General Degree of Periodic Spline Functions

In this paper we consider spline function of general degree m which has the same area as the function does in each partition of the sub-intervals. The existence and uniqueness in general of spline functions also been studied and obtained the result. : In this paper we consider spline function of general degree, i.e.,of degree m, m=2,3,.... We first give definitions and notations. We take 0 =x 0 <x 1 <...<x n−1 <x n =1 a subdivision of the interval [0,1]. The periodic spline function of degree m was defined by Ahlberg, Nilson and Walsh [1] in the following way: DEFINITION 1 : A function φ is said to be periodic spline function of degree m if it satisfies the following conditions: ( a ) In each sub-interval [x i−1 ,x i ] , i = 1,2,...,n , the function φ coincides with a polynomial of degree at most m i.e., φ(x)∈π m ; ( b ) its derivatives upto m-1 order are continuous, i.e., φ∈C (m−1) [0,1]; ( c ) φ holds the boundary conditions; φ (j) (0)=φ (j) (1) , j = 0,1,...,m-1. H. ter. Morsche [10] had defined the periodic spline function of degree m which is as follows: DEFINITION 2 By S( m,n ) we denote the set of spline functions φ, defined on [0,∞) , that have the following properties : ( a ) The restrictions of φ to an arbitrary sub-interval [x i−1 ,x i ] , i = 1,2,..., x i =ih, belongs to π m ; ( b ) φ∈C m−1 [0,∞). The set of periodic spline functions of degree m corresponding to the uniform sub-division of the interval [0,1] into n sub-intervals will be denoted by S 0 (m,n). DEFINITION 3: Truncated Power Function : The truncated power function is x m + . It is defined by x m + = [Sorry. Ignored \begin{cases} ... \end{cases}] where m is a positive real number.