For convenience we shall give scheme of representation of the approximation residual in general situation (see also [1]).

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 [2]) that the matrix is invertible. Hence the functions are defined uniquely and they are linear independent. If, , then the functions belong to, and functional system defined by formula

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. [1]).

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 [3], p. 43); it is equal to

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).