1 Introduction and statement of the Problem
There are algebraic and variational approaches in the spline theory. In the algebraic approach splines are considered as some smooth piecewise polynomial functions. In the variational approach splines are elements of Hilbert or Banach spaces minimizing certain functionals. The first spline functions were constructed from pieces of cubic polynomials. After that, this construction was modified, the degree of polynomials increased. The theory of splines based on variational methods studied and developed, for example, by J.H.Ahlberg et al. [1], C. de Boor [3], A.R.Hayotov, G.V.Milovanivić and Kh.M.Shadimetov [5], I.J.Schoenberg [6], L.L.Schumaker [7], S.L.Sobolev [8], V.A.Vasilenko [12] and others.
The present work is also devoted to the variational method of construction of optimal interpolation formulas.
Assume, we are given the table of the values , of a function at the points . It is required to approximate the function by another more simple function , i.e.
(1.1) 
which satisfies the following interpolation conditions
Here and () are the coefficients and the nodes of the interpolation formula (1.1), respectively. By we denote the class of all functions defined on [0,1] which posses an absolutely continuous first derivative on [0,1] and whose second derivative is in . The class under the pseudoinner product
is a Hilbert space if we identify functions that differ by and , where . Here we consider the norm
For the fixed the error
(1.2) 
of the interpolation formula (1.1) is a linear functional.
Here, in fixed ,
and is the error functional of the interpolation formula (1.1) belongs to the space . Here is the conjugate space to the space , is Dirac’s deltafunction.
By the CauchySchwarz inequality the absolute value of the error (1.2) is estimated as follows
where
Therefore, in order to estimate the error of the interpolation formula (1.1) on functions of the space it is required to find the norm of the error functional in the conjugate space . That is we get the following problem.
Problem 1.1. Find the norm of the error functional of the interpolation formula (1.1) in the space .
It is clear that the norm of the error functional depends on the coefficients and the nodes . The problem of minimization of the quantity by coefficients is the linear problem and by nodes is, in general, nonlinear and complicated problem. We consider the problem of minimization of the quantity by coefficients when nodes are fixed.
The coefficients (if there exist) satisfying the equality
(1.3) 
are called the optimal coefficients and corresponding interpolation formula is called the optimal interpolation formula in the space . Therefore, for construction of the interpolation formula we should solve the next problem.
Problem 1.2. Find the coefficients which satisfy equality (1.3) when the nodes are fixed.
It should be noted that Problems 1.1 and 1.2 were solved in [2] in the Hilbert space . There the optimal interpolation formulas, which are exact for any polynomial of degree and for the function , were obtained.
The rest of the paper is organized as follows. In Section 2, using the extremal function, the norm of the error functional is found. Existence and uniqueness of the optimal interpolation formula of the form (1.1) is discussed in Section 3. Section 4 is devoted to construction of the optimal interpolation formula. Finally, in Section 5 some numerical results are presented.
2 The extremal function and the norm of the error functional
Here we find explicit form of the norm of the error functional .
For finding the explicit form of the norm of the error functional in the space we use its extremal function which was introduced by Sobolev [8, 9]. The function from space is called the extremal function for the error functional if the following equality is fulfilled
According to the Riesz theorem any linear continuous functional in a Hilbert space is represented in the form of a inner product. So, in our case, for any function from space we have
(2.1) 
Here is the function from is defined uniquely by functional and is the extremal function.
It is easy to see from (2.1) that the error functional , defined on the space , satisfies the following equalities
(2.2)  
(2.3) 
The equalities (2.2) and (2.3) mean that our interpolation formula is exact for the functions and .
The equation (2.1) was solved in [5] and for the extremal function was obtained the following expression
(2.4) 
where
(2.5) 
is the operation of convolution which for the functions and is defined as follows
(2.6) 
Now we obtain the norm of the error functional . Since the space is the Hilbert space then by the Riesz theorem we have
(2.7) 
Hence, using (2.4) and (2.5), taking into account (2.6) and (2.7), we get
Hence, keeping in mind that , defined by (2.5), is the even function, we have
(2.8) 
Thus, Problem 1.1 is solved.
Further, we solve Problem 1.2.
3 Existence and uniqueness of the optimal interpolation formula
Assume that the nodes of the interpolation formula (1.1) are fixed. The error functional (1.2) satisfies the conditions (2.2) and (2.3). The norm of the error functional is a multivariable function with respect to the coefficients . For finding the point of the conditional minimum of the expression (2.8) under the conditions (2.2) and (2.3) we apply the Lagrange method.
Consider the function
Equating to 0 the partial derivatives of the function by , and , we get the following system of linear equations of unknowns
(3.1)  
(3.2)  
(3.3) 
where is defined by equality (2.5).
The system (3.1)(3.3) has a unique solution and this solution gives the minimum to under the conditions (3.2) and (3.3).
The uniqueness of the solution of the system (3.1)–(3.3) is proved as the uniqueness of the solution of the system (24)–(26) of the work [11].
Therefore, in fixed values of the nodes the square of the norm of the error functional , being quadratic function of the coefficients , has a unique minimum in some concrete value .
Remark 3.1.
It should be noted that by integrating both sides of the system (3.1)(3.3) by from 0 to 1 we get the system (3.1)(3.1) of the work [5]. This means that by integrating the optimal interpolation formula (1.1) in the space we get the optimal quadrature formula of the form (1.1) in the same space (see [5]).
Remark 3.2.
Below for convenience the optimal coefficients we remain as .
4 The algorithm for computation of coefficients of the optimal interpolation formula
In the present section we give the algorithm for solution of the system (3.1)(3.3). Below mainly is used the concept of discrete argument functions and operations on them. The theory of discrete argument functions is given, for instance, in [9, 11]. For completeness we give some definitions about functions of discrete argument.
Assume that the nodes are equal spaced, i.e. , .
Definition 4.1.
The function is a discrete argument function (or discrete function) if it is given on some set of integer values of .
Definition 4.2.
The inner product of two discrete functions and is given by
if the series on the right hand side converges absolutely.
Definition 4.3.
The convolution of two functions and is the inner product
Now we turn on to our problem.
Suppose that when and . Thus we have the following problem.
Further we investigate Problem 4.1 which is equivalent to Problem 1.2. Instead of we introduce the following functions
(4.1) 
Now we should express the coefficients by the function . For this we use the operator which satisfies the equality
(4.2) 
where is equal to 0 when and is equal to 1 when , i.e. is the discrete deltafunction.
In [4] the operator which satisfies equation (4.2) is constructed and its some properties are studied.
The following theorems are proved in [4].
Theorem 4.4.
The discrete analogue of the differential operator satisfying the equation (4.2) has the form
(4.3) 
where
Theorem 4.5.
The discrete analogue of the differential operator satisfies the following equalities
1)
2)
3)
4)
Then taking into account (4.2), (4.3), using Theorems 4.4 and 4.5, for optimal coefficients we have
(4.4) 
Thus if we find the function then the optimal coefficients will be found from equality (4.4).
In order to calculate the convolution (4.4) it is required to find the representation of the function at all integer values of . From equality (4.1) we get that when . Now we find the representation of the function when and .
Since when then
Now we calculate the convolution when .
Suppose and then taking into account equalities (2.5), (3.2) and (3.3), we have
(4.5) 
where and are unknowns.
From (4.5) when and we get
(4.6)  
(4.7) 
Thus, putting (4.6) and (4.7) to (4.5) we have the following explicit form of the function :
(4.8) 
In the last expression of the function we have only two unknowns and .
Unknowns and we find from the equation
(4.9) 
when and . From the last equation
From the system (4.9) in the case and after some simplifications we have the following system of equations
(4.10) 
where
Now, from (4.9) in the case doing some calculations we get the next equation
(4.11) 
where
Then solving the system (4.10), (4.11) of equations we get and . Finally, from (4.4) for we get the explicit formulas for optimal coefficients as claimed in the following theorem.
Theorem 4.6.
Coefficients of the optimal interpolation formula (1.1) with equal spaced nodes in the space have the following form
here
5 Numerical results
In this section we give some numerical results.
First, when using Theorem 4.6, we get the graphs of the coefficients of the optimal interpolation formulas
They are presented in Fig 1. These graphical results confirm Remark 3.2 for the case , i.e. for the optimal coefficients the following hold
where is the Kronecker symbol.
Now, in numerical examples, we interpolate the functions
by optimal interpolation formulas of the form (1.1) in the cases , using Theorem 4.6. For the functions , the graphs of absolute errors , , are given in Fig 2, Fig 3, Fig 4. In these Figures one can see that by increasing value of absolute errors between optimal interpolation formulas and given functions are decreasing.
Fig. 1. Graphs of coefficients of the optimal interpolation formulas (1.1) in the case .
Fig. 2. Graphs of absolute errors for and : .
Fig. 3. Graphs of absolute errors for and : .
Fig. 4. Graphs of absolute errors for and : .
The Figure 4 shows exactness our optimal interpolation formula for the function .
References
Список литературы
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[3]
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