The approximation evaluations by polynomial splines are well-known. They are obtained by the similarity principle; in the case of non-polynomial splines the implementation of this principle is difficult. Another method for obtaining of the evaluations was discussed earlier (see [1]) in the case of nonpolynomial splines of Lagrange type. The aim of this paper is to obtain the evaluations of approximation by non-polynomial splines of Hermite type. Considering a linearly independent system of column-vectors
For convenience we shall give scheme of representation of the approximation residual in general situation (see also [
We consider a linearly independent system of columnvectors (where m is a natural number)
in the space. The matrix composed of these columns is denoted by
Let be linear space.
Suppose that and
are columns with components belonging to the space; assume the relation
is valid; matrix A is defined by (1).
Let be vector with components
belonging to conjugate space.
For an element we consider a linear combination of elements:
From (2) and (3) it follows that
where denotes the column-vector innamely,. The outer round brackets in (4) mean the inner product of -dimensional vectors.
Theorem 1 The following relation holds:
where the second factor on the right-hand side is the determinant of a block-matrix of order.
Proof By (4), we have Hence
where is the cofactor of an entry of the matrix. By (6), we can represent the difference as the product of determinants, written as
The equality (7) is equivalent to the equality (5).
On we consider a grid of the form
We set
Let be -component vector-function with components in. We assume that Wronskian of the components is separated from zero.
Consider function, , and introduce notation
Let symbol denote the number of elements of a set.
We assume that natural numbers comply with relations, ,.
By definition, put
where. Obviously.
We introduce the functions by the approximate relations
Consider square matrix of the order (see notation (8)),
and vector-function
then the relations (9) may be rewritten as
It can be proved (for example, see [
is biorthogonal to the system so that
Rewrite the system (9) in the form
Under condition we have
Analogously on the adjacent interval we get
Discuss the linear space
where is the linear hull of the elements in the curly brackets and means the closure of the linear hull in the topology of pointwise convergence.
We call the space of elementary Hermite type -splines.
By definition, put
We consider the function defined by
Theorem 2 For, ,
where the second factor on the right-hand side is the determinant of the square matrix of order written in the block form.
Proof We can obtain the identity (12) by expanding the second determinant of right part of (12) and by usage of the relations (10)-(11) (cf. [
Let, be natural numbers with property; let be real numbers, which comply with inequalities . Let us put
,.
Lemma 1 For arbitrary -component vector-function the representation
is valid; here is a linear operator of integration over parallelepiped
with nonnegative kernel.
Proof We consider the case ,. Introduce value with property and use notation
so that.
Using the additivity property of determinants and integrals and applying the Newton?-Leibnitz formula, we find
where,
Similarly,
where
Finally
where
Integral operators can be rewritten in the form
where, and
.
It is obvious that
Since the lower limit is no more than the upper one in the integrals in (15)-(17), the result of integration is nonnegative for any nonnegative continuous function. Hence the integral operations, have nonnegative kernels By (17) we have
Recall that vector-function is continuously differentiable in neighborhood of the point, and passaging to limit as, we get
It follows easily that relation (20) can be written in the form
where, and the operator is defined by identity
By relations (18) and (21) we see that the integral operator may be represented in the form
where, and is nonnegative function Taking into account (14), we obtain , where, ,. Thus the assertion is true in discussed case.
Now consider the case of, ,.
Let is new variable,; by definition put
so that.
Under condition according to Taylor formula we have
whence we get
Thus by (21) we obtain
where
It follows in the standard way that
where,.
Passaging to limit under we obtain
taking into account (23), we rewrite the formula in the form
Thus
where
here and is nonnegative function.
Now recall notation (22); we obtain , where,. This completes the proof in discussed case.
For an arbitrary natural one can obtain a similar representation via multiple integrals with the lower integration limit less than the upper one. Analogously the assertion is proved for. This completes the proof.
Denote and introduce the function
Lemma 2 If suppositions of Lemma 1 are fulfilled, then
Proof Substituting vector-function for in (13), we have
The determinant on the right-hand side of (25) contains a lower triangular matrix with entries at the main diagonal so that right-hand side is equal to
The left-hand side contains the determinant of matrix, which appears in Hermite interpolation problem
where are prescribed numbers and
. Value of the mentioned determinant is known (see [
Equating of (26) to (27) gives (24). It completes the proof.
We assume that and
By the uniform continuity of the function under consideration on [a,b], from (28) we conclude that for any there exists such that for and
where.
By definition, put
Lemma 3 Under the assumption (29), for the inequality
is true; here, , ,.
Proof We use Lemma 1 and represent in the form (13) for, , , ,. As a result, we find
Using the estimate (29), the positiveness of the kernel of the integral operation, and the relation (24) obtained in Lemma 2, we derive the estimate (4.3) for.
Now we set
Lemma 4 If, then for the following inequality holds:
where
and the maximum is taken over
Proof By (31)-(33) the relation (13) may be written in the form
It is clear that conditions of Lemma 1 and Lemma 2 are fulfilled, and therefore the kernel of integral operator is nonnegative. By Lemma 2 we get evaluation (34)-(35).
Theorem 3 If and (29) holds, then for
where is defined by (35)
Proof Usage (34)-(35) in (12) gives the evaluation (36).
Corollary 1 Under the assumptions of Theorem 3, the interpolation of a function is exact on elements of the space, i.e.,
Proof If identity is fulfilled for a number, , then in (33) the determinant includes two identical rows; therefore. Thus the relation (37) is true.
The work is partially supported by the Russian Foundation for Basic Research (grant No. 13-01-00096).