ELEMENTARY SPLINE FUNCTIONS

. The aim of this paper is to deﬁne the elementary spline functions in analogy with the deﬁnition of the polynomial spline functions, given by I. J. Schonberg. Also, it is described a method for constructing the elementary spline functions. Finally, some examples are given. MSC 2000. 65D07.

Remark 2. As it is mentioned by I. J. Schoenberg, in [9], the polynomial spline functions were already used for approximation of functions by T. Popoviciu in [4].In particular, he showed that a continuous non-concave function of order n in a finite interval [a, b] is the uniform limit of elementary functions of order n that are also non-concave of order n in The elementary spline functions will be introduced here as an approximate solution of a Cauchy problem regarding linear differential equations, using the method described in [3].

One considers the Cauchy problem
(1) y (n) + a 1 y (n−1) + ... + a n y = 0 and (2) A method for approximating the solution y of the Cauchy problem (1)-( 2), on the interval [a, b], is that of attaching, on each subinterval [x k−1 , x k ], k = 1, ..., m, a Cauchy problem regarding a linear differential equation with constant coefficients and to approximate the solution y, on the corresponding subinterval, with the solution of the attached problem.
Let us consider the characteristic equation of (3): (5) The solution y k of the problem (3)-( 4) depends on the nature of the solution of the characteristic equation (5).
As a consequence of this statement is that a fundamental system of solutions y k1 , ..., y kn of the equation ( 3) depends on the nature of the roots of the equation (5).Thus, if the characteristic equation ( 5) has: • n real and distinct roots r k1 , ..., r kn , then y k1 (x) = e r k1 x , ..., y kn (x) = e r kn x .
• complex roots, for example, r = u ± iv, then the functions e ux cos vx and e ux sin vx are solutions of the equation ( 5).• multiple complex roots, for example, r = u + iv with multiplicity order µ, then the functions e ux cos vx, e ux sin vx xe ux cos vx, xe ux sin vx ...
x µ−1 e ux cos vx, x µ−1 e ux sin vx are solutions of the equation ( 5).If y k1 , ..., y kn is a fundamental system of solutions, generated by the roots of the characteristic equation ( 5), then is the solution of the Cauchy problem (3)-( 4), with the constants c 1 , ..., c n obtained by the conditions (4).Similarly to the definition of the polynomial spline function, as a piecewise polynomial function, given by I. J. Schonberg, we have the following definition.
Definition 5.The function ȳ, defined above, is called an elementary spline function.
Because ȳ was constructed as an approximation of the solution of the n-th order differential equation (1), Definition 5 can be completed by the following definition.Definition 6.The function ȳ is an elementary spline function of order n.
In the case of an uniform partition of the interval, for the approximation error we have the following estimation.
In the same way we get Remark 9.The solution of the problem (6), ȳ, has the following properties: ȳ ∈ C 1 [0, 1] and ȳ| is an exponential function, respectively, ȳ| y .
In Figure 1 we plot the graph of the solution ȳ, for Case 1.In Figure 2 we plot the graph of the solution ȳ, for Case 2.

Proposition 7 .
If x k = a + kh, k = 0, ..., m, with h = (b − a)/m, is an uniform partition of the interval [a, b] then the following estimation holds: