1. Introduction

In order to find an approximate representation of a function by elements of a certain finite dimensional space, it is possible to use values of this function at some finite set of points , . The corresponding problem is called the interpolation problem, and the points the interpolation nodes.

There are polynomial and spline interpolations. Now the theory of spline interpolation is fast developing. Many books are devoted to the theory of splines, for example, Ahlberg et al ^[1], Arcangeli et al ^[2], Attea ^[3], Berlinet and Thomas-Agnan ^[4], Bojanov et al ^[5], de Boor ^[7], Eubank ^[9], Green and Silverman ^[12], Ignatov and Pevniy ^[17], Korneichuk et al ^[18], Laurent ^[19], Mastroianni and Milovanovic ^[20], Nürnberger ^[21], Schumaker ^[23], Stechkin and Subbotin ^[29], Vasilenko ^[30], Wahba ^[32] and others.

Suppose the functions belong to the Hilbert space (see [^[1], Chapter 3])

equipped with the norm

(1.1)

and , where

(1.2)

here each () is in and does not vanish on . Let be a formal adjoint of and

The equality (1.1) is semi-norm and only for a solution of the equation . We give definition of generalized splines following [1, Chapter 6]. If is a mesh on , then a generalized spline (or -spline) of deficiency () with respect to is a function which is in and satisfies the differential equation

on each open mesh interval of . The ordinary spline (deficiency one) allows discontinuities in the th derivative, but only at mesh points.

If the exact values of an unknown smooth function at the set of points in an interval are known, it is usual to approximate by minimizing

(1.3)

in the set of interpolating functions (i.e., , ) of the Sobolev space . Here is the Sobolev space of functions with a square integrable -th generalized derivative. It turns out that the solution is a natural polynomial spline of degree with knots called the interpolating spline for the points . In non periodic case first this problem was investigated by Holladay ^[16] for and the result of Holladay was generalized by de Boor ^[6] for any . In the Sobolev space of periodic functions the minimization problem of integrals of type (1.3) was investigated by I.J.Schoenberg ^[22], M.Golomb ^[13], W.Freeden ^{[10, 11]} and others.

We consider the Hilbert space

equipped with the norm

(1.4)

where and

The equality (1.4) is semi-norm and if and only if .

Consider the following interpolation problem:

Problem 1. Find the function which gives minimum to the norm (1.4) and satisfies the interpolation condition

(1.5)

for any , where are the nodes of interpolation.

Following [^[30], p.45, Theorem 2.2] we get the analytic representation of the interpolation spline

(1.6)

where , , , are real numbers,

(1.7)

and is a fundamental solution of the operator , i.e., is a solution of the equation

here is Dirac’s delta function. It should be noted that the rule for finding a fundamental solution of a linear differential operator

where are real numbers, is given in [31, p.88]. Using this rule, it is found the function which is a fundamental solution of the operator and has the form (1.7).

Furthermore from [30, p.45-47] it follows that the solution of the form (1.6) of Problem 1 is exists, unique when and coefficients , , , of are defined by the following system of linear equations

(1.8)

(1.9)

(1.10)

where .

It should be noted that systems for coefficients of splines similar to the system (1.8)-(1.10) were investigated, for example, in ^{[2, 8, 17, 19, 30]}.

The main aim of the present paper is to solve Problem 1, i.e., to solve the system (8)-(10) for equal spaced nodes , , and to find analytic formula for coefficients , , and of .

The rest of the paper is organized as follows: in section 2 we give the algorithm for solution of system (1.8)-(2.10) when the nodes are equal spaced. Using this algorithm coefficients of the interpolation spline are computed in section 3. In section 4 some numerical results are presented.

2. The Algorithm for Computation of Coefficients of Interpolation Splines

In the present section we give the algorithm for solution of system (1.8)-(1.10) when the nodes are equal spaced. Here we use similar method suggested by S.L. Sobolev ^{[26, 28]} for finding the coefficients of optimal quadrature formulas in the space . Below mainly is used the concept of discrete argument functions and operations on them. The theory of discrete argument functions is given in ^{[27, 28]}. For completeness we give some definitions about functions of discrete argument.

Assume that the nodes are equal spaced, i.e., , .

Definition 2.1. The function is a function of discrete argument if it is given on some set of integer values of .

Definition 2.2. The inner product of two discrete functions and is given by

if the series on the right hand side of the last equality converges absolutely.

Definition 2.3. The convolution of two functions and is the inner product

Now we turn to our problem.

Suppose that when and . Using above mentioned definitions, we rewrite the system (1.8)-(1.10) in the convolution form

(2.1)

(2.2)

(2.3)

Thus we have the following problem.

Problem 2. Find the discrete function , and unknown constants , which satisfy the system (2.1)-(2.3).

Further we investigate Problem 2 which is equivalent to Problem 1. Instead of we introduce the following functions

(2.4)

(2.5)

In such statement it is necessary to express the coefficients by the function . For this we have to construct such operator which satisfies the equality

(2.6)

where is equal to 0 when and is equal to 1 when , i.e., is the discrete delta-function. In connection with this the discrete analogue of the operator , which satisfies equation (2.6) is constructed in ^[14] and its some properties were investigated. Following in ^[14] we have:

Theorem 2.1. The discrete analogue of the differential operator satisfying the equation.(2.6) has the form

(2.7)

where

(2.8)

and

(2.9)

is a zero of the polynomial

and and is a small parameter.

Theorema 2.2. The discrete analogue of the differential operator satisfies the following equalities:

,

,

,

,

.

Here is the function of discrete argument, corresponding to the function defined by (1.7) and is the discrete delta function.

Then taking into account (2.5), (2.6) and Theorems 2.1 and 2.2, for the coefficients we have

(2.10)

Thus if we find the function then the coefficients can be obtained from equality (10). In order to calculate the convolution (2.10) we need a representation of the function for all integer values of . From equality (2.1) we get that when . Now we need to find a representation of the function when and . Since when then Now we calculate the convolution when and .

Suppose then taking into account equalities (1.7), (2.2), (2.3), we have

Denoting

we get for

and for

Now, setting

we have the following problem:

Problem 3. Find the solution of the equation

(2.11)

in the form:

(2.12)

where , , , are unknown coefficients.

It is clear that

(2.13)

These unknowns , , , can be found from equation (2.11), using the function . Then the explicit form of the function and coefficients , , can be found. Thus Problem 3 and respectively Problems 2 and 1 can be solved.

In the next section we realize this algorithm for computation of coefficients , , and of the interpolation spline (1.6) for any .

3. Computation of Coefficients of Interpolation Spline (1.6)

In this section using the algorithm which is given in Section 2 we obtain explicit formulas for coefficients of interpolation spline (1.6) which is the solution of Problem 1.

It should be noted that the interpolation spline (1.6) which is the solution of Problem 1 is exact for the functions and .

The following holds

Theorem 3.1. Coefficients of interpolation spline (1.6) which minimizes the norm (1.4) with equal spaced nodes in the space have the following form

where , , and are defined by (2.8), (2.9),

(3.1)

(3.2)

, , , are defined by (3.3), (3.7).

Proof. First we find the expressions for and . From (2.12) when and we get

(3.3)

Now we have unknowns , . These unknowns we find from (2.11) when and .

Taking into account (2.12) and Definition 2.3 from (2.11) we have

where and .

Hence for , , taking into account (3.3) and (2.7), after some calculations we obtain

where

(3.4)

(3.5)

(3.6)

Hence we get

(3.7)

Combaining (2.13), (3.3) and (3.7) we obtain and which are given in the statement of Theorem 3.1.

Now we calculate the coefficients , . Taking into account (2.12) from (2.10) for we have

From here using (2.7), taking into account notations (3.1), (3.2) when for we get expressions which are given in the statemant of Theorem 3.1.

Theorem 3.1 is proved.

4. Numerical Results

As numerical examples we consider the following functions

Applying the interpolation spline (1.6) to the functions and , using Theorem 3.1 with we get corresponding interpolation splines denoted by , and . Graphs of absolute errors between functions and corresponding interpolation splines are displayed in the Figure 4.1 and Figure 4.2.

Figure 4.1. Graphs of the absolute errors for : a) , b) , c) .

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Figure 4.2. Graphs of the absolute errors for : a) , b) , c)

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In Figure 4.1, Figure 4.2 one can see that by increasing values of the absolute errors between interpolation splines and given functions are decreasing.

It should be noted that in ^[15] the optimal quadrature formula of the following form

(4.1)

was constructed in the space and the following was proved

Theorem 4.1 (Theorem 7 of ^[15]). The coefficients of the optimal quadrature formulas in the sense of Sard of the form (4.1) in the space are

where is given in Theorem 2.1 and .

In ^[15] in numerical results were considered the functions and corresponding integrals

Applying the optimal quadrature formula (4.1), with , to the previous integrals were obtained their approximate values denoted by , , and , respectively. The corresponding absolute errors are displayed in Table 4.1 (Table 4.1 of ^[15]). Numbers in parentheses indicate decimal exponents. Now applying the interpolation spline (1.6), with to the functions using Theorem 3.1 we get corresponding interpolation splines , and . Further integrating of the differences

Table 4.1. Absolute errors of quadrature approximations I_N, J_N, K_N

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and taking their absolute values we get the results of the Table 4.1, i.e.

Thus, we conclude that by integrating the interpolation spline of the form (1.6) which minimize the norm (1.4) in the space we obtain optimal quadrature formula in the sense of Sard of the form (4.1) in the same space.

References

[1]	Ahlberg, J.H., Nilson, E.N., Walsh, J.L.: The theory of splines and their applications, Mathematics in Science and Engineering, New York: Academic Press, (1967).
	In article
[2]	Arcangeli, R., Lopez de Silanes, M.C., Torrens, J.J.: Multidimensional minimizing splines, Kluwer Academic publishers. Boston, (2004).
	In article
[3]	Attea, M.: Hilbertian kernels and spline functions, Studies in Computational Matematics 4, C. Brezinski and L. Wuytack eds, North-Holland, (1992).
	In article
[4]	Berlinet, A., Thomas-Agnan, C.: Reproducing Kernel Hilbert Spaces in Probability and Statistics, Kluwer Academic Publisher, (2004).
	In article		CrossRef
[5]	Bojanov, B.D., Hakopian, H.A., Sahakian, A.A.: Spline functions and multivariate interpolations, Kluwer, Dordrecht, (1993).
	In article		CrossRef
[6]	de Boor, C.: Best approximation properties of spline functions of odd degree, J. Math. Mech. 12, (1963), pp.747-749.
	In article
[7]	de Boor, C.: A practical guide to splines, Springer-Verlag, (1978).
	In article		CrossRef
[8]	Duchon, J.: Splines minimizing rotation-invariant semi-norms in Sobolev spaces, (1977), pp. 85-100.
	In article
[9]	Eubank, R.L.: Spline smoothing and nonparametric regression. Marcel-Dekker, New-York, (1988).
	In article
[10]	Freeden, W.: Spherical spline interpolation-basic theory and computational aspects, Journal of Computational and Applied Mathematics, 11, 367-375, (1984).
	In article		CrossRef
[11]	Freeden, W.: Interpolation by multidimensional periodic splines, Journal of Approximation Theory, 55, 104-117 (1988).
	In article		CrossRef
[12]	Green, P.J., Silverman: Nonparametric regression and generalized linear models. A roughness penalty approach. Chapman and Hall, London, (1994).
	In article		CrossRef
[13]	Golomb, M.: Approximation by periodic spline interpolants on uniform meshes, Journal of Approximation Theory, 1, (1968), pp. 26-65.
	In article		CrossRef
[14]	Shadimetov Kh.M., Azamov S.S.: Construction of discrete analogue of the differential operator d4/dx4+d2/dx2+1 and its properties (Russian). Uzbek Math. Zh. 2010, no 1, 181-188. (2010).
	In article
[15]	Shadimetov, Kh.M., Hayotov, A.R., Azamov, S.S.: Optimal quadrature formula in K2(P2)space, Applied Numerical Mathematics, 62, 1893-1909 (2012).
	In article		CrossRef
[16]	Holladay, J.C.: Smoothest curve approximation, Math. Tables Aids Comput. V.11. (1957) 223-243.
	In article		CrossRef
[17]	Ignatev, M.I., Pevniy, A.B.: Natural splines of many variables, Nauka, Leningrad, (in Russian) (1991).
	In article
[18]	Korneichuk, N.P., Babenko, V.F., Ligun, A.A.: Extremal properties of polynomials and splines, Naukovo dumka, Kiev, (in Russian) (1992).
	In article
[19]	Laurent, P.-J.: Approximation and Optimization, Mir, Moscow, (in Russian) (1975).
	In article
[20]	Mastroianni, G., Milovanovic, G.V.: Interpolation Processes – Basic Theory and Applications, Springer Monographs in Mathematics, Springer – Verlag, Berlin – Heidelberg (2008).
	In article
[21]	Nürnberger, G.: Approximation by Spline Functions, Springer, Berlin (1989).
	In article		CrossRef
[22]	Schoenberg, I.J.: On trigonometric spline interpolation, J. Math. Mech. 13, (1964), pp.795-825.
	In article
[23]	Schumaker, L.: Spline functions: basic theory, J. Wiley, New-York, (1981).
	In article
[24]	Sobolev, S.L.: Formulas of Mechanical Cubature in n- Dimensional Space, in: Selected Works of S.L.Sobolev, Springer, (2006), pp. 445-450.
	In article		CrossRef
[25]	Sobolev, S.L.: On Interpolation of Functions of n Variables, in: Selected Works of S.L.Sobolev, Springer, (2006), pp.451-456.
	In article		CrossRef
[26]	Sobolev, S.L.: The coefficients of optimal quadrature formulas, in: Selected Works of S.L.Sobolev. Springer, (2006), pp.561-566.
	In article		CrossRef
[27]	Sobolev, S.L.: Introduction to the Theory of Cubature Formulas, Nauka, Moscow, (in Russian) (1974).
	In article
[28]	Sobolev, S.L., Vaskevich, V.L.: The Theory of Cubature Formulas. Kluwer Academic Publishers Group, Dordrecht (1997).
	In article		CrossRef
[29]	Stechkin, S.B., Subbotin, Yu.N.: Splines in computational mathematics, Nauka, Moscow, (in Russian) (1976).
	In article
[30]	Vasilenko, V.A.: Spline functions: Theory, Algorithms, Programs, Nauka, Novosibirsk, (in Russian) (1983).
	In article
[31]	Vladimirov, V.S.: Generalized functions in mathematical physics. -M.: Nauka, (in Russian) (1979).
	In article
[32]	Wahba, G.: Spline models for observational data. CBMS 59, SIAM, Philadelphia, (1990).
	In article		CrossRef