Ibrahim, A. and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....An important feature of the Morgan Scott basis is that each basis function is supported at most in the star of a vertex, i.e. the union of all triangles sharing the vertex. For many further results on the dimension and bases for the spaces S r q ( Delta) and their superspline subspaces, see [1 5,9,10,12,13,15] and references therein. As it has been shown recently [7] the property of local linear independence of a system of functions plays an important role in the problems of multivariate spline interpolation. Because of this, the question of existence of locally linearly independent systems of ....

Ibrahim, A., and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

....operator s f : C 2r( Omega Gamma S r;ae q ( Delta) of Section 3 is bounded by a constant that depends only on r; q and the smallest angle Delta in Delta. We note that there is some interrelation between our basis fs 1 ; s n g and the basis for S r;ae q ( Delta) constructed in [16] by using Bernstein B ezier techniques. Particularly, the supports of basis functions are the same. However, the minimal determining set of [16] cannot be transformed by standard Bernstein B ezier arguments into a Hermite interpolation scheme of our type. 2 Nodal Functionals Given a regular ....

.... Delta in Delta. We note that there is some interrelation between our basis fs 1 ; s n g and the basis for S r;ae q ( Delta) constructed in [16] by using Bernstein B ezier techniques. Particularly, the supports of basis functions are the same. However, the minimal determining set of [16] cannot be transformed by standard Bernstein B ezier arguments into a Hermite interpolation scheme of our type. 2 Nodal Functionals Given a regular triangulation Delta, we denote by N the number of triangles, by V the number of vertices, by V I and VB the number of interior and boundary ....

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A. Kh. Ibrahim and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

....S r;ae q ( Delta) is the superspline subspace of 16 O. Davydov, G. Nurnberger, F. Zeilfelder S r q ( Delta) q 3r 2, S r;ae q ( Delta) fs 2 S r q ( Delta) s 2 C ae (v) for all vertices v of Deltag; with ae = r Theta r 1 2 : The dimension of S r;ae q ( Delta) is given in [14]. Since restrictions of a spline s 2 S r;ae q ( Delta) to every triangle of Delta are polynomials, we are allowed to use derivatives of order greater than ae, but in this case a particular triangle T 2 Delta has to be chosen so that the derivative information comes from sj T . Let f 2 C ....

....noncollinear, 1; otherwise, and i is the angle between e and e i , i = 1; 2. It follows from Theorem 5. 1 that the fundamental functions s 1 ; s N of the above Hermite interpolation scheme form a basis for S r;ae q ( Delta) We note that a basis for this space has been constructed in [14] by using Bernstein B ezier techniques. Although there exists some interrelation between two bases, particularly, the supports of basis functions are the same, the minimal determining set of [14] cannot be transformed by standard Bernstein B ezier arguments into a Hermite interpolation scheme of ....

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A. Ibrahim and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

.... surface are discussed in [5] Even though we are working in IR 3 , because of the nature of homogeneous polynomials which are essentially bivariate functions the entire development is closely modelled after the analysis of the bivariate spaces of splines S r d ( Delta) carried out in [7, 15, 16]. 2. Homogeneous Spline Spaces We begin by introducing some notation, closely following [4] Definition 2.1. Let fv 1 ; v 2 ; v 3 g be a set of linearly independent unit vectors in IR 3 . We call T = fv 2 IR 3 : v = b 1 v 1 b 2 v 2 b 3 v 3 with b i 0g (2:1) the trihedron generated ....

....In view of these properties, we say that such splines have local support. The duality property (2. 9) assures that the splines s for 2 are linearly independent, and since there are precisely #(G) of them, they form a basis for H 0 d (T ) To obtain analogous results for H r d (T ) we follow [7, 15, 16]. To get an upper bound on dimension, we construct a determining set Gamma ae such that if s 2 H r d (T ) fls = 0 for all fl 2 Gamma implies s j 0: 2:10) Then as shown in [7] dim H r d (T ) is bounded above by the cardinality of Gamma. We can get a lower bound for the dimension (and ....

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Ibrahim, A., and L. L. Schumaker, Superspline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

.... surface are discussed in [6] Even though we are working in IR 3 , because of the nature of homogeneous polynomials which are essentially bivariate functions the entire development is closely modelled after the analysis of the bivariate spaces of splines S r d ( Delta) carried out in [8, 15, 17]. 2. Homogeneous spline spaces. We begin by introducing some notation, closely following [4] Definition 1. Let fv1 ; v2 ; v3g be a set of linearly independent unit vectors in IR 3 . We call (3) T = fv 2 IR 3 : v = b1v1 b2v2 b3v3 with b i 0g the trihedron generated by fv1 ; v2 ; v3g. ....

....In view of these properties, we say that such splines have local support. The duality property (11) assures that the splines s for 2 are linearly independent, and since there are precisely #(G) of them, they form a basis for H 0 d (T ) To obtain analogous results for H r d (T ) we follow [8, 15, 17]. To get an upper bound on dimension, we construct a determining set Gamma ae such that if s 2 H r d (T ) 12) fls = 0 for all fl 2 Gamma implies s j 0: Then as shown in [8] dimH r d (T ) is bounded above by the cardinality of Gamma. We can get a lower bound for the dimension (and construct ....

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A. Ibrahim and L. L. Schumaker, Superspline spaces of smoothness r and degree d 3r + 2, Constr. Approx., 7 (1991), pp. 401--423.

....We denote such a cell by 4 v . Since the Clough Tocher split of a triangle T is defined using its barycentric center, it is clear that the smallest angle in 4CT is equal to one half of the smallest angle in T . To construct stable local bases, we make use of Bernstein B ezier techniques as in [1,2,4,5,10]. For any given d and triangulation 4, let D d;4 : T24 D d;T ; be the set of domain points, where D d;T : f T ijk : iv 1 jv 2 kv 3 ) d ; i j k = dg; and T : hv 1 ; v 2 ; v 3 i. Then every polynomial p of degree d can be written uniquely in the B form p = X i j k=d c T ....

....and it follows that we need d 2 3r Gamma 1 2 1 = ae 6m 1; if r = 2m, 6m 3; if r = 2m 1. 7:2) We now examine what values can be chosen for the super smoothness ae at the center v of the cell 4 v . Let S r;ae d (4 v ) fs 2 S r d (4 v ) s 2 C ae (v)g: By Lemma 3. 2 of [5], dimS r;ae d (4 v ) ae 2 2 3 h d Gamma r 1 2 Gamma ae Gamma r 1 2 i oe; 13 where oe : d Gammar X j=ae Gammar 1 (r j 1 Gamma je) and e is the number of edges attached to the center vertex with different slopes. For stability of dimension, we ....

Ibrahim, A. and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

....problems. Finding stable local bases for spline spaces S r d (4) is a nontrivial task for r 0, and for general triangulations can only be done when d 3r 2, see Remark 11.4. The first constructions were for very special superspline subspaces of S r d (4) and can be found in [ 2] [6]] A construction for arbitrary spline spaces S r d (4) and corresponding superspline subspaces was discovered only very recently, see [3] To get stable bases for spline spaces with d 3r 2, we have to restrict ourselves to classes of triangulations with a special structure. In this paper ....

....r. The best elements will have the lowest degree possible, will have stable dimensions, and will use the least number of degrees of freedom. We begin by examining the dimension of the superspline space S : fs 2 S r d (4 v ) s 2 C ae v (v)g defined on a Powell Sabin cell. By Lemma 3. 2 of [6], dimS = ae v 2 2 6 d Gamma r 1 2 Gamma ae v Gamma r 1 2 oe; where oe : d Gammar X j=aev Gammar 1 (r j 1 Gamma je) and e is the number of edges attached to the center vertex v with different slopes. Since 3 e 6 for a Powell Sabin split, we ....

Ibrahim, A., Schumaker, L. L., Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx., 7(1991), 401--423.

....; vn ; wn ; vn 1 in counterclockwise order, where we identify vn 1 : v1 if v is an interior vertex. We denote the triangles of which share the vertex v by T i = hv; v i ; w i i and T 0 i = hv; w i ; v i 1 i, i = 1; Delta Delta Delta ; n. The first lemma is a rephrasing of Lemma 3. 1 of [10]. See also [1] splines on triangulated quadrangulations 151 Lemma 6. Let r be even, and set = 3r Gamma 2) 2. Suppose v is a boundary vertex of of index n, and let Gamma v consist of the domain points 1 ) T1 ijk , i j k = d with i d Gamma Gamma 1, 2 ) T i ....

....Then Gamma v forms a minimal determining set for any s 2 S r; d (star(v) with d 1, on D 1 (v) Proof. Since on D 1 (v) the smoothness conditions for s 2 S r; d (star(v) are the same as those for s 2 S r; 1 (star(v) it suffices to consider s in the latter space. By Lemma 3. 2 of [10], this space has dimension D : Gamma 2 2 Delta 2n[ Gamma 2 Gammar 2 Delta Gamma Gamma 1 Gammar 2 Delta ] oe, with oe = P Gammar 2 j= Gammar 1 (r j 1 Gamma je) where e is the number of edges in with different slopes attached to v. Since e 4 and = 3 Gamma ....

A. Ibrahim and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Construct. Approx, 7(1991), pp. 401--423.

....leads to sparse linear systems. We call such splines star supported. In [1] they are referred to as minimally supported, while in [8] they are called vertex splines. It is easy to see that for all d 1, the spaces S 0 d (4) have star supported bases. In addition, for r 1, it is known [9,10] that the spaces S r d (4) possess bases of star supported splines for all d 3r 2. The following complement to this result is the main result of this paper. Theorem 1. Suppose r 1 and d 3r 1. Then there are triangulations 4 for which S r d (4) does not have a star supported basis. Our ....

....d r since in this case S r d (4) j P d , and clearly there are no star supported splines in the space. We give a proof of Theorem 2 for r 1 d 2r in Sect. 2, and for 2r 1 d 3r 1 in Sect. 4. Throughout the paper we assume familiarity with the Bernstein B ezier machinery as used e.g. in [1 10]. In particular, given a vertex v of 4, we recall that the j th ring R j (v) around v is the set of domain points at a distance j from v, while the j th disk D j (v) around v is the set of domain points at a distance of at most j from v. For each domain point P , we write P s for the associated ....

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A. Ibrahim and L.L. Schumaker, Superspline spaces of smoothness r and degree d 3r + 2, Constr. Approx., 7 (1991), pp. 401--423.

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Ibrahim, A. and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

....would be a nontrivial polynomial of degree q q Gamma 2 Gamma 1 which vanishes q Gamma times at 0 and q Gamma times at fl. This is impossible, and we conclude that C cannot be zero. Various weaker versions of Lemma 2.1 can be found in the literature for example, see Lemma 3. 3 of [6] in the special case where q = q. We now illustrate the lemma for the case = Gamma1, which is the only case we use here) Example 2.2. Let d = 10, m = 8, q = 2, q = 3, and = Gamma1 in Lemma 2.1. Discussion: Fig. 1 shows the domain points of two adjoining triangles. We assume that we ....

Ibrahim, A. and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

....r. The best elements will have the lowest degree possible, will have stable dimensions, and will use the least number of degrees of freedom. We begin by examining the dimension of the superspline space S : fs 2 S r d (4 v ) s 2 C ae v (v)g defined on a Powell Sabin cell. By Lemma 3. 2 of [4], dimS = ae v 2 2 6 d Gamma r 1 2 Gamma ae v Gamma r 1 2 oe; where oe : d Gammar 1 X j=aev Gammar 1 (r j 1 Gamma je) and e is the number of edges attached to the center vertex v with different slopes. Since 3 e 6 for a Powell Sabin split, we ....

Ibrahim, A. and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

....We denote such a cell by 4 v . Since the Clough Tocher split of a triangle T is defined using its barycentric center, it is clear that the smallest angle in 4CT is equal to one half of the smallest angle in T . To construct stable local bases, we make use of Bernstein B ezier techniques as in [1,2,4,5,10]. For any given d and triangulation 4, let D d;4 : T24 D d;T ; be the set of domain points, where D d;T : f T ijk : iv 1 jv 2 kv 3 ) d ; i j k = dg; and T : hv 1 ; v 2 ; v 3 i. Then every polynomial p of degree d can be written uniquely in the B form p = X i j k=d c T ....

....and it follows that we need d 2 3r Gamma 1 2 1 = ae 6m 1; if r = 2m, 6m 3; if r = 2m 1. 7:2) We now examine what values can be chosen for the super smoothness ae at the center v of the cell 4 v . Let S r;ae d (4 v ) fs 2 S r d (4 v ) s 2 C ae (v)g: By Lemma 3. 2 of [5], dimS r;ae d (4 v ) ae 2 2 3 h d Gamma r 1 2 Gamma ae Gamma r 1 2 i oe; 13 where oe : d Gammar X j=ae Gammar 1 (r j 1 Gamma je) and e is the number of edges attached to the center vertex with different slopes. For stability of dimension, we ....

Ibrahim, A. and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

....for S r;ae v (4 v ) in the case where 4 v is a boundary cell. x6. A stable basis for S r;ae d (4) In this section we combine the constructions of the two previous sections to create stable local bases for the spaces of supersplines S r;ae d (4) defined in (1.1) for all d 3r 2. As in [23], we assume that k v ku d for each pair of neighboring vertices v; u 2 V, where k v : maxfae v ; g; v 2 V; with as in (3.2) Given a triangle T = hu; v; wi, let C T : C T n [D T ku (u) D T kv (v) D T kw (w) Associated with u, let A T (u) A T (u) n D T ku ....

....absolute value, which leads to a contradiction and completes the proof. Note that the above proof also applies to the superspline spaces S r;ae d (4) whenever ae v 2r for some near singular vertex v. On the other hand, if d 4r 1 and ae v 2r for all vertices, then the basis constructed in [23] is both stable and LLI. x8. Remarks Remark 8.1. It is well known [5 8,22,23] that the dimension of spline spaces and superspline spaces (when ae v 2r) generally depends on the exact geometry of the triangulation, and in particular may change as certain near singular vertices become singular. ....

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Ibrahim, A. and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

....0 more carefully. The basis constructed in Algorithm 4 is not locally linearly independent. To get an LLI basis, step 4) has to be modified in a different way, see [4] Remark 4. Star supported bases were constructed for general spline spaces srd S r d (4) for d 4r 1 in [1] and for d 3r 2 in [10,11]. The constructions were based on Bernstein B ezier techniques, and are not stable for triangulations that contain near degenerate edges and or near singular vertices. Remark 5. In [7] we use Bernstein B ezier techniques to construct stable local general bases for general spline spaces S r d ....

Ibrahim, A. and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

....We denote such a cell by 4 v . Since the Clough Tocher split of a triangle T is defined using its barycentric center, it is clear that the smallest angle in 4CT is equal to one half of the smallest angle in T . To construct stable local bases, we make use of Bernstein B ezier techniques as in [1,2,4,5,10]. For any given d and triangulation 4, let D d;4 : T24 D d;T ; be the set of domain points, where D d;T : f T ijk : iv 1 jv 2 kv 3 ) d ; i j k = dg; and T : hv 1 ; v 2 ; v 3 i. We recall that if M is a minimal determining set of domain points for a linear space S S r d ....

....and it follows that we need d 2 3r Gamma 1 2 1 = ae 6m 1; if r = 2m, 6m 3; if r = 2m 1. 7:2) We now examine what values can be chosen for the super smoothness ae at the center v of the cell 4 v . Let S r;ae d (4 v ) fs 2 S r d (4 v ) s 2 C ae (v)g: By Lemma 3. 2 of [5], dimS r;ae d (4 v ) ae 2 2 6 h d Gamma r 1 2 Gamma ae Gamma r 1 2 i oe; 13 where oe : d Gammar 1 X j=ae Gammar 1 (r j 1 Gamma je) and e is the number of edges attached to the center vertex with different slopes. For stability of dimension, we ....

Ibrahim, A. and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

.... They are also of interest since an LLI basis B for S r d (4) is a least supported basis in the sense that it is optimal with respect to the size of the supports of the B i , see [7] Locally supported bases have been constructed for the spline spaces S r d (4) and their superspline subspaces in [3,4,16,17,20,24], but they are mostly not LLI, 1) Mathematical Institute, Justus Liebig University, D 35392 Giessen, Germany, oleg.davydov math.uni giessen.de 2) Department of Mathematics, Vanderbilt University, Nashville, TN 37240, s mars.cas.vanderbilt.edu. Supported by the National Science Foundation under ....

....8.2 below. Recently, LLI bases have been constructed for the spaces S 1 d (4) see [10] and for certain superspline spaces (see [11,14] The main purpose of this paper is to construct LLI bases for S r d (4) for all d 3r 2, and for the entire scale of superspline spaces discussed in [17]. The paper is organized as follows. In Sect. 2 we treat the space S 0 d (4) separately as a means of introducing some needed notation. Then in Sect. 3 we show how to use Bernstein B ezier techniques to handle the spaces S 1 d (4) the results in [10] are based on nodal methods) In Sects. 4 ....

[Article contains additional citation context not shown here]

Ibrahim, A. and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

....r. The best elements will have the lowest degree possible, will have stable dimensions, and will use the least number of degrees of freedom. We begin by examining the dimension of the superspline space S : fs 2 S r d (4 v ) s 2 C ae v (v)g defined on a Powell Sabin cell. By Lemma 3. 2 of [4], dimS = ae v 2 2 6 d Gamma r 1 2 Gamma ae v Gamma r 1 2 oe; where oe : d Gammar 1 X j=aev Gammar 1 (r j 1 Gamma je) and e is the number of edges attached to the center vertex v with different slopes. Since 3 e 6 for a Powell Sabin split, we ....

Ibrahim, A. and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

....leads to sparse linear systems. We call such splines star supported. In [1] they are referred to as minimally supported, while in [5] they are called vertex splines. It is easy to see that for all d 1, the spaces S 0 d (4) have star supported bases. In addition, for r 1, it is known [6,7] that the spaces S r d (4) possess bases of star supported splines for all d 3r 2. The following complement to this result is the main result of this paper. Theorem 1.1. Suppose r 1 and d 3r 1. Then there are triangulations 4 for which S r d (4) does not have a star supported basis. Our ....

....clearly there are no star supported splines in the Fig. 1. A determining set for V 1 4 (4 H ) space. We give a proof of Theorem 1.2 for r 1 d 2r in Sect. 2, and for 2r 1 d 3r 1 in Sect. 4. Throughout the paper we assume familiarity with the Bernstein B ezier machinery as used e.g. in [1 7]. In particular, given a vertex v of 4, we recall that the j th ring R j (v) around v is the set of domain points at a distance j from v, while the j th disk D j (v) around v is the set of domain points at a distance of at most j from v. For each domain point P , we write P s for the associated ....

[Article contains additional citation context not shown here]

Ibrahim, A. and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

....coefficients, then the computed coefficients are bounded by K 15 C, where K 15 is a constant depending only on d, the smallest angle 4 in the triangulation, and the size of ff Gamma1 and fl Gamma1 , where ff; fi; fl are as in (6. 7) Proof: Versions of the first assertion can be found in [5,8,11]. Since the coefficients c 0 ; c r Gamma2q can be computed from the smoothness conditions, Lemma 6.1 provides a bound on their size in terms of the known coefficients. To bound the size of the remaining computed coefficients, we recall from Lemma 3.3 of [11] that the vector x : c ....

....can be found in [5,8,11] Since the coefficients c 0 ; c r Gamma2q can be computed from the smoothness conditions, Lemma 6.1 provides a bound on their size in terms of the known coefficients. To bound the size of the remaining computed coefficients, we recall from Lemma 3. 3 of [11] that the vector x : c r Gammaq ; c r Gamma2q 1 ; c r Gamma2q 1 ; c r Gammaq ) is uniquely determined by a system of equations of the form Mx = y, where M is a nonsingular matrix with det M = ff i 1 fl i 2 fi fi fi fi fi fi fi 1 q 1 (q Gamma1) Delta Delta Delta ....

[Article contains additional citation context not shown here]

Ibrahim, A. and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

....; vn ; wn ; vn 1 in counterclockwise order, where we identify vn 1 : v 1 if v is an interior vertex. We denote the triangles of which share the vertex v by T i = hv; v i ; w i i and T 0 i = hv; w i ; v i 1 i, i = 1; Delta Delta Delta ; n. The first lemma is a rephrasing of Lemma 3. 1 of [10]. Lemma 4.2. Let r be even, and set = 3r Gamma 2) 2. Suppose v is a boundary vertex of of index n, and let Gamma v consist of the domain points 1) T1 ijk , i j k = d with i d Gamma Gamma 1, 2) T i d Gamma Gamma1;j; 1 Gammaj , for j = 0; Gamma r and i = 2; ....

.... Gamma v forms a minimal determining set for any s 2 S r; d (star(v) with d 1, on D 1 (v) Proof: Since on D 1 (v) the smoothness conditions for s 2 S r; d (star(v) are the same as those for s 2 S r; 1 (star(v) it suffices to consider s in the latter space. By Lemma 3. 2 of [10], this space has dimension D : Gamma 2 2 Delta 2n[ Gamma 2 Gammar 2 Delta Gamma Gamma 1 Gammar 2 Delta ] oe, with oe = P Gammar 2 j= Gammar 1 (r j 1 Gamma je) where e is the number of edges in with different slopes attached to v. Since e 4 and = 3 ....

Ibrahim, A. and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

....i 2 L : fr Gamma 2q 1; r Gamma qg, and are bounded by KC, where K is a constant depending only on d, the smallest angle 4 in the triangulation, and the size of ff Gamma1 and fl Gamma1 , where ff; fi; fl are as in (6. 7) Proof: Versions of the first assertion can be found in [5,8,11]. To bound the size of the computed coefficients, we recall from Lemma 3.3 of [11] that the vector x : c r Gammaq ; c r Gamma2q 1 ; c r Gamma2q 1 ; c r Gammaq ) is uniquely determined by a system of equations of the form Mx = y, where M is a nonsingular matrix with det M = ....

....constant depending only on d, the smallest angle 4 in the triangulation, and the size of ff Gamma1 and fl Gamma1 , where ff; fi; fl are as in (6.7) Proof: Versions of the first assertion can be found in [5,8,11] To bound the size of the computed coefficients, we recall from Lemma 3. 3 of [11] that the vector x : c r Gammaq ; c r Gamma2q 1 ; c r Gamma2q 1 ; c r Gammaq ) is uniquely determined by a system of equations of the form Mx = y, where M is a nonsingular matrix with det M = ff i 1 fl i 2 fi fi fi fi fi fi fi 1 q 1 (q Gamma1) Delta Delta Delta ....

[Article contains additional citation context not shown here]

Ibrahim, A. and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7 (1991), 401--423.

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Ibrahim, A. K. and L. L. Schumaker, Super spline spaces of smoothness r and degree d 3r + 2, Constr. Approx. 7, 401--423.

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