1. Representation of Approximation Residual
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
(1)
Let 
be linear space.
Suppose that 
and
are columns with components belonging to the space
; assume the relation
(2)
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
:
(3)
From (2) and (3) it follows that
(4)
where 
denotes the column-vector in
namely,
. The outer round brackets in (4) mean the inner product of 
-dimensional vectors.
Theorem 1 The following relation holds:
(5)
where the second factor on the right-hand side is the determinant of a block-matrix of order
.
Proof By (4), we have 
Hence
(6)
where 
is the cofactor of an entry 
of the matrix
. By (6), we can represent the difference 
as the product of determinants, written as
(7)
The equality (7) is equivalent to the equality (5).
2. Representation of the Remainder of Approximation by Elementary Hermite Type Splines
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
(8)
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
(9)
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
(10)
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
(11)
Theorem 2 For
, 
,
(12)
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]).
3. Some Auxiliary Assertions
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
(13)
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
(14)
so that
.
Using the additivity property of determinants and integrals and applying the Newton?-Leibnitz formula, we find
where
,
(15)
Similarly,
where
(16)
Finally
where
(17)
Integral operators 
can be rewritten in the form
where
, and
.
It is obvious that
(18)
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
(19)
Recall that vector-function 
is continuously differentiable in neighborhood of the point
, and passaging to limit as
, we get
(20)
It follows easily that relation (20) can be written in the form
where
, and the operator 
is defined by identity
(21)
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
(22)
so that
.
Under condition 
according to Taylor formula we have
whence we get
Thus by (21) we obtain
where
(23)
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
(24)
Proof Substituting vector-function 
for 
in (13), we have
(25)
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
(26)
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
(27)
Equating of (26) to (27) gives (24). It completes the proof.
4. Evaluations of Approximation by Splines of Hermite Type
We assume that 
and
(28)
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 
(29)
where
.
By definition, put
Lemma 3 Under the assumption (29), for 
the inequality
(30)
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
(31)
(32)
(33)
Lemma 4 If
, then for 
the following inequality holds:
(34)
where
(35)
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 
(36)
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.,
(37)
Proof If identity 
is fulfilled for a number
, 
, then in (33) the determinant 
includes two identical rows; therefore
. Thus the relation (37) is true.
5. Acknowledgements
The work is partially supported by the Russian Foundation for Basic Research (grant No. 13-01-00096).