C. de Boor, K. H ollig (1983): Approximation order from bivariate C -cubics: a counterexample. Proc. Amer. Math. Soc., 87: 649--655.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....let us consider the scaled spline spaces hS k of functions f such that f h defined by f h (x) f(hx) are in the space S k on the 3 direction mesh in R 2 . If OE is one of the three box splines M 221 , M 212 , M 122 , then an order of O(h 3 ) is obtained, and de Boor, Hollig [5] also showed that this is the highest order. Thus, it would seem that there is a gap between univariate and bivariate approximation However, if we use the result in Theorem 3 with OE = M 212 , we see that if f is a function of one variable x 1 , say, then since Gamma 1 n Gamma = f(4; 0) 3; ....

C. de Boor and K. Hollig, Approximation order from bivariate C 1 -cubics: A counterexample, Proc. Amer. Math. Soc. 87 (1983), 13-28.

....a result of Zeni sek [167] see Section 5) that S r q ( has optimal approximation order provided that q 4r 1; r 0. However, in contrast to the univariate case, the approximation order of a bivariate spline space is not always optimal. This was rst proved in 1983 by de Boor and H ollig [25], who considered the space S 1 3 ( 1 ) Here, 1 is a triangulation obtained from a rectangular partition by adding the same diagonal to each rectangle (see Figure 4.1) Extensions of this result, were given by Jia [90, 91, 92] In 1993, de Boor and Jia [27] proved the following theorem. ....

C. de Boor and K. Hollig, Approximation order from bivariate C 1 cubics: a counterexample, Proc. AMS 87 (1983) 649-655.

....3 (4) successfully, we have to restrict our attention to special classes of triangulations. Indeed, for general triangulations, at this point it is not known whether interpolation at all of the vertices of 4 is even possible, and the dimension of S 1 3 (4) is also unknown. Moreover, it is known [3] that the space is defective in the sense that it does not give full approximation power on some triangulations (including the very regular type I triangulations) This implies that in general it does not have a stable local basis. There are several classes of triangulations where the situation ....

Boor, C. de and K. Hollig, Approximation order from bivariate C 1 -cubics: a counterexample, Proc. Amer. Math. Soc. 87 (1983), 649--655.

....: r k, Gamma also includes the last i Gamma r points on R i (v) in the triangles T [2] T [3] T [4] The numbers of these points are shown in the third column of the table. If r = 2m, Gamma also includes the last 2(i Gamma m) Gamma 1 points on R r i (v) in the triangles T [5] for i = m 1; k. These points are shown in the fourth column of the table. Finally, if r = 2m 1, Gamma also includes the last 2(i Gamma m Gamma 1) points on R r i (v) in the triangles T [5] for i = m 2; k. These points are shown in the fifth column of the table. ....

.... Gamma also includes the last 2(i Gamma m) Gamma 1 points on R r i (v) in the triangles T [5] for i = m 1; k. These points are shown in the fourth column of the table. Finally, if r = 2m 1, Gamma also includes the last 2(i Gamma m Gamma 1) points on R r i (v) in the triangles T [5] for i = m 2; k. These points are shown in the fifth column of the table. non existence of spline bases 7 We now show how to select a subset G of Gamma which is a determining set for V r d (4H ) Since the cardinality of G is an upper bound on dim V r d (4H ) our proof of ....

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C. de Boor and K. H ollig, Approximation order from bivariate C 1 - cubics : a counterexample, Proc. Amer. Math. Soc., 87 (1983), pp. 649--655.

....(h ) k # h G 1 h # h ,h h 0 , are bounded in L# (B) for some neighborhood B of the origin. 3. An application: bivariate C cubics on 3 direction mesh In this section only, let S denote the space of bivariate C cubics on the 3 direction mesh. This space was shown in [BH1] to provide approximation order 3 only, even though # 3 is contained locally in it. This result made clear that the approximation order of a FSI space might be harder to ascertain than originally thought. It is therefore worthwhile to show how Theorem 2.2 provides the exact approximation order for ....

....locally in it. This result made clear that the approximation order of a FSI space might be harder to ascertain than originally thought. It is therefore worthwhile to show how Theorem 2.2 provides the exact approximation order for this space. In the interest of brevity, we refer the reader to [BH1], BH2] as a source for any missing details and for prior literature concerning this particular S, which consists of all piecewise cubic functions in C (IR ) for the 3 direction mesh, i.e. with breaklines IR i 1 , i 2 , i 3 ZZ involving the three directions i 1 : 1, 0) i 2 : ....

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C. de Boor, K. H ollig (1983): Approximation order from bivariate C -cubics: a counterexample. Proc. Amer. Math. Soc., 87: 649--655.

....f j of degree j, all j. This example also illustrates the limits of the Strang Fix conditions (see Section 4) For it shows that (S h ) has positive approximation order even though none of the S h contains # 0 . In fact, # h S h = 0 . Example 2 This disturbing example comes from [BH83] The space S : # 1 3,# of C 1 cubics on the three direction mesh # contains all cubic polynomials. It even contains them locally in the sense that any cubic polynomial on one of the triangles of the partition # can be extended to an element of S with compact support. However, the ....

....h ) related in the previous section gave rise to the hope that the approximation order of (S(#) h ) would be the largest m for which # m # S(#) Even the determination of such an m would be nontrivial, but less involved than finding a best possible #. Unfortunately, any such hope was dashed in [BH83] where it is shown that the approximation order of the scale obtained from the space of C 1 cubics on the three direction mesh is only 3 even though its subspace S(#) with # consisting of the two box splines M 221 and M 122 contains # 4 . SF73] as well as [DM84] speak of controlled L p ....

C. de Boor, K. Hollig, Approximation order from bivariate C 1 -cubics: a counterexample, Proc. Amer. Math. Soc. 87 (1983), 649--655.

....0. It turns out that the upper bound to be proven here already holds when # is a very simple triangulation, viz. the three direction mesh, i.e. the mesh # : 3 [ i=1 IRe i ZZ 2 with e 1 : 1, 0) e 2 : 0, 1) e 3 : 1, 1) e 1 e 2 . A first result along these lines was given in [BH83 1 ] where it was shown that the approximation order of # 1 3,# (with # the three direction mesh) is only 3, which was surprising in view of the fact that all cubic polynomials are contained locally in this space. J83] showed the corresponding result for C 1 quartics on the three direction mesh ....

....p , 1 # p # #) is at best k when k 3# 2, # 0 and # is the three direction mesh. In this section, we outline the proof, leaving the verification of certain technical Lemmata to a subsequent section. The proof uses the same ideas with which the special cases # = 1 and 2 were handled in [BH83 1 ] J83] and [BH88] respectively, i.e. the construction of a local linear functional which vanishes on # # k,# but does not vanish on some homogeneous polynomial of degree k 1 and whose integer translates add up to the zero linear functional. But the construction of the specific linear ....

C. de Boor and K. Hollig, Approximation order from bivariate C 1 -cubics: A counterexample, Proc. Amer. Math. Soc. 87 (1983), 649--655.

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