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
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
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 ,
where is the Dirac delta-function and
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
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 the unknowns certain quantities.
- 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.