Optimal Quadrature Formula of Hermite Type in the Space of Differentiable Functions

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Introduction: Statement of the Problem
It is well known that numerical integration formulas or quadrature formulas are a method of approximate estimation of definite integrals.They are used when the initial functions of the functions under the integral cannot be expressed by elementary functions, when the integral exists only at discrete points, or when some special types of integrals with the property of singularity are approximated, for example: The effectiveness of quadrature formulas is usually classified according to its degree of accuracy and the order of approximation of the error.Most problems of applied sciences and mathematical physics are brought to the calculation of integrals.In particular, in the numerical-analytical solution of integral equations, in the calculation of the center of mass, the moment of inertia or various properties of physical systems, in the calculation of integrals related to image and signal analysis and filtering, the construction of quadrature formulas and the evaluation of their errors is one of the targeted scientific studies.
We consider the following quadrature formula with error functional ) ''( ) 12 where The norm in space () 2 (0,1) m L is defined by the following form: So, for the existence of a quadrature formula of the form (1.1), condition 2 Nm − must be fulfilled, that is, starting from 3 m = , unknown coefficients of the quadrature formula 1 [] C  can be found.
From the definition of the functional norm Therefore, the estimation of the error of the quadrature formula (1.1) is related to the minimization of the norm of the error functional () N x .In computational mathematics, quadrature formulas are constructed mainly in three directions: the spline method, the method of  − functions, and the Sobolev method is based on using discrete analog of the linear differential operator.I. J. Shoenberg ( [3,4]) constructed a quadrature formula in the ()  2 (0, ) m Ln space by spline method.From among the following formulas: In the work of S.A. Michelli [5] it was shown that () 2 m W is the best formula, when m is an odd natural number; b) A.A. Jensikbayev [6] proved this formula is optimal in the () 2 (0,1) m L space; c) T. Catinash, G.T. Koman [7] constructed an optimal quadrature formula using  − function method in the (2)  2 (0,1) L space; d) Kh.M. Shadimetov [8] constructed the optimal quadrature formula in the () 2 (0,1) m L space when 0  = and calculated the norm of the error function; e) Kh.M.Shadimetov, A.R.Hayotov, F.A.Nuraliev [9] constructed the optimal quadrature formula for 1  = and estimated its error.
Article [10] presents new and effective quadrature formulas, which merge function and first derivative estimation at equally-spaced data points, with a particular emphasis on improving computational efficiency in terms of both cost and time.The objective of the research presented in work [11] is to simplify the computation of the components involved in the integral transformation, denoted as m F and 0. m  The analytical expressions for these components encompass definite integrals.Instead of the Newton-Cotes formulas, it is proposed to use non-trivial quadrature formulas with unevenly distributed integration points.The quadrature method is essential in the approximate solution of integral equations.In [12], the trapezoidal numerical integration formula is used to solve the Fredholm-Hammerstein integral equations.In [13], the perturbed Milne quadrature rule was derived for n -fold differentiable functions.Thus, in order to construct an optimal quadrature formula of the form (1.1), we need to solve the following problems.Problem 1. Find the norm of the error functional (1.2) of the quadrature formula (1.1).

CN  =
, which give the smallest value to the norm of the error functional (1.2), that is, calculating the value It is important to mention that, the coefficients .If such coefficients exist, these are called optimal coefficients, denoted as

Known Definitions and Theorems
In this section, we provide definitions and formulas necessary to prove the main results.
Assume that  and  are real-valued functions of real variable and are defined in real line .

Definition 2.1. Function () h
 is a function of a discrete argument if it is defined for a set of integer values of  .
When calculating sums, we use the following equations derived from the sum formula of geometric progression [14] where ik   is the finite difference of order i of k  .
We use the following formula to calculate some sums [15] For any continuous functions, the operation of convolution is defined as follows

 
where k q are the roots of the Euler-Frobenius polynomial 26 () The proof of this Lemma is given in [9].

The Expression of the Error Functional Norm
In this section, we find general representation of the norm of the (1.2).We utilize the extremal function for this purpose ( [1,2]).The function  is called an extremal function of the error functional (1.2) if the following equality holds ( ) In the () 2 (0,1) m L space it is defined as follows where is a solution of the equation In addition, the extremal function satisfies the following relationships.(Riesz theorem) [16] ( ) ( ) and ( ) Based on equations (3.2) and (3.5), the square of the norm of any linear continuous space can be written as follows


Using this equation, we get a general representation of the square of the norm of the error Thus, Problem 1 was resolved.

Optimal Coefficients of the Optimal Quadrature Formula Form (1)
In this section, we will consider the problem of finding the minimum of the equation (3.6) under the conditions (1.3) for coefficients 1 [ ], 0,1,..., CN  = .Using the method of Lagrange unknown multipliers, we find the conditional extremum of multivariable functions.Therefore, we construct the Lagrange function where   unknown multipliers.The  function is a multivariable function with respect to the coefficients of 1 [] C  and   .By equalizing the derivatives of the function  with respect to 1 [] C  and   to zero, we obtain the following system of equations   The system of equations (4.1)-(4.2) is the discrete Wiener-Hopf system used to find optimal coefficients.This system has a unique solution, and this solution gives a minimum value to To do this, we rewrite equation (4.1) in the form of convolution, taking into Also, instead of the left side of equation (4.4), we introduce functions First of all, 1 [] C  coefficients should be expressed by () uh  function.For this we need the (1 Considering these theorems, we obtain the following equality for the 1 [] C  optimal coefficients ( ) So, we need overview of the function () uh  all integer values of  to calculate the (4.5) We find the () 2), we have we enter the following function So, the general representation of the () uh  function is as follows where , we obtain from (4.13) ( ) ( ) Now we find the 1 [] C  optimal coefficients when = 1,2,...We introduce the following equalities ( ) where , kk ab are defined by (4.14), k q are given in Theorem 4.2.Proof.When = 1,2,..., 1 N  − , using equations (4.3), (4.7), (4.9) and (4.12), we can write the After some simplifications, we get the following space are determined as follows where k a satisfy the following system of 3 m − linear equations Proof.First we use equation  We first calculate the sum of (4.1) 2), (4.2) and (4.15) and taking into account that k q is the root of the Euler-Frobenius polynomial of degree (2 6) m − , we get the following for S 2 5 2 5 3 2 5 3 [0] = , 1 ( 1) () ( ) 0 ( 25 )! ! ( 1) From (4.2), we get the following for 0 Using (2.2) and (4.15) for the left side of equation (4.27), we get the following ( ) ( 1) 0 ( 1) Taking equality (4.28) into account, we subtract the left and right sides of (4.27) Equating the corresponding degrees of h from (4.29), we get the following system It can be seen that (4.31) is a part of system (4.25).Thus, using (4.30), (4.25) and Lemma 2.1 , we obtain the new system for the unknowns k a and when  is even numbers, from this the equation (4.32) can be written as follows If we subtract (4.33) from (4.25), we get the following system of equations k m − .So, from (4.25) we will have a system of equations to find k a unknowns.Theorem 4.4 is proved.From Theorem 4.4, we get the following results: Result 4.1.The coefficients of the optimal quadrature formula (1.1) in the (3)  2 (0,1) L space are determined as follows The coefficients of the optimal quadrature formula (1.1) in the (4)  2 (0,1) L space are determined as follows ( ) where 1 120( 1) N a q = + , 32 q =− .

High Estimate of the Error of the Optimal Quadrature Formula
In this section, we calculate the square of the norm of the error function.

Numerical Results
We numerically analyze the analytical results obtained in this section and compare them with other works.
Determining the absolute value of
The exact value of the integral

Conclusion
In this research work, the derivative optimal quadrature formula was built using the values of the function up to the second derivative at the nodal points for the approximate calculation of the exact integrals.We found the representation of the error functional corresponding to the difference between the quadrature sum and the exact integral.The error functional   1 C  is a multivariate function with respect to the coefficients.To find the conditional extremum of a multivariable function, we constructed the Lagrange function and obtained the system of equations.By solving the system of equations, we found the analytical representation of the coefficients.Using the optimal coefficients, we calculated the norm of the error function and numerically analyzed the order of its approximation.We proved that this quadrature formula accurately integrates polynomials of 1 m − degree.We analyzed the error of the proposed quadrature formula in numerical experiments using degree, exponential and logarithmic functions.

Conflicts of Interest:
The authors declare that there are no conflicts of interest regarding the publication of this paper.


In addition, the error functional (1.2) is required to satisfy the following conditions ([1,2])

2 (
seen from this inequality that the error of the quadrature formula (1.1) is estimated by the norm of the error functional () N x obtained from the conjunction space()

2 N.
To solve this system, we use an approach based on the 2 operator, which constracted in[17].

Theorem 4 . 1 .
The discrete analogue of the differential operator2

Theorem 4 . 2 .
7) where 25 () m Eq − is the Euler-Frobenius polynomial of degree 25 m − , k q are the roots of the Euler-Frobenius polynomial 26 () The monomials () k h have the following relation with the discrete operator 2 ()

Theorem 4 . 4 .
The coefficients of the optimal quadrature formula of the form (1.

( 4 . 2 )
to find the representation of the coefficients 1

From ( 4 .
Substituting the expression (4.22) into (4.1),we get the following equation with respect to h 23) we get the system of equations to find the coefficient   ) and (4.24), we find   km − from equation (4.23), we obtained the system of equations (4.25), and we find the remaining 3 m − using equation (4.2)

Theorem 5 . 2 . 1
The squared norm of the error functional (1.3) of the optimal quadrature formula (

TABLE 1 .
Squared norm of error functional of optimal quadrature formula

TABLE 2 .
Error of optimal quadrature formula

TABLE 3 .
[10]r of optimal quadrature formula L space and the error of the derivative formula constructed in the work[10]are numerically analyzed in TABLE 4 using the following 5 functions.

TABLE 4 .
Error of optimal quadrature formula