## 1 Introduction

It is known, that quadrature and cubature formulas, are methods for the approximate evaluation of definite integrals. In addition and even more important, quadrature formulas provide a basic and important tool for the numerical solution of differential and integral equations. The theory of cubature formulas consists mainly of three branches dealing with exact formulas, formulas based on functional-analytic methods, and formulas based on probabilistic methods Sobolev74 ; SobVas

. In the functional-analytic methods the error between an integral and corresponding cubature sum is considered as a linear functional on a Banach space and it is estimated by the norm of the error functional in the conjugate Banach space. The norm of the error functional depends on coefficients and nodes of the formula. 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 the obtained formula is called

*the optimal formula in the sense of Nikol skii*(see, for instance, Nik88 ). Minimization of the norm of the error functional by coefficients when the nodes are fixed is called

*Sard s problem*. And the obtained formula is called

*the optimal formula in the sense of Sard*. First this problem was studied by A. Sard Sard . Solving these problems in different spaces of differentiable functions various type of optimal formulas of numerical integration are obtained.

There are several methods for constructing the optimal quadrature formulas in the sense of Sard such as the spline method, the function method (see e.g. BlaCom , SchSil ) and the Sobolev method. It should be noted that the Sobolev method is based on using a discrete analog of a linear differential operator (see e.g. Sobolev06 ; Sobolev74 ; SobVas ). In different spaces based on these methods, the Sard problem was studied by many authors, see, for example, IBab ; BlaCom ; CatCom ; HayMilShad10 ; HayMilShad15 ; Koh ; FLan ; Sard ; SchSil ; ShadHay11 ; Sobolev06 ; Sobolev74 ; SobVas ; Zag and references therein.

Among these formulas the Euler-Maclaurin type quadrature formulas are very important for numerical integration of differentiable functions and are widely used in applications. In different spaces the optimality of the Euler-Maclaurin type quadrature and cubature formulas were studied, for instance, in works CatCom ; FLan ; Mic74 ; Schoen65 ; ShadHayNur13 ; ShadHayNur16 ; ShadNur18 ; Zhen81 .

The Euler-Maclaurin quadrature formulas can be viewed as well as an extension of the trapezoidal rule by the inclusion of correction terms. It should be noted that in applications and in solution of practical problems numerical integration formulas are interesting for functions with small smoothness.

The present paper is also devoted to extension of the trapezoidal rule.

We consider a quadrature formula of the form

(1) |

where are coefficients of the trapezoidal rule, i.e.

(2) |

are unknown coefficients of the formula (1) and they should be found, , is a natural number. We suppose that an integrand belongs to , where by we denote the class of all functions defined on which posses an absolutely continuous first derivative and whose second derivative is in . The class under the pseudo-inner product

is a Hilbert space if we identify functions that differ by a linear combination of a constant and (see, for example, Ahlb67 ). Here, in the Hilbert space , we consider the corresponding norm

(3) |

The difference

(4) |

is called *the error* and

(5) |

is said to be *the error functional* of the quadrature formula (1), where is the indicator of the interval and is Dirac’s delta function.
The value of the error functional at a function is defined as

In order that the error functional (5) is defined on the space it is necessary to impose the following conditions for the functional

(6) | |||||

(7) |

The last two equations mean that the quadrature formula (1) is exact for any constant and . We have chosen the coefficients , such that the equality (6) is fulfilled. Therefore we have only condition (7) for coefficients , .

The error functional of the formula (1) is a linear functional in , where is the conjugate space to the space .

By the Cauchy-Schwarz inequality we have the following

Hence we conclude that the error (4) of the formula (1) is estimated by the norm

(8) |

of the error functional (5).

The main aim of this work is to find the minimum of the absolute value of the error (4) by coefficients for given in the space . That is the problem is to find the coefficients that satisfy the following equality

(9) |

The coefficients which satisfy the last equation are called *optimal* and are denoted as .

Thus, to obtain the optimal quadrature formula of the form (1) in the sense of Sard in the space , we need to solve the following problems.

Problem 2. *Find the coefficients that
satisfy equality (9).*

Here we solve Problems 1 and 2 by Sobolev’s method using the discrete analog of the differential operator .

The paper is organized as follows: in Section 2 using the extremal function of the error functional the norm of this functional is calculated, i.e. Problem 1 is solved; Section 3 is devoted to solution of Problem 2. Here the system of linear equations for the coefficients of the optimal quadrature formulas (1) is obtained in the space . In Subsection 3.1 using the discrete analog of the operator the explicit formulas for the coefficients of optimal quadrature formula of the form (1) are obtained. Furthermore, it is proved that the obtained quadrature formula of the form (1) is exact for any function of the set . Finally, in Subsection 3.2 in the space the square of the norm of the error functional of the constructed quadrature formula is calculated. It is shown that the error of the obtained optimal quadrature formula is less than the error of the Euler-Maclaurin quadrature formula on the space .

## 2 The norm of the error functional (5)

In this section we study Problem 1. To calculate the norm of the error
functional (5) in the space
we use *the extremal function* for this functional (see, Sobolev74 ; SobVas ) which satisfies the equality

We note that in ShadHay14 for a linear functional defined on the Hilbert space the extremal function was found and it was shown that the extremal function is the solution of the boundary value problem

(10) | |||

(11) | |||

(12) |

That is for the extremal function the following was proved.

###### Theorem 2.1 (Theorem 2.1 of ShadHay14 )

Furthermore, there were shown that and

(14) |

From Theorem 2.1, in the case , we get the extremal function for the error functional (5) and it has the form

(15) |

where

(16) |

and are any real numbers.

Then, from (14), in the case , using (5) and (15), taking into account equations (6) and (7), we get

(17) |

where is defined by (16), and are derivatives of , i.e.

(18) |

It is easy to see from (13) and (18) that

(19) |

Thus Problem 1 is solved.

In the next section we study Problem 2.

## 3 Minimization of the norm (17)

Now we consider the minimization problem of the expression (17) by the coefficients under the condition (7). For this we use the Lagrange method of conditional extremum.

Consider the Lagrange function

Taking partial derivatives from the function by , then equating them to 0 and using the condition (7), we get the following system of linear equations with unknowns

(20) | |||

(21) |

where

(22) | |||||

(23) |

Here , are defined by (2), , and are unknowns.

The system (20)-(21) has a unique solution for any fixed natural number and this solution gives the minimum to the expression (17). Here we omit the proof of the existence and uniqueness of the solution of this system. These statements can be proved similarly as the proof of the existence and uniqueness of the solution of the discrete Wiener-Hopf type system for the optimal coefficients of quadrature formulas with the form in the space (see Sobolev06 ; Sobolev74 ; SobVas ).

3.1. The coefficients of the optimal quadrature formula (1)

In this subsection we solve the system (20)-(21) and find the explicit forms for optimal coefficients , .

Here we use the concept of discrete argument functions and operations on them. The theory of discrete argument functions is given in Sobolev74 ; SobVas . We give some definitions about functions of discrete argument.

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

A function is called
*a function of discrete argument* if it is defined on some set
of integer values of .

*The inner product* of two
discrete functions and is defined as the
following number

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

The convolution of two discrete argument functions and is the inner product

Furthermore, for finding the coefficients of the optimal quadrature formula (1) we need the discrete analog of the differential operator . It should be noted that in the work ShadHay04 the discrete analog of the differential operator was constructed. In particular, when from the result of the work ShadHay04 we get the following

###### Theorem 3.1

The discrete analog of the differential operator satisfying the equation

has the form

(24) |

where and

Furthermore, it is easy to check that

(25) |

Suppose when and . Then we rewrite the system (20)–(21) in the following convolution form

(26) | |||

(27) |

Here, calculating the right hand sides of equalities (22) and (23), for and we get

(28) | |||||

(29) |

Denoting by

(31) |

the left hand side of the equation (26) we get

(32) |

Indeed, if the function is defined at all integer values of , then taking into account Theorem 3.1 and using properties (25) of the function , we have

Thus, if we find the function for all integer values of then the optimal coefficients will be found from the equality (32).

The following is true.

###### Theorem 3.2

The coefficients of the optimal quadrature formula of the form (1) in the sense of Sard in the space have the following form

(33) | |||||

Let then from (31), using the form (18) of the function and equality (27), we have

Similarly, for we obtain

Then, keeping in mind the last two equalities and denoting by

for we get the following

(34) |

Here, in the equality (34), and are unknowns. These unknowns can be found from the conditions of consistency of values of the function at the points and . Therefore from (34) when and we obtain the system of linear equations for and . Then, using (28) and (29), after some calculations, we have

(35) |

Finally, from (32) for , using (24) and (34) and taking into account (35), by directly calculations we get (33). Theorem 3.2 is proved.

Remark 1. Using (2) and (33), one can get that and . These equalities mean that the optimal quadrature formula of the form (1) with the coefficients (2) and (33) is also exact for functions and . Therefore, keeping in mind equalities (6) and (7), we conclude that the optimal quadrature formula of the form (1.1) with coefficients (2) and (33) is exact for any linear combinations of functions and , i.e. it is exact for elements of the set .

3.2. The norm of the error functional of the optimal quadrature formula (1)

In this subsection we study the order of convergence of the optimal quadrature formula of the form (1) with coefficients (2) and (33), i.e. we calculate the square of the norm (17) of the error functional for the optimal quadrature formula (1).

The following holds

###### Theorem 3.3

*Proof.* We rewrite the expression (17) as follows

where is defined by (22).

Hence, taking into account (35), we have

(37) |

here

Now we need the following sums which are obtained by using (2) and (33)

(38) |

Taking into account (28) and (16), using (38) for , , and we get

Further, putting the last equalities to (37) and after some simplifications we have

Hence, using well known formula , we get (36).

Theorem 3.3 is proved

Remark 2. It should be noted that optimality of the classical Euler-Maclaurin was proved and the error of this quadrature formula was calculated in , where is the space of functions which are square integrable with -th generalized derivative (see, for instance, CatCom ; Schoen65 ; ShadHayNur13 ). In particular, when from Corollary 5.1 of the work ShadHayNur13 we get optimality of the Euler-Maclaurin formula

(39) |

in the space . Furthermore for the square of the norm of the error functional the following is valid

(40) |

Comparison of equalities (36) and (40) shows that the error of the optimal quadrature formula of the form (1.1) on the space is less than the error of the Euler-Maclaurin quadrature formula (39) on the space .

## Acknowledgments

This work has been done while A.R.Hayotov was visiting Department of Mathematical Sciences at KAIST, Daejeon, Republic of Korea. A.R.Hayotov is very grateful to professor Chang-Ock Lee and his research group for hospitality. A.R. Hayotov’s work was supported by the ’Korea Foundation for Advanced Studies’/’Chey Institute for Advanced Studies’ International Scholar Exchange Fellowship for academic year of 2018-2019

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