_{1}

^{*}

Approximation theory experienced a long term history. Since 50’ last century, the rise of spline function as well as the advance of calculation promotes the growth of classical approximation theory and makes them develop a profound theory in maths, and application values have shown among the field of scientific calculation and engineering technology and etc. At present, the study of spline function had made a great progress and had a lot of fruits, as for that, the reader could look up the book [1] or [2]. Nevertheless, the research staff pays less attention to exponential spline function, since polynomial spline function is a special case of that, so it is much essential and meaningful for one to explore the nature of exponential spline function.

At the beginning, we introduce the definition of exponential spline function. From literature [

the definition: if function

spline function, where L is a differential operator

_{d} distinct roots of characteristic poly-

nomial and

Theorem 1:

If the differential operator’s characteristic polynomial is

Proof:

Let

Since there exists order

Hence

So that

Furthermore,

Therefore

We get

put

In terms of this idea, we obtain

Theorem 2: The dimension of the exponential spline function space is

Proof:

Suppose

We have

Since

So that

When characteristic polynomial has single real root, the linear space can be written as

Next we prove that

Set

have

According to theorem 1. 4. 23 of the book [

Corollary: There exists the

Theorem 3: If condition of interpolation and boundary satisfy:

then there exist the 3^{rd} degree exponential spline function satisfied with condition. And we have formula of error evaluation

Proof:

Suppose ^{rd} degree polynomial spline function, let

Hence

Both of them can be denoted by:

This lead to

Since

If boundary condition is

Since one of 3^{rd} degree polynomial spline function meet the constraint of interpolation

So that exponential spline function satisfied with condition (1) exists. That is

Next we prove formula of error evaluation. Suppose^{rd} degree exponential spline function satisfied with condition (1).

Let ^{rd} degree polynomial spline function)

Since

By formula of error evaluation for 3^{rd} degree polynomial spline function, we can have

In terms of book [

Since

Hence

Furthermore

By above expressions, we can conclude that

Supported partly by National Natural Science Foundation of China (11126140, 11201007) and partly by Beijing Talents Training Program (2011D005002000006) and partly by Science and Technology Development Plan Project of Beijing Education Commission (KM20121000-9013) and partly by Scientific Research Personnel Promotion Plan of North China University of Technology (BJRC201309).

Ge Yang, (2015) The Property of a Special Type of Exponential Spline Function. Advances in Pure Mathematics,05,804-807. doi: 10.4236/apm.2015.513074