5.33) is the part of system (5.27). Thus from system (5.32) and (5.33) for unknowns and we get only new system (5.32). Therefore, applying Theorem 2.1 and using (5.27), from (5.32) we have

(5.34)

Since equation (5.26) is the combination of the equations (5.27) and (5.34) when then from (5.27) and (5.34) we get the following system of linear equations for unknowns and ():

(5.35)

(5.36)

Adding (5.35) to (5.36) we obtain

(5.37)

Hence clear that

Taking into account the last equalities from (5.35) we get the system (5.22) for , which is given in the statement of Theorem 5.4.

Theorem 5.4 is proved.

5.2. The Norm of the Error Functional of Optimal Quadrature Formulas of the Form (1.5)

For the square of the norm of the error functional (1.6) of optimal quadrature formulas of the form (1.5) the following holds

Theorem 5.5. For the square of the norm of the error functional (1.6) of the optimal quadrature formula of the form (1.5) on the space the following holds

where , are determined in Theorem 5.4, are Bernoulli numbers, is given in Theorem 2.1, are given in Theorem 5.1.

Beforehand we give the following results without proof. They are used in the proof of Theorem 5.5.

Lemma 5.1. For the Bernoulli numbers the following identities hold

(5.38)

here , and

(5.39)

here .

Lemma 5.2. The following equalities are true

(5.40)

(5.41)

(5.42)

Proof of Theorem 5.5. We rewrite equality (3.5) in the following form

(5.43)

From (4.1), taking into account (4.3), we have

Keeping in mind the last equality, from (5.43) we get

Hence, denoting and using (2.4), (2.5), (4.2), (4.3), (5.28), (5.35), Theorem 5.4 and Lemmas 5.1, 5.2, after some calculations, we get

(5.44)

where

By direct calculation one can show that and Then taking into account that , from (5.44) we get the statement of Theorem 5.5.

Theorem 5.5 is proved.

In particular, from Theorems 5.4 and 5.5 for the cases we get the following corollaries which confirm the optimality of the classical Euler-Maclaurin quadrature formula.

Corollary 5.1. In the space among quadrature formulas of the form (1.5) with the error functional (1.6) there exists unique optimal formula whose coefficients are determined by the following formulas

Furthermore for the square of the norm of the error functional the following is valid

Corallary 5.2. In the space among quadrature formulas of the form (1.5) with the error functional (1.6) there exists unique optimal formula whose coefficients are determined by the following formulas

Furthermore for the square of the norm of the error functional the following is valid

Remark 5.1. It should be noted, that the result of Corollary 5.2 was already obtained by using - function method in [4, Theorem 10].

6. Numerical Results

We note that constructed optimal quadrature formulas of the form (1.5) with the error functional (1.6), the coefficients which are determined by formulas (5.16)-(5.21) are exact for monomials , . This statement is also checked numerically.

Clearly, that the optimal coefficients (5.16)-(5.21) 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 for 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 ^[23].

Below we consider some particular cases.

We consider the case In this case we obtain the optimal quadrature formulas of the form (1.5) in the space which are exact for the monomials and . Here we need the roots of the Euler-Frobenius polynomial , which . From (1) we get

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

(6.1)

For solving the system (5.22) and using (6.1) from (5.16)-(5.21) we get the following optimal quadrature formula of the form (1.5)

(6.2)

Using Theorem 5.5 we get the following estimation of the formula (6.2)

For solving the system (5.22) and using (6.1) from (5.16)-(5.21) we get the following optimal quadrature formula of the form (1.5)

(6.3)

Using Theorem 5.5 we get the following estimation of the formula (6.3)

Now we give tables of values of the coefficients of optimal quadrature formulas of the form (1.5) for the cases and .

For and solving the system (5.22) and using (6.1) from (5.16)-(5.21) we get the following optimal quadrature formula of the form (1.5) in the space

(6.4)

The coefficients of the optimal formula (6.4) are presented in Table 6.1.

Using Theorem 5.5 we get the following estimation of the formula (6.4)

The case . In this case we obtain the optimal quadrature formulas of the form (1.5) in the space which are exact for the monomials and . We need the roots of the Euler-Frobenius polynomial , which . From (2.1) we get

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

(6.5)

For solving the system (5.22) and using (6.5) from (5.16)-(5.21) we get the following optimal quadrature formula of the form (1.5) in the space

(6.6)

The coefficients of the optimal formula (6.6) are presented in Table 6.2.

Using Theorem 5.5 we get the following estimation of the formula (6.6)

Remark 6.1. In the work ^[32] the optimal quadrature formulas in the sense of Sard of the form

(6.7)

were constructed in the Sobolev space . The coefficients , , and of the optimal formulas (6.7) are expressed by the roots of the Euler-Frobenius polynomial of degree and by the solution of the system of linear equations (see Theorem 5.4, i.e. formulas (5.24)-(5.30) of ^[32]). But the coefficients , of the optimal quadrature formulas (1.5) are expressed by the roots of the Euler-Frobenius polynomial of degree and by the solution of the system linear equations (see Theorem 5.4, i.e. formulas (5.19)-(5.22) of the present work). Furthermore, the order of convergence of the optimal quadrature formulas (1.5) and (6.7) are the same (see Theorem 5.5 and Numerical results of ^[32] and Theorem 5.5 and Numerical results of the present work).

Thus, when by solving only the system of linear equations (5.22) with respect to , using Theorems 5.4 and 5.5 we obtain new optimal quadrature formulas of the form (1.5) in the sense of Sard in the space

All calculations were performed in MAPLE with 150 decimal digits. All decimals listed in Tables 6.1 and 6.2 are correct.

Acknowledgements

The part of this work has been done in the University of Santiago de Compostela, Spain. A.R. Hayotov thanks the program Erasmus Mundus Action 2, Stand 1, Lot 10, Marco XXI for financial support (project number: 204513 –EM -1-2011 -1-DE-ERA MUNDUS-EMA21).

A.R. Hayotov thanks professor A.Cabada for discussion of the results and for hospitality.

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