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)
where 
and 
(
) are coefficients and nodes of the interpolation formula (1), respectively. We suppose that the functions 
belong to the Hilbert space
| 
is absolutely continuous and
,
equipped with the norm
(2)
and 
. The equality (2) is semi-norm and 
if and only if 
.
The difference 
is called the error of the interpolation formula (1). The value of this error at some point 
is the linear functional on functions 
,
(3)
where 
is the Dirac delta-function and
(4)
is the error functional of the interpolation formula (1) and belongs to the space 
. The space 
is the conjugate space to the space 
. By the Cauchy-Schwarz inequality
the error (3) of formula (1) is estimated with the help of the norm
of the error functional (4).
Therefore from here we get the first problem.
Problem 1. Find the norm of the error functional 
of interpolation formula (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
(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 (5) when the nodes 
are fixed.
In this work using Sobolev’s [1, 2] method we give the algorithm for finding the coefficients of the optimal interpolation formula (1);
The following main result is valid.
Theorem 1. Coefficients of optimal interpolation formula with equal spaced nodes in the space 
have the following form:
,
,
,
Where
,
where the unknowns 
certain quantities.
References:
- S. L. Sobolev, V. L. Vaskevich. The Theory of Cubature Formulas. Kluwer Academic Publishers Group, Dordrecht (1997).
- S. L. Sobolev, The coefficients of optimal quadrature formulas, in: Selected Works of S. L. Sobolev. Springer, 2006, pp.561–566.
