1. Introduction. Setting goals

Abstract:In this paper, in the space - is the space of functions, the - th generalized derivative of which is summed with a square on the segment , we consider the construction of an optimal lattice quadrature formula. Further, finding the extremal function of the error functional of quadrature formulas, by virtue of the Riesz theorem on the general form of a linear continuous functional, are reduced to solving a boundary value problem for ordinary differential equations. By solving this boundary value problem, an explicit expression for the extremal function of the error functional of the quadrature formulas is obtained. In addition, by virtue of the Riesz theorem, the square of the norm of the error functional of quadrature formulas in the Sobolev space is explicitly calculated, where we obtain a number of interesting and new results that are unattainable in the classical sense. Some theorems are presented and established to theoretically support computational simplifications that reduce costs.

1. Introduction. Setting goals. An important problem in the theory of quadrature formulas is to find the maximum error of a quadrature formula over a given class of functions.

Consider a quadrature formula of the form

, (1)

here are the coefficients of the quadrature formula (1), is an element of the space at .

The error of the quadrature formula (1) is the difference

,

where

, (2)

here - is the indicator of the segment [0,1], - is the Dirac delta function.

In this article, for the quadrature formula (1) with the error functional (2), an extremal function is found in the space , and the square of the norm of the error functional is calculated. By minimizing this norm over the coefficients , a system of linear algebraic equations is obtained. The uniqueness of the optimal coefficients is proved.

The Sobolev space is the Hilbert space of classes of real functions , differing by the largest one by a polynomial of degree with derivatives (in the sense of generalized functions) of order , square integrable in the interval (0,1) and the scalar product

Then the norm of the function in the space is determined by the formula

The error of the quadrature formula will be a linear functional in , where - is the dual space of , that is,

.

For the error functional of the quadrature formulas (1) to belong to the space , it is necessary and sufficient that

. (3)

It is natural to evaluate the quality of the quadrature formula (1) using the maximum error of this formula on the unit ball of the Hilbert space , that is, using the norm of the functional :

.

Obviously, the norm of the error functional depends on the coefficients .

If

, (4)

then the functional is said to correspond to the optimal quadrature formula in the space . If it is required to find the maximum possible error over the space of the constructed quadrature formula, then it is sufficient to solve the following problem.

Problem 1. Find the norm of the error functional of the considered quadrature formula (1) in the space . If it is required to find the optimal quadrature formula by the coefficients and it is necessary to solve the following problem.

Problem 2. Find values ​​of the coefficients , such that equality (4) holds.

From the formulated problems, the theory of approximate calculations of integrals is transferred to the section of extremal problems of functional analysis, which formed in the scientific direction in the 30-50s of the last century and is associated with the name of A.N. Kolmogorov.

In the multidimensional case, the formulation of problems 1 and 2 were posed by S.L. Sobolev. Then he gives an algorithm for constructing optimal lattice cubature formulas in the Sobolev space [1,2].

Subsequently, Sobolev's research on optimal lattice cubature formulas and asymptotic formulas was developed by his students [3-14].

2. Main results.

In the section, we solve Problem 1. For finding the norm of the error functional (2) we use the extremal function for the error functional _{ } that satisfies the following equality

(5)

In the space , using the Riesz theorem on the general form of the linear continuous functional in the Hilbert spaces, the extremal function is expressed in terms of the given functional and

(6)
Therefore, from (5) and (6) we conclude that


On the other hand, by the same theorem, for any element of the space , we obtain

where

(7)
is the inner product in the space . Let is a finite infinitely differentiable function, then


Integrating over parts of times the right-hand side of equality ( 7 ), for the functions we obtain

( 8 )