1. Introduction
At the beginning, we introduce the definition of exponential spline function. From literature [3] , we could learn
the definition: if function satisfies equation, we describe it as exponential
spline function, where L is a differential operator. Here, are constant coefficient and represent kth-order derivative. By this definition, we learn that exists continuous derivative and in each interval is linear combination of
, where the’s are the Nd distinct roots of characteristic poly-
nomial and is of order. As exists a single root 0 for characteristic polynomial, is polynomial spline function. Next we will deal with the case of there being unique real root.
2. Main Result
Theorem 1:
If the differential operator’s characteristic polynomial is , where is a root of multiplicity. Then the expression for exponential spline function of this special case is
Proof:
Let be on interval, Suppose
And we have
Since there exists order continuous derivatives for,
Hence
So that
Furthermore, is polynomial of nth degrees.
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 is continuous at the knot, hence has order continuous derivatives on interval.
When characteristic polynomial has single real root, the linear space can be written as
Next we prove that is linearly independent
Set On the interval, above equation become, we
have On the interval, we can get, so that, For the interval, By means of the same technique, we can obtain, hence is linearly independent. So that we conclude.
According to theorem 1. 4. 23 of the book [4] , we can prove next conclusion is true.
Corollary: There exists the for every f belonging to, such that
Theorem 3: If condition of interpolation and boundary satisfy:
(1)
then there exist the 3rd degree exponential spline function satisfied with condition. And we have formula of error evaluation
Proof:
Suppose is 3rd degree polynomial spline function, let
Hence
Both of them can be denoted by:, , , so that A is invertible matrix.
This lead to (2)
Since, hence, we can get is exponential spline function.
If boundary condition is, , by matrix relation (2), let
and
Since one of 3rd degree polynomial spline function meet the constraint of interpolation, boundary condition is and.
So that exponential spline function satisfied with condition (1) exists. That is.
Next we prove formula of error evaluation. Suppose, is 3rd degree exponential spline function satisfied with condition (1).
Let (where is 3rd degree polynomial spline function)
Since
By formula of error evaluation for 3rd degree polynomial spline function, we can have
In terms of book [5] , we have
Since
Hence
Furthermore
By above expressions, we can conclude that
.
Fund
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).