Construction of a Normalized Basis of a Univariate Quadratic C 1 Spline Space and Application to the Quasi-interpolation

: In this paper, we use the ﬁnite element method to construct a new normalized basis of a univariate quadratic C 1 spline space. We give a new representation of Hermite interpolant of any piecewise polynomial of class at least C 1 in terms of its polar form. We use this representation for constructing several superconvergent and super-superconvergent discrete quasi-interpolants which have an optimal approximation order. This approach is simple and provides an interesting approximation. Numerical results are given to illustrate the theoretical ones.


Introduction
In the classical Hermite interpolation, for each one of the knots x 0 < . . . < x n we are given a set of interpolation values f (j) i , j = 0, . . . , k then we need to find a polynomial function S of degree 2k + 1 in every interval [x i , x i+1 ], such that S(x i ) = f (j) i , i = 0, . . . , n, j = 0, . . . , k. Many authors have been working in this direction like Schoenberg [31], Lee [16] and Mummy [19], who has derived an explicit formula for the control points in terms of the interpolation data. He has used the de Boor-Fix dual functionals as an effective tool for solving this problem. In [33], Seidel gave another simple and elegant proof of Mummy's result using polar forms. Another interesting paper in this area of work is Schumaker's, he gave a general treatment of the use of quadratic splines for solving a similiar Hermite interpolation problem. The author showed exactly when it was necessary to add knots to a subinterval and where they can be placed. Let τ := (a = x 0 < x 1 < · · · < x n = b) be a partition of a bounded interval I := [a, b]. For the sake of the paper we assume that we know the values and the derivatives of a function f. Our problem of interpolation is to find a piecewise polynomial function S of class C 1 and degree two such that To solve this problem we need to add a new knot in each subinterval [x i , x i+1 ], i = 0, . . . , n − 1, and imposing the C 1 smoothness at this knot. The associated spline space is P 1 2 (I, τ 1 ), where τ 1 is a refinement of τ which is of dimension 2(n + 1). The Hermite spline interpolants' construction by adding some additional knots to the initial partition and increasing the polynomial pieces' number is not new. This method has been recently studied in the literature (see [17,32]) also by Lamnii et al [11]. The added knots can be chosen to preserve certain geometric shape such as monotonicity and convexity.
Various methods have been developed for building a positive and stable basis of B-splines, (see [15,35], for instance). Originally Dierckx [9] presented in the bivariate case, an algorithm for calculating a suitable normalized B-spline reprentation for Powell-Sabin spline. Similar B-spline representation for bivariate Powell-Sabin splines with higher smoothness have been considered in [37] and for bivariate Clough-Tocher splines in [36]. These B-splines have been used in [18,25] for constructing some interesting quasi-interpolants with optimal approximation order. Such a representation is given in trivariate setting by Sbibih et al [24] for C 1 quadratique B-splines over a Worsey-Piper split of a tetrahedral partition. In the recent years, Speleers [38] presented a method for constructing a normalized basis for the multivariate quadratic spline space defined over a generalized Powell-Sabin refinement of a triangulation in R s (s ≥ 1). He has indicated that the univariate version is related to the well-known classical B-splines of degree two. In addition these results, our main objectif is to construct a compact normalized basis of univariate quadratic C 1 spline space on the real interval refined by τ 1 . Thanks to the new constracted normalized B-splines we were able to find the classical quadratic B-splines of class C 1 without using any recurrence formula or divided differences. The main motivation is that we establish a new B-spline representation of Hermite interpolant Hf of any function f of class C 1 in P 1 2 (I, τ 1 ). Particularly we obtain the Marsden Identity.
A quasi-interpolant for a given function f is obtained as a linear combination of some elements of a suitable set of basis functions. In order to achieve stability and local control, these functions are required to be positive and to have small local supports. The coefficients of the linear combination are the values of linear functionals, depending on f and (or) its derivatives or integrals. Many works concerning the construction of quasi-interpolant are developed in the literature (see [1,2,3,8,4,5,14,13,21,22,23]). The main gain of these operators is that they have a direct construction without solving any system of equations and with the minimum possible computation time.
In numerical analysis, the superconvergence is a phenomenon where the order of convergence of the approximant error at certain special points is higher than the order of convergence of the approximant error over the definition's domain (see [6,26,27,28,29]). Then by considering a local linear polynomial operator in the neighborhood of the support of the B-splines that reproduces the space of polynomials of degree at most m ≥ 2, we propose a method to bluid superconvergent discrete quasi-interpolants of a function f. It satisfies an interesting property that these quasi-interpolants are globally of order 3 and of order m + 1 at the knots of the initial partition τ . This property is not only true for approximating function values but also for approximating first derivative.
To improve the numerical results given by our operator, we introduce a new concept, called the supersuperconvergence, when the local polynomial approximant is even. Thanks to this phenomenon, the quadratic spline quasi interpolant provides an improvement of the approximation order at the knots, it is of order m + 2 instead of m + 1.
The paper is organized as follows. In Section 2, we first define the finite element used in the construction of the normalized B-splines. In Section 3, we introduce a B-spline representation of the Hermite interpolant of any piecewise polynomial on the refinement τ 1 of class at least C 1 in terms of its polar form. In Section 4, we show how to construct superconvergent discrete quasi-interpolants. In Section 5, we give an estimate of the errors between the function f and the superconvergent discrete quasi-interpolants and between their first derivatives at the knots of τ . In Section 6, we introduce a new technique allowing us Define τ 1 := ∪ n−1 i=0 τ i,1 as a refinement of τ . The space of C 1 quadratic splines on the interval I endowed with τ 1 P 1 2 (I, τ 1 ) : In [32], Schumaker proved that for a given data f (x i ) and f ′ (x i ), i = 0, . . . , n, there exists a unique spline S ∈ P 1 2 (I, τ 1 ) solution of the following Hermite interpolation problem: Therefore, the dimension of the space P 1 2 (I, τ 1 ) equals 2(n + 1). 2.2. Hermite basis of space P 1 2 (I, τ 1 ) Let ϕ i and ψ i be the solution functions of the problem (2.1) in P 1 2 (I, τ 1 ) which satisfy the conditions ., n, where δ ij stands for the Kronecker symbol. We can easily verify that the supports of ϕ i and ψ i are supp ϕ i = supp ψ i = [x i−1 , x i+1 ] and that the spline S solution of the problem (2.1) can be written as Furthermore, the functions ϕ i , ψ i , i = 0, . . . , n, constitute the Hermite basis of the space P 1 2 (I, τ 1 ). This basis presents a major disadvantage which is the instability caused by the non-positivity of its elements (see Figure 2). Consequently, it is in practice undesirable especially in the construction of approximants. To remedy this problem we are going to build a normalized basis of the spline space P 1 2 (I, τ 1 ). 2.3. Normalized basis of P 1 2 (I, τ 1 ) In this subsection, we present the construction of new normalized B-splines. For i = 0, . . . , n, let [a i , b i ] be an interval such that x i ∈ ]a i , b i [ , i = 1, . . . , n − 1, a 0 = x 0 and b n = x n . Let φ j,1 , j = 0, 1, be the Bernstein polynomials of degree 1 associated with the interval [a i , b i ], where they have been defined in subsection 2.1. Then, we construct new B-splines H i,j as follows It is easy to see that H i,s , i = 0, . . . , n, s = 0, 1, are linearly independents and therefore form a basis of the space P 1 2 (I, τ 1 ). We will prove that this basis can be constructed in such a way that its elements have a local support [x i−1 , x i+1 ] with modifications when i = 0, n,, are nonnegative and form a partition of unity, i.e., for each x ∈ I H i,s (x) ≥ 0, The following theorem gives the non-negativity conditions.
Proof. The B-coefficients of the basis elements H i,s are easily computed on the subintervals of the support. They appear in Table 1.
Consequently, with the boundary conditions x 0 = a 0 and x n = b n , the normalized B-splines coefficients in the Bernstein basis on their support are given in Table 2.
For s = 0, 1, the B-splines H i,s are nonnegatives since all their B-cofficients are no-negatives, if From Table 2, we see that these conditions are also necessary. ✷ In Figure The form of the boundary B-splines change according to the boudary conditions x 0 = a 0 and x n = b n (see Figure 4).
represents the minimal segment ensuring the positivity of the corresponding B-splines.

Basis of the classical B-splines
In this subsection, We prove that for a particular choice of the normalized B-splines, we can construct the classical quadratic B-splines of class C 1 . If we cancel the first non zero B-coefficient of H i,1 and the last non zero B-coefficient of H i,0 , we obtain Then for i = 0, . . . , n we have Hence, defining the partition with knots X i given by X 2i := x i and X 2i+1 : An interesting and powerful tool in the approximation which is based on new polar approach has been emerged from the work of de Casteljau, Ramshaw and others (see [7,20,34]). The polar form of a polynomial is a transformation that reduces its complexity by adding new variables while having a certain symmetry property. In this subsection, we review some basic properties of the blossoming principle. • Multi-affine: for any index i and any real number λ, it holdŝ where λ := 1 − λ.
• Diagonal:p reduces to p when evaluated on its diagonal for each real number u, i.e., p(u, . . . , u) = p(u).
In order to express the polar form of a product of polynomials of the first degree, we have the following result.
Hence the result. ✷

Quadratic Hermite interpolation
By using the polar form approach, we give some result to represent the Hermite interpolant Hf of any function f of class C 1 in P 1 2 (I, τ 1 ). Let m be an integer greater than or equal to 2 and let A i and B i be two points such that It is well known, see [10] page 5, that every polynomial q of degree ≤ k defined on a segment [a i , b i ] can be written in the Bernstein basis of P k as follows : where φ s,k are the Bernstein polynomials of degree k defined in subection 2.1.
Theorem 3.3. For any f ∈ C 1 (I), the Hermite interpolant Hf of f in the space P 1 2 (I, τ 1 ) is given by for all x ∈ I and m ≥ 2.
Normalized Basis of a Univariate Quadratic C 1 Spline Space 9 Proof. Define For 0 ≤ l ≤ n, we have From (3.1) and the partition of the unity of the Bernstein basis, we have Using (3.2), we obtain Then, On the other hand, we have Then, 0 ≤ l ≤ n, we get By using (3.1), we get Finally, by uniqueness of the Hermite interpolant in the space P 1 2 (I, τ 1 ), we deduce that ψ = Hf, which completes the proof.
Proof. Let S be a polynomial spline of degree m ≥ 2 and of class C 1 on the interval I endowed with the refinement τ 1 . From Theorem 3.3, the Hermite interpolant in P 1 2 (I, τ 1 ) of S can be written in the form Then From Taylor expansion, we have For k = 2, . . . , m, we put By using the Proposition 3.2, one can see that, for m ≥ 2 where p(x) = m j=0 l j (x). Thus Therefore

✷
In particular if the spline S is a polynomial in P m (R), we have the following results : 4. Superconvergent discrete quasi-interpolants in P 1 2 (I, τ 1 ) In this section, we propose a method to build quasi-interpolants based on discrete values of a function f . We are interested on the quasi-interpolants of the form Proof. Let f ∈ P m , then we have p i,s = I i,s (f ) = f for i = 0, .., n, s = 0, 1. According to (4.1) and Corollary 3.5, we get Then, To build a superconvergent discrete spline quasi-interpolant, it suffices to take m+1 distinct interpolation points in the support of H i,s for i = 0, . . . , n and s = 0, 1. Let t i,s,k , k = 0, . . . , m, be these points and consider the interpolation polynomial of f at t i,s,k i.e., where L i,s,k are the Lagrange basis functions of P m associated with the points t i,s,k . Then, the quasiinterpolant defined by (4.1) and (4.2) satisfies Q m p = Hp, ∀p ∈ P m .
In the following theorem, we give an explicit formula of the coefficients µ i,s (f ) in terms of the data values f (t i,s,k ) for k = 0, . . . , m.
We have In order to compute the value of q i,s,k k = 0, . . . , m, we use Proposition 3.2, then . Since, it holds that

Error estimate of superconvegent discrete quasi interpolants
In this section, we will prove that the constructed quasi-interpolant is superconvergent at knots x i , i = 0, . . . , n. Let f be a function in C 3 (I). Since the operators Q m , m ≥ 2 reproduce the space P 2 , there exists constants C k > 0, k = 0, 1, independent of m such that where . ∞,I denotes the infinity norm on the interval I. In the following proposition we give the error estimates associated with Q m and its first derivative at the knots.
Proof. Let f ∈ C m+1 (I) the Taylor expansion of f around x i , i = 1, . . . , m, is given by Denote by R m the polynomial part of the Taylor expansion. Then for each point x in the support of From Theorem 4.1, we have HR m = Q m R m , and by use the fact that R m (x i ) = HR m (x i ), we get By (4.5), we assume that q i,s,k are bounded by a constant C. Then, from (4.4) we obtain Then, Thus, In a similar way, we prove that which completes the proof. ✷ We present now the results of the numerical experiments for the classical case. To illustrate the superconvergence characteristic of the C 1 quadratic spline quasi-interpolants Q m , we choose arbitary the interpolation points in the interval [x i , A s i B 1−s i ] for s = 0, 1 such that β i,s,k ∈ [0, 1], k = 0, . . . , m shown in Table 3.  Table 3: The values of t i,s,k for m = 2, 3, 4, 5 and k = 0, .., m.
We consider the following test functions defined on I = [0, 1] by We define the local error between a function g and the quasi-interpolant Q m g at the knots of τ by the following relation: , k = 0, 1, and the numerical convergence order by where, m = 2, 3, 4, 5.

Approximating function values
To illustrate numerically the result, we give in Table 4, for different values of n, the maximum absolute errors at knots E (0) m,n (f 1 ) associated with the operator Q m for m = 2, 3, 4, 5. In the case of the function f 2 the same errors (i.e., E (0) m,n (f 2 ), m = 2, 3, 4, 5) are given in Table 6. Also, we list in Tables 5 and 7 respectively the numerical convergence orders NCO m of the maximum absolute errors at the knots.     From the above examples, we remark that when we increase n or m, we get a quasi-interpolant with small errors and the numerical convergence order is in good agreement with the theoretical one.

Approximating derivative values
As in above, we illustrate numerically in Tables 8 and 10

Super-superconvergence phenomenon
In this section, we add a concept called the super-superconvergence phenomenon. From a numerical observation , we have remarked that the approximation order is O(h m+2 ) at knots when the degree m of the local polynomials is even. This is what we call a super-convergence phenomenon, it leads to an improvement of approximation properties. Unfortunately, this phenomenon does not happen for an arbitrary data site. In the following result, we sum up how to choose quasi-interpolation points in order to achieve the super-superconvergence phenomenon.
For each i = 1, . . . , n − 1, let x i be the midpoint of [a i , b i ].
Theorem 6.1. Let f ∈ C m+2 (I) such that m is even. If the set of the local interpolation points corresponding to H i,0 is symmetric to the one corresponding to H i,1 with respect to x i for i = 1, . . . , n − 1, and if the local polynomial approximant p 0,0 (resp. p n,0 ) interpolates f at x 0 (resp. x n ), then the quasiinterpolant Q m is super-superconvergent at x i , and Proof. Let f ∈ C m+2 (I), the Taylor expansion of f around x i for i = 1, . . . , n − 1 is given by Denote by R m the polynomial part of the Taylor expansion and by g m+1 its last term Using a similar way as in the proof of Proposition 5.1, we get and This implies that H j,s (x i ) = 1, then for any function g ∈ C 1 (I) we have By considering that a i = x i − d and b i = x i + d with d ≥ 0, we get First of all, we show that the coefficients q i,s,k , s = 0, 1 are the same for two symmetrical interpolation points with respect to x i . i.e., t i,0,k + t i,1,k = 2x i .
Also, as A i and B i are symmetric with respect to x i , we have x i = Ai+Bi 2 and we deduce that β i,0,k = β i,1,k . From Theorem 4.2, we easily obtain q i,0,k = q i,1,k = q i,k , for k = 0, . . . .m Then Q m g(x i ) = 1 2 m k=0 q i,k (g(t i,0,k ) + g(t i,1,k )) .
Therefore, for i = 1, . . . , n − 1, Q m g m+1 (x i ) = 0. Similarly, for i = n we get H n,0 (x n ) = 0, H n,1 (x n ) = 1, then Q m f (x n ) = p n,0 (A n , x m−1 n )H n,0 (x n ) + p n,1 (x m n )H n,1 (x n ) = p n,1 (x n ) = f (x n ), and the proof is complete. ✷ Remark 6.2. If the data sites t i,0,k , k = 0, . . . , m, are symmetric with respect to x i , to achieve the supersuperconvergence, it suffices to take {t i,0,k , k = 0, . . . , m} = {t i,1,k , k = 0, . . . , m} for i = 1, . . . , n − 1 i.e., we take the same interpolation points for p i,0 and p i,1 .   Through these examples, we remark that the numerical convergence order is in good agreement with the theoretical one. A comparison with the previous results allow us to see that when m is even, the associated errors of the super-superconvergence phenomenon are smaller than the ones of the superconvergence phenomenon.

Conclusion
In this paper, we have shown how to construct a new normalized B-spline basis of a C 1 continuous spline space of degree two. The basis functions have a local support, they are nonnegative, and they form a partition of unity. The classical C 1 quadratic B-splines are a particular case of our Hermite B-splines. Moreover, we used some results on blossoming to establish the B-spline representation of Hermite interpolant of any C 1 continuous spline of degree 2 in terms of its polar form. Hence we used this representation for constructing several superconvergent and super-superconvergent discrete quasiinterpolants. This new approach provides an interesting approximation and it can be used for solving some numerical analysis problems. In futur works, the generalization of the proposed results to higher degrees will be studied.