﻿ Approximation by Splines of Hermite Type

Applied Mathematics
Vol.4 No.11C(2013), Article ID:38300,6 pages DOI:10.4236/am.2013.411A3002

Approximation by Splines of Hermite Type

Yuri K. Dem’yanovich, Irina G. Burova

Faculty of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia

Email: Yuri.Demjanovich@gmail.com, burovaig@mail.ru

Copyright © 2013 Yuri K. Dem’yanovich, Irina G. Burova. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received July 24, 2013; revised August 24, 2013; accepted August 31, 2013

Keywords: Splines; Errors of Approximations

ABSTRACT

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,. Let be square matrix. Supposing that and are columns with components from the linear space such that. Let be vector with components belonging to conjugate space. For an element we consider a linear combination of elements: By definition, put. 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.

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 innamely,. 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).

REFERENCES

1. Yu. K. Dem’yanovich, “Approximation by Minimal Splines,” Journal of Mathematical Sciences, Vol. 193, No. 2, 2013, pp. 261-266.
2. I. G. Burova and Yu. K. Dem’yanovich, “Theory of Minimal Splines,” St.-Petersburg University Press, St.-Petersburg, 2000.
3. A. O. Gelfond, “Calculation of Finite Differences,” Nauka Press, Moscow, 1967.