1. Introduction. Statement of the Problem.
In order to find an approximate representation of a function 
by elements of a certain finite collection, it is possible to use values of this function at some finite set of points 
, 
. The corresponding problem is called the interpolation problem, and the points 
the interpolation nodes.
In the present paper we deal with optimal interpolation formulas. Now we give the statement of the problem of optimal interpolation formulas following by S. L. Sobolev. Now following we consider interpolation formula of the form
(1.1)
where 
and 
(
) are coefficients and nodes of the interpolation formula (1.1), respectively. We suppose that the functions 
belong to the Hilbert space
equipped with the norm 
(1.2)
and 
. The equality (1.3) is semi-norm and 
if and only if 
. The difference 
is called the error of the interpolation formula (1.1). The value of this error at some point 
is the linear functional on functions 
, i.e.
(1.3)
where 
is the Dirac delta-function and 
(1.4)
is the error functional of the interpolation formula (1.1) and belongs to the space
. The space 
is the conjugate space to the space 
.
By the Cauchy-Schwarz inequality 
the error (1.3) of formula (1.1) is estimated with the help of the norm
of the error functional (1.4).
Therefore from here we get the first problem.
Problem 1. Find the norm of the error functional 
of interpolation formula (1.1) in the space 
.
Obviously the norm of the error functional 
depends on the coefficients 
and the nodes 
. The interpolation formula which the error functional in given number 
of the nodes has the minimum norm with respect to 
in the space 
is called the optimal interpolation formula. The main goal of the present paper is to construct the optimal interpolation formula in the space 
for fixed nodes 
, i.e. to find the coefficients 
satisfying the following equality 
(1.5).
Thus in order to construct the optimal interpolation formula in the space 
we need to solve the next problem.
Problem 2. Find the coefficients 
which satisfy equality (1.5) when the nodes 
are fixed.
In this work we give the solution of Problem 1.
2. The extremal function and representation of the norm of the error functional
In this section we solve Problem 1, i.e. we find explicit form of the norm of
.For finding the explicit form of the norm of the error functional 
in the space 
we use concept of its extremal function which was introduced by S. L. Sobolev. The function 
from 
space is called the extremal function for the error functional 
if the following equality is fulfilled
The space 
is the Hilbert space and the inner product in this space is given by the formula 
(2.1). According to the Riesz theorem any linear continuous functional 
in a Hilbert space is represented in the form of a inner product. So, in our case we have
(2.2) for arbitrary function 
from 
space. Here 
is the function from 
is defined uniquely by functional 
and is the extremal function. Now we solve equation (2.2) and find 
. Suppose 
belongs to the space 
, where 
is the space of functions, which are infinity differentiable and finite in the interval 
. Then from (2.1), integrating by parts, we obtain
(2.3).
Keeping in mind (2.3) from (2.2) we get 
(2.4). So, when 
the extremal function 
is a solution of equation (2.4). But, we have to find the solution of equation (2.2) when the functions 
belong to the space 
. Since the space 
is dense in the space 
, then we can approximate arbitrarily exact functions of the space 
by a sequence of functions of the space 
. Next for any 
we consider the inner product 
and, integrating by parts of (2.1), we have
Hence from arbitrariness of 
and uniqueness of the function 
(up to the function 
and polynomials of degree 
), taking into account (2.4), it must be fulfilled the following equation
(2.5)
with boundary conditions
(2.6);
(2.7)
Thus, we conclude that the extremal function
is the solution of the boundary value problem (2.5)-(2.7).The following holds
Theorem 1. The solution of the boundary value problem (2.5)-(2.7) is the extremal function 
of the error functional 
of the interpolation formula (1.1) and has the following form 
where 
(2.8) is a solution of the equation
(2.9) 
is any real number, 
is a polynomial of degree 
.
(2.9),
(2.10). Now we obtain representation for the norm of the error functional 
. Since the space 
is a Hilbert space then by the Riesz theorem we have 
Hence, using (1.4) and Theorem 1, taking into account (2.9), (2.10), we get
Hence and taking account that 
is the even function, we have
(2.11).
Thus Problem 1 is solved.
References:
- J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The theory of splines and their applications, Mathematics in Science and Engineering, New York: Academic Press, 1967.
- S. L. Sobolev, Introduction to the Theory of Cubature Formulas, Nauka, Moscow, 1974, 808 p.
