Approximation by New Families of Univariate Symmetrical B-splines

In this paper we prove that there exists a unique positive symmetrical univariate B-spline with minimal support. It is obtained as linear combination of a minimal number of successive classical B-splines with multiple knots in the space, S d , of cardinal polynomial splines of class C and degree d. Next, we show that the approximation order in the space generated by the integer translates of this B-spline is not optimal. However it can be used for geometrical design where the small support is appreciated but the approximation order is not crucial. To have a higher approximation order, we define the B-splines of high order by recurrence and by convolution with the characteristic function of the interval [0, 1]. We use these B-splines to study the cardinal interpolation and we show that it is correct in the sense of Schoenberg. Finally, we give the explicit expression of interplant operators associated with some of these B-splines. Résumé. Dans cet article, nous montrons l’existence et l’unicité d’une spline symétrique à support minimal qui s’écrit comme combinaison linéaire d’un nombre minimal de B-spline successives de l’espace S d des splines polynomiales cardinales de degré d et de régularité r. Nous démontrons que l’ordre d’approximation dans l’espace engendré par les translatés entières de cette B-spline n’est pas optimal. Cependant, leur utilisation dans le dessin géométrique, où l’ordre d’approximation n’est pas crucial mais où un support de longueur réduit est recommondé, pourrait être très utile. Pour avoir un ordre d’approximation élevé, nous définissons par récurrence de nouvelles familles de B-splines cardinales symétriques. Ensuite, nous étudions l’unisolvance du problème d’interpolation basé sur ces B-splines. Nous donnons enfin, des exemples de calcul des coefficients des splines fondamentales associées à quelques éléments de degrés faibles de ces nouvelles familles. Introduction It is well known that the classical cardinal B-spline of order d+1 and degree d with simple knots 0, 1, . . . , d+1 is of class Cd−1 and of support [0, d+1]. Moreover, it is symmetric and satisfies other properties (see e.g. [4,5,15]). The degree, class and length of support of this B-spline depend all on the integer d. However, a high regularity leads to a B-spline with a large support. Consequently, the approximation in the space spanned by its integer translates requires the resolution of a linear system with a very large size, and a full associated matrix. Therefore the linear system is difficult to solve. So, one way to construct B-splines of high regularity and small support is to use the classical cardinal B-splines with multiple knots. In this case, (see e.g. [4, 6, 15]), one can obtain a finite family of B-splines which form a partition of unity, but in general they are not symmetric. Our aim is to define new polynomial symmetrical B-splines, associated with the uniform knot sequence τ = Z, with small 1 Département d’Informatique, Faculté des Sciences et Techniques, Errachidia, Maroc;e-mail: eb [email protected] 2 Université Mohammed I, Ecole Supérieure de Technologie , Laboratoire MATSI, Oujda, Maroc; e-mail: [email protected] & e-mail: hamid [email protected] c © EDP Sciences, SMAI 2007 Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx.doi.org/10.1051/proc:072003 30 ESAIM: PROCEEDINGS supports, which depend only on d, but their class and degree depend on d and another nonnegative integer r. This idea is generalized to the bivariate case, (see [9–11]), the authors construct B-splines of class Cr and small supports on three and four directional meshes of the plane. Let φ be one of these B-splines and φ̂ its Fourier transform. We denote by S(φ) the space generated by the integer translates of φ, and for h > 0, Sh(φ) = {σhf, f ∈ S(φ)}, where (σhf)(x) = f( h ). In this paper, we study cardinal interpolation problem with new families of univariate cardinal B-splines of high regularity and small support which are symmetric and form a partition of unity. A cardinal interpolant to a sequence f on Z is a spline function in S(φ) of the form ∑ i∈Z ai(φ)φ(· − i) and which matches f on Z, i.e., ∑ i∈Z ai(φ)φ(j − i) = f(j), for all j ∈ Z, (1) where f is a sufficiently smooth function. As in [14], we say that the cardinal interpolation with φ is correct if for any bounded function f there exists a unique bounded solution a = {ai(φ), i ∈ Z} of equation (1). The solution L = ∑ i∈Z ci(φ)φ(·− i) of (1) which interpolates the data {δ0,j}, where δ0,j is the Kronecker symbol, is called cardinal fundamental spline function associated with φ. The cardinal interpolant to a sequence f on Z given by (1) can be written in Lagrange form (If)(x) = ∑ j∈Z f(j)L(x− j). Therefore, solving equation (1) is equivalent to determine the cardinal fundamental spline L. On the other hand, if we denote φ̃(w) = ∑ j∈Z φ(j)e−ıwj (resp. ã(w) = ∑ j∈Z aj(φ)e−ıwj) the symbol of the function φ (resp. the sequence a), then, according to [14], the cardinal interpolation with φ is correct if and only if the symbol φ̃ of φ does not vanish. In this case, the cardinal fundamental spline L has an exponential decay and the symbol of the sequence {ci(φ)} is given by c̃(w) = 1 φ̃(w) . Moreover, for h > 0, the optimal approximation order of the interpolant Ih = σhIσ 1 h in Sh(φ) coincides with the following so-called Fix-Strang condition of order n of φ (see [16]) { φ̂(0) = 1, Dφ̂(2πj) = 0, for j ∈ Z\{0} and k = 0, . . . , n. The paper is organized as follows. In Section 2, we give some preliminary results about the classical cardinal B-splines with multiple knots in the space of polynomial spline functions Sr d of class Cr and degree d associated with the uniform knot sequence Z. In Section 3, we prove the existence and the uniqueness of a symmetrical B-spline with minimal support. This B-spline, noted M and obtained as a linear combination of a minimal number of successive classical cardinal B-splines with multiple knots, is positive and its integer translates form a partition of unity, i.e., ∑ j∈ZM(· − j) = 1. We show that approximation order in the space S(M) is not optimal, so these splines can be useful for the geometrical design where the order of convergence is not crucial but small supports are recommended. In Section 4, we construct, by recurrence and convolution with the the characteristic function of [0, 1], a new family of B-splines with higher approximation orders and based on M . Next, we study the cardinal interpolation problem in the space S(φ) when φ is one of these later B-splines. We prove that the symbol of φ is positive and therefore the cardinal interpolation with φ is correct. Finally, as examples, we give the explicit expressions of fundamental splines associated with some of these B-splines. 1. Preliminary results Let r, d be two positive integers such that r < d. We denote by Sr d the space of cardinal polynomial splines of class C and degree d, with the set Z of all integers as the knot sequence. Let μ = d− r be the multiplicity ESAIM: PROCEEDINGS 31 of each knot. We assume that d ≥ 1 and 0 ≤ r ≤ d− 1, then 1 ≤ μ ≤ d− 1. In the following, we put e = bd+1 μ c and ρ = d + 1− μe, where bxc = max{n ∈ Z : n ≤ x}. In order to define the B-splines basis for the space Sr d , we consider the following knot sequence · · · , x−1 = −1, x0 = 0, · · · , xμ−1 = 0, xμ = 1, · · · , x2μ−1 = 1, x2μ = 2, · · · Let ngi = #{j ∈ Z : j ≤ i and xj = xi} and ndi = #{j ∈ Z : j ≤ i + d + 1 and xj = xi+d+1}. Then we have the following result. Theorem 1.1. There exists a unique B-splines basis (Ni)i∈Z for the space Sr d which satisfies the following properties (i) suppNi = [xi, xi+d+1], (ii) Ni is positive on ]xi, xi+d+1[, (iii) Ni is of class Cr−1+ngi at xi and of class Cr−1+ndi at xi+d+1, (iv) ∑ i∈ZNi = 1. Proof. It derives from the classical properties of B-splines, see [15]. ¤ From the above properties, we deduce that for each integer i, there are μ− ρ B-splines Niμ+j , j = 0, · · · , μ− ρ − 1, which have as support [i, i + e] and ρ B-splines Niμ+j , j = μ − ρ, · · · , μ − 1, with support [i, i + e + 1]. Moreover, these B-splines are all of class C at i, but the μ− ρ first ones are of class Cd−ρ−1−j at i + e and the ρ later ones are of class Cd+μ−ρ−1−j at i + e + 1. It is well known that when μ = 1 (simple knots), the B-splines Ni are symmetrical w.r.t. the middle of their supports and they can be easily obtained by translation, i.e., Ni(x) = N0(x − i). When μ > 1, there are also some specific properties of symmetry and periodicity. Proposition 1.2. For μ ≥ 1, the B-splines Ni satisfy (i) Niμ+j(x) = Nj(x− i), for all x ∈ R, i ∈ Z and 0 ≤ j ≤ μ− 1; (ii) Nj(x) = Nμ−ρ−j−1(e− x), 0 ≤ j ≤ bμ−ρ−1 2 c; (iii) Nj(x) = N2μ−ρ−j−1(e + 1− x), μ− ρ ≤ j ≤ μ− ρ + bρ−1 2 c. Proof. (i) derives from the fact that the knots are uniform and of multiplicity μ. (ii) and (iii) can be easily proved by using Theorem 2.1 or the recurrence relation of B-splines. ¤ 2. Symmetrical B-splines with minimal support In this section we show that there exists a unique symmetrical B-spline with minimal support in Sr d . It is obtained as linear combination of a minimal number of successive B-splines satisfying interesting properties. To establish the main result, we need the following lemma. Lemma 2.1. Let L be a spline function in S d of support [0,m], m ∈ N∗. If L is a linear combination of successive l B-splines Ni, then ∑ i∈Z L(· − i) = 1 implies that l ≥ μ. Proof. As L does not vanish on ]0, 1[, and it is linear combination of successive l B-splines Ni, it can be written in the form L = l+k−1 ∑ i=k aiNi , where k ∈ {0, . . . , μ− 1} and ai ∈ R∗. Then, for l = n ∗ μ + θ, with 0 ≤ θ ≤ μ− 1, we obtain