Interpolation Splines Minimizing Semi-Norm in K2(P2) Space
1Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan
2Tashkent Institute of Railway Engineers, Tashkent, Uzbekistan
In the present paper using S.L. Sobolev’s method interpolation splines minimizing the semi-norm in K2(P2) space are constructed. Explicit formulas for coefficients of interpolation splines are obtained. The obtained interpolation spline is exact for the functions and . Also we give some numerical results where we showed connection between optimal quadrature formula and obtained interpolation spline in the space K2(P2).
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Keywords: interpolation spline, Hilbert space, the norm minimizing property, S.L. Sobolev’s method, discrete argument function
American Journal of Numerical Analysis, 2014 2 (4),
Received May 16, 2014; Revised May 26, 2014; Accepted May 26, 2014Copyright © 2013 Science and Education Publishing. All Rights Reserved.
Cite this article:
- Shadimetov, Kholmat M., Abdullo R. Hayotov, and Azamov S. Siroj. "Interpolation Splines Minimizing Semi-Norm in K2(P2) Space." American Journal of Numerical Analysis 2.4 (2014): 107-114.
- Shadimetov, K. M. , Hayotov, A. R. , & Siroj, A. S. (2014). Interpolation Splines Minimizing Semi-Norm in K2(P2) Space. American Journal of Numerical Analysis, 2(4), 107-114.
- Shadimetov, Kholmat M., Abdullo R. Hayotov, and Azamov S. Siroj. "Interpolation Splines Minimizing Semi-Norm in K2(P2) Space." American Journal of Numerical Analysis 2, no. 4 (2014): 107-114.
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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 , Arcangeli et al , Attea , Berlinet and Thomas-Agnan , Bojanov et al , de Boor , Eubank , Green and Silverman , Ignatov and Pevniy , Korneichuk et al , Laurent , Mastroianni and Milovanovic , Nürnberger , Schumaker , Stechkin and Subbotin , Vasilenko , Wahba  and others.
Suppose the functions belong to the Hilbert space (see [, Chapter 3])
equipped with the norm
and , where
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
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  for and the result of Holladay was generalized by de Boor  for any . In the Sobolev space of periodic functions the minimization problem of integrals of type (1.3) was investigated by I.J.Schoenberg , M.Golomb , W.Freeden [10, 11] and others.
We consider the Hilbert space
equipped with the norm
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
for any , where are the nodes of interpolation.
Following [, p.45, Theorem 2.2] we get the analytic representation of the interpolation spline
where , , , are real numbers,
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
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
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
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
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  and its some properties were investigated. Following in  we have:
Theorem 2.1. The discrete analogue of the differential operator satisfying the equation.(2.6) has the form
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
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
we get for
we have the following problem:
Problem 3. Find the solution of the equation
in the form:
where , , , are unknown coefficients.
It is clear that
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),
, , , are defined by (3.3), (3.7).
Proof. First we find the expressions for and . From (2.12) when and we get
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
Hence we get
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.
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  the optimal quadrature formula of the following form
was constructed in the space and the following was proved
Theorem 4.1 (Theorem 7 of ). 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  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 ). 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
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.
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