Approximation by Splines of Hermite Type

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     0,1, , j j m  a  , 1 m j R   a  . Let be square matrix. Supposing that and are columns with components from the linear space such that  def 0 1 , , , m A  a a a     def T 0 1 , , , m       A  0 1 , , , m      def  F    def g g  . Let be vector with components  , , , g   m g 0 1 j g belonging to conjugate space  F . For an element we consider a linear combination of elements uF     0,1, , j j m    : def m 0 , . j j g u j u      By definition, put  0 1 , , , , m  T , def , , g u g  u g u g u  . The discussions are based on the next assertion. The following relation holds: 1 det A T et , , A u u d g u u             where the second factor on the right-hand side is the determinant of a block-matrix of order m + 2. Using this assertion, we get the representation of residual of approximation by minimal splines of Hermite type. Taking into account the representation, we get evaluations of the residual and calculate relevant constants. As a result the obtained evaluations are exact ones for components of generated vector-function .   t 

that and are columns with components from the linear space such that be vector with components  , , , .The discussions are based on the next assertion.The following relation holds: where the second factor on the right-hand side is the determinant of a block-matrix of order m + 2. Using this assertion, we get the representation of residual of approximation by minimal splines of Hermite type.Taking into account the representation, we get evaluations of the residual and calculate relevant constants.As a result the obtained evaluations are exact ones for components of generated vector-function .
  t  Keywords: Splines; Errors of Approximations

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    0,1, , (where m is a natural number) in the space .The matrix R A composed of these columns is denoted by , , , .
is valid; matrix A is defined by (1).
Let be vector with components For an element g belonging to conjugate space .
From ( 2) and (3) it follows that where , g u denotes the column-vector in . The outer round brackets in (4) mean the inner product of -dimensional vectors. 1 m  Theorem 1 The following relation holds: where the second factor on the right-hand side is the determinant of a block-matrix of order . 2 m  Proof By (4), we have where ij A is the cofactor of an entry ij of the matrix a A .By (6), we can represent the difference as the product of determinants, written as The equality (7) is equivalent to the equality (5).

Representation of the Remainder of Approximation by Elementary Hermite Type Splines
On  ,    we consider a grid of the form 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 , , , , ,   We introduce the functions by the approximate relations Consider square matrix and vector-function then the relations (9) may be rewritten as It can be proved (for example, see [2]) that the matrix k A is invertible.Hence the functions are defined uniquely and they are linear independent.If , and functional system   , j i g defined by formula , , , 0,1, , 1 .

Analogously on the adjacent interval we get
Open Access AM Discuss the linear space where is the linear hull of the elements in the curly brackets and Cl means the closure of the linear hull in the topology of pointwise convergence.
We call , H X   the space of elementary Hermite type We consider the function   where the second factor on the right-hand side is the determinant of the square matrix of order written in the block form.

Some Auxiliary Assertions
Let , be natural numbers with property 0 1 ; let 0 be real numbers, which comply with inequalities .Let us put is valid; here is a linear operator of integration over parallelepiped , , , , , 1,2, , and use notation Using the additivity property of determinants and integrals and applying the Newton?-Leibnitz formula, we find Similarly, Integral operators can be rewritten in the form 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   f  .
Hence the integral operations , have nonnegative kernels , 1,2,3 i i   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
By relations (18) and (21) we see that the integral operator     1 y  may be represented in the form , , d d ,


, and . Thus the assertion is true in discussed case.Now consider the case of 1 n  , , so that .
It follows in the standard way that

Now recall notation (22); we obtain
, where , . This completes the proof in discussed case.
For an arbitrary natural 0 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 Proof Substituting vector-function for 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   .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.

Evaluations of Approximation by Splines of Hermite Type
We assume that , , , , where Using the estimate (29), the positiveness of the kernel of the integral operation   , p z  , and the relation (24) obtained in Lemma 2, we derive the estimate (4.3) for , for 1 .
and the maximum is taken over

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]).
taking into account (23), we rewrite the formula in the form continuity of the function sideration on [a,b], from (28) we conclude t 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