A NEW METHOD FOR THE CONSTRUCTION OF SPLINE BASIS FUNCTIONS FOR SAMPLING APPROXIMATIONS

A method for constructing a new kind of spline basis functions (φ2n(x)) with compact support on R was described by Ueno et al. (2007). These basis functions for sampling approximations consist of a linear combination of the cardinal B splines, but the construction is complicated and therefore were constructed basis functions φ2n(x) only for n ≤ 5. We discuss a new construction of the basis functions and its approximation properties are considered. AMS Subject Classification: 65D, 41A15


Introduction
In [5] the spline basis functions ϕ 2n (x) which are symmetric with narrower compact support are constructed satisfying the following conditions In [5] it is also shown that if ϕ 2n (x) is a specific linear combination of {N 2n+2 , • • • , N n+2 } and satisfies (C3), it enjoys the condition (C4) which is equivalent to the moment condition.As mentioned in [5], the construction of such basis functions is interesting from both the sampling and interpolating approximation point of view.But the general rule for choosing the specific linear combination mentioned above is complicated.(C1)-(C4) properties can not guarantee a unique basis depending on how a specific linear combination of the cardinal B-spline is chosen.
The aim of this paper is to give a new construction of functions ϕ 2n that can overcome the above mentioned complication.The main idea of our construction is the direct applications of conditions (C1), (C2) and (C4), but not the condition (C3) in general.
The rest of the paper is organized as follows.In Section 2, we propose a new construction of basis splines without (C3) condition.In Section 3, we consider the approximation properties of a local cubic splines.Finally, in Section 4, we present results of numerical tests which confirm the theoretical order of convergence.

Construction of Basis Splines
We are looking for ϕ 2n which satisfies the (C1), (C2) and (C4).If ϕ satisfies Strang-Fix (SF) and moment (M) conditions, then the sampling approximation obtained by the translation of ϕ gives us a good approximation [1].We need the following property: Proposition 1.The following are equivalent: 1. ϕ satisfies (SF) and (M) of order r,

2.
k∈Z Proof (see [1], for example.) The fulfilment of condition (C4) indicates that ϕ 2n (x) is differentiable sometimes.So we can differentiate (C4) s times (s ≤ 2n) When i = s, setting x = 0 in (1), we obtain Analogously, when i = s, setting x = 0 in (1), we obtain From (C1) it is clear that Setting x = 0 in (4) we conclude that ϕ (s) 2n (0) = 0, when s is odd. ( If we take into account (C2) and (4), the equalities (2) and (3) can be rewritten as  We summarise the following algorithm to construct the basis splines: 1. Set a value of n.

Find the ϕ
), and (6c) (when 2 ≤ s ≤ n).In this step we can find n 2 + n values of ϕ which consists 2n + 2 equations with 2n + 2 unknowns a i,j , where c k are parameters and p i (0 then the spline ϕ 2n (x) coincides with one, constructed in [5].
(b) find c k using p Let us explain some cases of the proposed algorithm.For STEP 2, from ( 5) and ( 6), the direct calculations for n = 1, 2 and 3 give us the following values: These turn out to correspond to the well known 5-point finite difference scheme for the first and second derivatives [2].
which correspond to the well known 7-point central difference formulas of order 4 for derivatives of order 1-4 respectively [2].
For STEP 3, we consider the case n = 1.Assume that ϕ 2 (x) has a form In order to determine the coefficients a 1,i in (11) we use ( 7) and ( 8), i,e., As a result we have Similarly, in order to determine the coefficients of ϕ 2 (x) on x ∈ [1, 2] we will use the conditions which give us Thus, we find ϕ 2 (x) in the form It is easy to show that It means that ϕ 2 (x) ∈ C 1 [−2; 2] when c 1 = 1 6 .When c 1 = 0 the spline ϕ 2 (x) given by (12) coincides with one, constructed in [5].When c 1 = 1 6 the function ϕ 2 (x) belongs to C 2 [−2, 2] and it coincides with well-known cubic B-spline of class C 2 [3], [7].Similarly we find ϕ 2n (x) for n = 2, 3, 4, 5, 6.For brevity we present some basis functions in the form: To compare the profiles of ϕ 2 , ϕ 4 , ϕ 6 , and ϕ 8 , we sketch the graph of these functions in Fig. 1 and 2.  As mentioned above, in [5] was proposed general scheme to construct basis splines, but their construction (method 1) is more complicated and time consuming than our construction.Comparison of these two approaches was made by CPU time (in seconds, processor Intel Core i5 CPU 2.67GHz (4 CPUs)).From Table 1, it is clear that our approach (Algorithm 1) is preferable and very easy to implement.The construction also shows that the basis splines in [5]

Approximate Splines and their Properties
We now consider the sampling approximation By virtue of ( 8), (12), and (C1)-(C2) one can obtain at point We find the coefficients of sampling approximation (13) such that Then it is easy to show that Then from ( 16) we obtain From this and from (16) it follows that where Using in ( 17) we obtain where It remains to determine the coefficients α i for i = −1, 0 and i = N, N + 1.If we use the well-known one-sided approximation formulas [2] in ( 16) for k = 0 then we obtain formulas with accuracy O(h 4 ).Analogously, we obtain From ( 19), (20) it is clear that instead of (13) one can consider a local cubic spline where α k are given by (20).When c 1 = 0 the spline S j (x) coincides with sampling approximation given in [5].When c 1 = 1 6 , S j coincides with a local cubic one given in [7], [8].It is well known that for sampling approximation (22) holds the following error estimation [4], [5].
Theorem 1.Let ϕ satisfy (SF) and (M) of order r.If f belongs to W N,p (R), N ≤ r, then for 1 ≤ p ≤ ∞ we have where C ϕ,p,N is a constant depending only on ϕ, p, and N .
It means that for ϕ = ϕ 2 (x) we have We shall show that for smooth function f (x) ∈ C 4 (R) the order of accuracy increases.More precisely, we have the following theorem.
Theorem 2. If f belongs to C 4 (R), then for local approximation (21) holds the following estimation where l is given by (18).
Proof.Using Taylor expansion of f k , f k±1 at point x in (20) it is easy to show that in which we have used x k = kh, h = 2 −j .

Table 1
belong to ϕ 2n ∈ C n [a, b], while our construction without the requirement of (C3) improved the smoothness of spline.