Interval estimation of trigonometrical splines

Quite often, it is necessary to quickly determine variation range of the function. If the function values are known at some points, then it is easy to construct the local spline approximation of this function and use the interval analysis rules. As a result, we get the area within which the approximation of this function changes. It is necessary to take into account the approximation error when studying the obtained area of change of function approximation. Thus, we get the range of changing the function with the approximation error. This paper discusses the features of using polynomial and trigonometrical splines of the third order approximation to determine the upper and lower boundaries of the area (domain) in which the values of the approximation are contained. Theorems of approximation by these local trigonometric and polynomial splines are formulated. The values of the constants in the estimates of the errors of approximation by the trigonometrical and polynomial splines are given. It is shown that these constants cannot be reduced. An algorithm for constructing the variation domain of the approximation of the function is described. The results of the numerical experiments are given.


Introduction
It is useful to determine the lower and upper bounds of the values of functions, eigenvalues of operators, solutions of systems of linear and nonlinear equations without without calculating a detailed numerical solution of the corresponding problems. The solution of such problems is considered in many papers published recently.
In the paper [1] a new approach for solving non linear systems of equations was proposed. This approach is based on Interval-Newton and Interval-Krawczyk operators and B-splines. The proposed algorithm is making great benefits of the geometric properties of B-spline functions to avoid unnecessary computations. For eigenvalue problems of self-adjoint differential operators, a universal framework is proposed to give explicit lower and upper bounds for the eigenvalues (see [2]).
To improve the calculation accuracy and reduce the computational cost, the interval analysis technique and radial point interpolation method are adopted in [3] to obtain the approximate frequency response characteristics for each focal element, and the corresponding formulations of structural-acoustic system for interval response analysis are deduced.
This paper continues the series of papers on approximation by local polynomial and non-polynomial splines and interval estimation see [5], [6], [7].
For constructing this interval extension, we use techniques from interval analysis. * e-mail: i.g.burova@spbu.ru

Approximation with the left splines
Suppose a, b be real numbers. Let the set of nodes X j be such that a < ... < X j−1 < X j < X j+1 < ... < b.

Trigonometric splines
First, we consider the approximation of a function f (x) by the left trigonometric splines (see [6]). We construct an approximation We obtain basic functions w j−1 (x), w j (x), w j+1 (x), x ∈ [X j , X j+1 ], solving the following system: The solution of this system can be written as follows: Suppose the set of nodes {X j } such, that h = X j+1 − X j , X j − X j−1 = Ah, A > 0, and let x = X j + th, t ∈ [0, 1]. Now the solution of system (2) can be written in the form: It is not difficult to see that the solution of system (2) also can be written as follows: .
can be written now also as follows: The last form of the basic splines will be used to construct the approximation.

Comparison with polynomial splines
As it is was shown in paper [6] we can use the following approximation: . This approximation will be a polynomial one if we obtain basic functions , solving the following system: The solution of this system is the following: , Using the notation x = X j + th, we get Easily it can be shown that there are relations between trigonometrical and polynomial splines: where Proof. Using the results from the paper [7] we get where c i , i = 1, 2, 3 are some arbitrary constants. Using expression (1), the solution of the system (2) and expression (7) we receive the estimation (6) of error of approximation with the trigonometrical splines. Using expression (1), the solution of the system (4) and the Taylor series for f (x) we receive the the estimation (5) of error of approximation with the polynomial splines.

Remark.
There are examples that show that the constants K 1 and K 2 can't be reduced. They are the following: f 1 (x) = x 3 /6 for the polynomial splines and f 2 (x) = sin(x) − cos(x) + x for the trigonometrical splines. Let us take h = 1, X j = 0, X j+1 = 1, X j−1 = −1 and construct the trigonometrical approximation F(x) and polynomial approximation G(x) using (1).
We have for these functions:

Approximation with right splines
Now we consider the approximation of a function f (x), x ∈ [X j , X j+1 ], with right trigonometric splines. We can construct an approximation of function f (x), also in the form: 0.573e-3 0.642e-3 sin(x) − cos(x) + x 0.642e-4 0.713e-4 We obtain basic functions w j (x), w j+1 (x), w j+2 (x) from the following system: The solution of this system can be written as follows: where S j = (sin(X j −X j+2 )−sin(−X j+1 +X j )−sin(X j+1 −X j+2 )).

Interval extention
Theorem 1 helps us to choose the correct length h = X j+1 − X j of the interval [X j , X j+1 ]. Suppose we know the values of function f (x) at nodes {X k }. Using formulas (1), (3) or (8) with trigonometrical splines and the technique of interval analysis [4] we can construct the upper and lower boundaries for every interval Y = [X j , X j+1 ]. Thus we avoid the calculations of approximation f (x) in many points of every interval [X j , X j+1 ] if we need to know the boundaries of the interval, where the function f varieties. In order to obtain the boundaries of variety f (x) we construct the approximation F(x), x ∈ Y in form (1) and consider F(Y).
In order to get the narrowest estimation interval we transform formulas (3). First, we consider the estimate of the lower bound of the estimating interval of the basis spline w j−1 (x).
Then the upper boundary of w j−1 will be the following After calculating the upper boundaries of w j−1 , w j and w j+1 we can calculate the upper boundary of F(x). Now the upper boundary of F(x) will be the following:  Suppose we know the values function sin(7x)−cos(9x) in points X j = j · π/15, j = 0, 1, . . . , 15. The information about the function can be represented in more then one possibility. At first, we can draw the pointplot of the points. The plot of points sin(7X j )−cos(9X j ), X j = j·π/15, j = 0, 1, . . . , 15 on [0, π] is given in figure 5.
After this is done, we could connect the points with lines. This plot is given in figure 6. But instead of the last one we could use the presentation of the function with our method of interval extension on every set interval. The result is given in figure 6. The plots of the upper and lower boundaries of the function sin(7x) − cos(9x) and points sin(7X j ) − cos(9X j ), X j = j·π/15, j = 0, 1, . . . , 15 on [0, π] are given in figure 7.
The plot of the error of approximation of the function sin(7x) − cos(9x) with the left trigonometrical splines on [0, π] is given on figure 11. The practical error of approximation is the following: max

Conclusion
In this paper we calculate the constants that can't be reduced in the theorem of approximation with trigonometrical splines and present the results of working the program of constructing interval extension. The results show that in many cases connected with approximation trigonometrical functions application approximations with trigonometrical splines gives better results then approximation with polynomial splines. To avoid calculation in many points we can use interval extension if we need to know only the upper and the lower boundaries of variation of the function. But we have to hold in mind the theorem of approximation.