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
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.
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
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
with boundary conditions
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 .
(2.10). 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
Thus Problem 1 is solved.
- 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.