STABILITY OF OPTIMAL-ORDER APPROXIMATIONBY BIVARIATE SPLINES OVER ARBITRARY TRIANGULATIONSC. K. Chui1, D. Hong2, and R. Q. Jia3Abstract. Let � be a triangulation of some polygonal domain in R2 and Srk(�), the space ofall bivariate Cr piecewise polynomials of total degree � k on �. In this paper, we construct alocal basis of some subspace of the space Srk(�), where k � 3r+2, that can be used to providethe highest order of approximation, with the property that the approximation constant of thisorder is independent of the geometry of � with the exception of the smallest angle in thepartition. This result is obtained by means of a careful choice of locally supported basisfunctions which, however, require a very technical proof to justify their stability in optimal-order approximation. A new formulation of smoothness conditions for piecewise polynomialsin terms of their B-net representations is derived for this purpose.1. IntroductionThe objective of this paper is to describe the approximation properties of certain bi-variate spline spaces over arbitrary triangulations of a polygonal domain in R2 and toconstruct the approximants that achieve the highest order of approximation. Let � be a2-dimensional simplicial complex [9, p. 131]. We assume throughout that � is pure; thatis, each maximal simplex has dimension 2. Then � is called a triangulation of a polygonalregion in R2 . As usual, for any nonnegative integers k and r, Srk(�) denotes the space ofall Cr functions which are piecewise polynomials of total degree at most k separated by�. The approximation order of the space Srk(�) is de�ned to be the largest integer � forwhich dist(f; Srk(�)) � C j�j� (1)holds for all su�ciently smooth functions f , where the smallest constant C, called theapproximation constant (of optimal-order), depends only on f and the smallest angle in�. Also j�j := sup�2� diam � denotes the mesh size of �, and the distance is measured1991 Mathematics Subject Classi�cation. Primary 41A25, 41A63; Secondary 41A05, 41A15, 65D07.Key words and phrases. bivariate splines, triangulations, B-net representations, approximation order,local bases, stability.1;2 Research supported by NSF Grant No. DMS 92-06928 and ARO Contract DAAH 04-93-G-0047.3 Research supported in part by NSERC Canada under Grant OGP 121336. Typeset by AMS-TEX1

2 TRANS. AMS, 347 (1995), 3301-3318in the supremum norm k k. It is clear that the approximation order of Srk(�) cannot behigher than k + 1, regardless of r, and is trivially k + 1 in case r = 0. On the otherhand, it is also well known that for r � 1 the approximation order from Srk(�) not onlydepends on k and r, but also on the geometric structure of the partition �. Accordingto the well-known results in �nite element theory (cf. [11]), the full approximation orderof k + 1 is obtained provided that k � 4r + 1. Extension of this property of optimalapproximation to k � 3r+2 is more recent. An abstract proof based on the Hahn-Banachtheorem was given by de Boor and H�ollig [2]. However, as was already pointed out by deBoor [1] (see also Schumaker [10, p. 547]), the proof given in [2] does not fully support theclaim that the approximation constant in (1) depends only on the smallest angle in thetriangulation. Although constructive proofs were also given in [5] and [6], yet the behaviorof the approximation constants still depends on the measurement of \near-singularity" of�; i.e., the constant becomes large for near-singular vertices. Observe that when � isre�ned so that j�j ! 0 in (1), the standard re�nement algorithms are mainly concernedwith the smallest angle in the partitions, but not with the \near-singularity" of suchre�nement. Therefore, it is important to give an approximation scheme in order to showthat the spline space Srk(�), k � 3r + 2, admits optimal approximation order of k + 1 insuch a way that the approximation constant C in (1) does not depend on the geometry(such as near-singularity), with the exception of the smallest angle in �.The main purpose of this paper is to construct a stable local basis of the super splinesubspace Sr;�k (�) of Srk(�), where k � 3r + 2 and � = � 3r+12 � (see [4, p. 73] and [10]),and to show that the full order of approximation can be achieved via a quasi-interpolationscheme using this basis, and that the approximation constant C in (1) of this optimalorder depends only on the smallest angle in the triangulation �. Unlike the techniquesintroduced in [2] (see also [1]), which are based on determining the smoothness conditions interms of the domain points on two triangles that share a common edge to \disentangle therings" of smoothness conditions, our approach is to inductively determine the smoothnessconditions on the rings of the domain points of all vertices; that is, we determine thesmoothness conditions in terms of the points on all of the triangles attached to a commonvertex.This paper is organized as follows. In Section 2, in order to facilitate our procedureof constructing a stable super spline basis, we give a new formulation of the smoothness

STABILITY OF OPTIMAL-ORDER APPROXIMATON 3conditions in terms of the B-net representations. In Section 3, we demonstrate how tochoose a minimum determining set and provide an explicit scheme of approximation fromSr;�k (�) that attains the optimal approximation order. Finally, in Section 4, we will givean explicit scheme for constructing some stable local basis of Sr;�k (�).2. PreliminariesThroughout this paper, we will always assume, without loss of generality, that � isconnected. For a vertex v of �, we denote by St(v) the closed star of vertex v in �[9, p. 135]; i.e., the cell formed by all the triangles in � with v as a common vertex. IfSt(v)nfvg is connected for every vertex v of �, then � is called strongly connected. If � isstrongly connected, then each boundary vertex has exactly two boundary edges attachedto it. For simplicity, we will always assume that � is strongly connected, though ourdiscussion is also valid otherwise.Let � = [u; v; w] be a triangle with vertices u; v and w. For any x 2 R2 , denote by�(x) = (�u; �v; �w) the barycentric coordinates of x with respect to � ; that is,x = �uu+ �vv + �ww; �u + �v + �w = 1:For � = (�u; �v; �w) 2 Z3+, the Bernstein-B�ezier polynomial B�;� is de�ned byB�;� (x) = �j�j� ���uu ��vv ��ww ;where j�j = �u + �v + �w and �j�j� � = j�j!�u!�v!�w! . Moreover, we de�ne the points x�;�on � to be (�uu+ �vv + �ww)=j�j. It is well-known that any p 2 �k, the space of allpolynomials of total degree � k, can be written in a unique way asp = Xj�j=k b�;�B�;� :This gives rise to a mapping b: x�;� 7! b�;� , j�j = k, and this mapping is called the B-netrepresentation of p with respect to � .Let X be the collection fx�;� : � 2 �; j�j = kg. To any f 2 S0k(�) there corresponds aunique mapping bf from X to R such that on each � 2 �,f j� = Xj�j=k bf (x�;� )B�;� :

4 TRANS. AMS, 347 (1995), 3301-3318This bf is called the B-net representation of f with respect to �.In our investigation, it is essential to represent Cr-smoothness conditions of spline func-tions in terms of B-net representations. Suppose that a spline function f is de�ned overtwo triangles, say � = [u; v; w] and ~� = [u; v; ~w], with a common edge [u; v]. Let S; Su; Svand Sw denote the oriented areas of the triangles � , [ ~w; v; w], [u; ~w;w], and ~� , respectively.For instance, if u is the origin of R2 , v = (v1; v2); w = (w1; w2) and ~w = ( ~w1; ~w2), thenS = 12(v1w2 � v2w1) (2)and Sv = 12( ~w1w2 � ~w2w1); Sw = 12(v1 ~w2 � v2 ~w1): (3)The following lemma describes Cr-smoothness conditions on a spline function f in termsof its B-net representation (cf. [8]).Lemma 1. Suppose that a function f is de�ned on � [ ~� byf j� = Xj�j=k b(x�;� )B�;� ;f j~� = Xj�j=k b(x�;~� )B�;~� :Then f 2 Cr(�[~�) if and only if for all nonnegative integers ` � r and = ( u; v; 0) 2 Z3+with j j = k � `,b(x +`e3;~� ) = Xj�j=` b(x +�;�)����SuS ��u �SvS ��v �SwS ��w ; (4)where � = (�u; �v; �w) 2 Z3+ and e3 = (0; 0; 1).We remark that the quantities SuS , SvS , and SwS are all bounded by some constantwhich depends only on the smallest angle in the partition �. Hence, as we will see, theapproximation constant for optimal-order approximation has to depend on this smallestangle.For later reference, we need another form of the smoothness conditions which plays animportant role to prove the stability of the local basis. For � = (�u; �v; �w) 2 Z3+ withj�j = k, letC�;� := �vX�v=0 �wX�w=0(�1)�v��v+�w��w��v�v���w�w�b�x(k��v��w;�v;�w);��: (5)Then we have the following result.

STABILITY OF OPTIMAL-ORDER APPROXIMATON 5Lemma 2. A function s 2 S0k(�[~�) is in Cr if and only if the corresponding terms fC�;�gand fC�;~�g satisfy the conditions:C�;~� = �wX=0C(�u;�v+`;�w�`);���w�SvS�w�`wS�w ; (6)for 1 � �w � r where � = (�u; �v; �w) 2 Z3+ with j�j = k.Proof. Without loss of generality, we may assume that u is the origin of R2 . Let � =[u; v; w], ~� = [u; v; ~w], and consider an s 2 S0k(� [ ~�) which agrees with some p 2 �k on� and ~p 2 �k on ~� . For 0 � m � k, let pm and ~pm be the homogeneous components ofdegree m of p and ~p, respectively. Also, let sm be the corresponding piecewise polynomialfunction which agrees with pm on � and ~pm on ~� . Clearly, sm 2 S0k(� [ ~� ), m = 0; 1; : : : ; k.Moreover, since we may assume that the mesh line [u; v] is on the x-axis, it is not di�cultto see that s is in Cr if and only if each sm is in Cr, m = 0; 1; : : : ; k. Note thatp(x) = X�v+�w�k b(x�;� ) k!(k � �v � �w)!�v!�w! (1� �v � �w)k��v��w��vv ��ww ;where (1� �v � �w; �v; �w) is the barycentric coordinate of x with respect to the triangle� = [u; v; w]. By the multinomial theorem, we have(1� �v � �w)k��v��w = X�v+�w�k��v��w (k � �v � �w)!(k � �v � �w � �v � �w)!�v!�w! (�1)�v+�w��vv ��ww :Therefore, by setting �v + �v and �w + �w to be �v and �w, respectively, we havep(x) = X�v+�w�k X�v��v�w��w b(x�;� ) k!(k � �v � �w)!�v!�w!��v�v���w�w�(�1)�v��v+�w��w��vv ��ww :Recall that u = (0; 0). So, writing v = (v1; v2), w = (w1; w2), and x = (x1; x2), we see that�v = area[u; x; w]area[u; v; w] = x1w2 � x2w1v1w2 � v2w1and �w = area[u; v; x]area[u; v; w] = v1x2 � v2x1v1w2 � v2w1are homogeneous linear functions of x; hence, we havepm(x) = X�v+�w=m X�v��v�w��w b(x�;� ) k!(k �m)!�v!�w!��v�v���w�w�(�1)�v��v+�w��w��vv ��ww :

6 TRANS. AMS, 347 (1995), 3301-3318Taking (5) into account, we deduce thatpm(x) = X�v+�w=m k!(k �m)!�v!�w!C�;���vv ��ww : (7)Similarly, we have ~pm(x) = X�v+� ~w=m k!(k �m)!�v!� ~w!C�;~� ~��vv ~�� ~w~w ; (8)where (1� ~�v � ~� ~w; ~�v; ~� ~w) is the barycentric coordinate of x with respect to [u; v; ~w].Let us now express �v and �w in terms of ~�v and ~� ~w. Suppose ~w = ( ~w1; ~w2). Then wehave ~�v = x1 ~w2 � x2 ~w1v1 ~w2 � v2 ~w1 ; ~� ~w = v1x2 � v2x1v1 ~w2 � v2 ~w1 :These two equalities together with (2) and (3) yield�w = SwS ~� ~wand �v = ~�v + SvS ~� ~w:Finally, replacing �v by ~�v + SvS ~� ~w and �w by SwS ~� ~w in (7), we obtainpm(x) = X�v+�w=m k!(k �m)!�v!�w!C�;� �~�v + SvS ~� ~w��v �SwS ~� ~w��w= X�v+�w=m �vX=0 k!(k �m)!�v!�w! �v!`!(�v � `)!C�;� �SvS ��v�` �SwS ��w ~�v ~��v�`+�w~w :Let � ~w = �v + �w � `; �v = ` and q = �v � `. Then we have �w = � ~w � q, andpm(x) = X�v+� ~w=m k!(k �m)!�v!� ~w! � ~wXq=0�� ~wq �C(k�m;�v+q;� ~w�q);� SqvS� ~w�qwS� ~w ~��vv ~�� ~w~w :Comparing this with the expression for ~pm(x) in (8), we conclude that sm 2 Cr if andonly if C�;~� = � ~wXq=0�� ~wq �C(�u;�v+q;� ~w�q);� SqvS� ~w�qwS� ~wfor � = (�u; �v; � ~w) 2 Z3+ with 1 � � ~w � r and �v + � ~w = m. This completes the proof ofthe lemma. �

STABILITY OF OPTIMAL-ORDER APPROXIMATON 73. Main ResultsTo investigate the approximation properties of bivariate spline spaces, it is convenientto introduce the notion of super splines. Given a triangulation � and nonnegative integersk; r and � with k � � � r, a super spline is a piecewise polynomial of degree at most k on� which is Cr across each edge and C� around each vertex. Let Sr;�k (�) be the space of allsuch splines. Then Sr;�k (�) is a subspace of Srk(�). In this section, we describe an explicitquasi-interpolation scheme and prove that the super spline space Sr;�k (�), � = � 3r+12 �,k � 3r + 2 admits the optimal approximation order of k + 1 with the approximationconstant dependent only on the smallest angle in the partition �.Let us introduce a natural pairingh�; bi := Xx2X �(x)b(x); �; b 2 RX ;on RX . Now choose and �x an orientation for each interior edge of �. Let e = [u; v] bean oriented interior edge with two triangles [u; v; w] and [u; v; ~w] attached to it. If theorientation of e is from u to v, then we assume that the points u; v and w are ordered inthe counterclockwise direction. In this case, we say that the orientation of the triangle �agrees with the orientation of the edge e. Let � = (�u; �v; � ~w) 2 Z3+ with j�j = k and� ~w � 1. Bearing Lemma 1 in mind, we de�ne a function �e;� on X as follows:�e;�(x) := 8><>: 1; if x = x�;~� ;��� ~w� �S�uu S�vv S�wwS� ~w ; if x = x(�u;�v;0)+�;� for � 2 Z3+ with j�j = � ~w;0; elsewhere. (9)The points x�;� and x�;~� will be called the tips of �e;�.In the sequel, we always assume that k � 3r+2 and consider � := � 3r+12 �. For a vertexu and an oriented interior edge e attached to u, we consider the collections �nu;e de�ned by�nu;e = f�e;�: �u = k � ng; n = 1; 2; : : : ; �; (10)and �nu;e := f�e;�: �u = k � n; �v; � ~w � rg; n = �+ 1; : : : ; 2r: (11)Furthermore, let �nu := [e3u�nu;e; n = 1; 2; : : : ; 2r; (12)�u := 2r[n=1�nu;

8 TRANS. AMS, 347 (1995), 3301-3318

Fig. 1. The points \�" in Xnu;e; n = �+ 1; : : : ; 2r and Xnv;e; n = �+ 1, and\?" in Ye for nonsingular vertex u (k = 26; r = 8; � = 12).and for an oriented interior edge e, let�e := f�e;�: 1 � � ~w � r < �u; �v < k � �g: (13)Finally, let � := [u2V �u! [ [e2E �e! ; (14)where V and E denote the collections of vertices and oriented interior edges of �, respec-tively. By Lemma 1, we see that f 2 Sr;�k (�) if and only if its B-net representation bfsatis�es h�; bf i = 0; � 2 �:A subset Y of X is called a determining set for the super spline space Sr;�k (�), if thelinear mapping f 7! bf jY de�ned on Sr;�k (�) is one-to-one. Our goal is to �nd a minimumdetermining set for this super spline space.An interior vertex u is said to be singular, if there are exactly four edges attached to itand these edges lie on two straight lines. Otherwise, u is called nonsingular. In particular,

STABILITY OF OPTIMAL-ORDER APPROXIMATON 9a boundary vertex is regarded as nonsingular.For a vertex u and a triangle � = [u; v; w] attached to u, letXnu;� := fx�;� : �u = k � ng ; n = 0; 1; : : : ; �;Xnu := [�3uXnu;� ; n = 0; 1; : : : ; �: (15)We associate with each vertex u a triangle � attached to u and de�neY nu := Xnu;� ; n = 0; 1; : : : ; �: (16)Let e be any oriented interior edge with a given u and some v as two of its vertices. Alsolet � and ~� be the two triangles attached to e, such that the orientation of � agrees withthat of e; moreover, denote by w and ~w the third vertices of � and ~� , respectively. Forn = �+ 1; � � � ; 2r, if u is a nonsingular vertex, we de�ne Xnu;e to be the union of the twosets fx�;~� : �u = k � n; n� r � � ~w � rgand fx�;� : �u = k � n; 2n� 3r � 1 � �w � rg(see Fig. 1). If u is a singular vertex and � = [u; v; w] is a triangle attached to u, we de�neXnu;� := fx�;� : �u = k � n; n� r � �w � rg; n = �+ 1; : : : ; 2r;(see Fig. 2). If e is an oriented edge attached to a singular vertex u, we setXnu;e := Xnu;� [Xnu;~� ; n = �+ 1; : : : ; 2r;where � and ~� are the two triangles with common edge e; also, setY nu := Xnu;� ; n = �+ 1; : : : ; 2r;where � is an arbitrarily chosen triangle attached to u. For any vertex u, singular orotherwise, we de�ne Xnu := [e3uXnu;e; n = �+ 1; : : : ; 2r: (17)

10 TRANS. AMS, 347 (1995), 3301-3318

Fig. 2. The points in Xnu;e, n = � + 1; : : : ; 2r for a singular vertex u (k =26; r = 8; � = 12).Furthermore, we associate with each oriented interior edge e three setsX+e := fx�;� : 0 � �w � r < �u; �v < k � �g;X�e := fx�;~� : 1 � � ~w � r < �u; �v < k � �g;and Ye := X+e n 2r[n=�+1(Xnu;e [Xnv;e): (18)Finally, for each triangle � , we de�neX� := fx�;� : �u; �v; �w > rg :From the preceding construction we see that X is the disjoint union 2r[n=0 [u2V Xnu! [ [e2E(Ye [X�e )! [ [�2�X�! ;

STABILITY OF OPTIMAL-ORDER APPROXIMATON 11where � denotes the collection of all triangles in �.Suppose now that u is a nonsingular vertex. Then for each integer n between �+1 and2r, we choose a subset Znu of Xnu such that the cardinality #Znu of Znu is equal to #�nu,and j det(�(x))�2�nu;x2Znu j � j det(�(x))�2�nu;x2Z j (19)for any subset Z of Xnu with #Z = #�nu. It is known that the matrix (�(x))�2�nu;x2Xnuhas full (row) rank (see [2, Prop. 6] and [7]); hencedet(�(x))�2�nu;x2Znu 6= 0;and we will write Y nu := Xnu n Znu ; n = �+ 1; � � � ; 2r: (20)For each triangle � 2 �, we de�ne Y� := X� :Finally, we set Yv := 2r[n=0Y nv ;and Y := [v2V Yv! [ [e2E Ye! [ [�2�Y�! : (21)Then from the following theorem, we see that Y is a minimum determining set for Sr;�k (�).Theorem 1. For each b: Y 7! R, there exists a unique f 2 Sr;�k (�) such that the B-netrepresentation bf of f satis�es bf jY = b:Proof. Let �? := fb 2 RX : h�; bi = 0, � 2 �g, where � is given by (14). Then f 2 Sr;�k (�)if and only if bf 2 �?; hence, it su�ces to show that for a given mapping b : Y 7! R, thereexists a unique bb 2 �? such that bbjY = b.We shall �rst extend b to Su2V Xnu for n = 0; 1; � � � ; �. For each u 2 V , (16) tells us thatY nu = Xnu;� for n = 0; 1; � � � ; �, where � is a triangle attached to u. On the other hand,there exists a polynomial pu 2 �k such that its B-net representation bpu on � satis�es

12 TRANS. AMS, 347 (1995), 3301-3318

Fig. 3. The classi�cation of point set X on a triangle.bpu(x) = b(x) for all x 2 S�n=0 Y nu . We extend b to S�n=0Xnu by setting bb(x) := bpu(x) forevery x 2 S�n=0Xnu . Evidently, h�;bbi = 0 for all � 2 S�n=1 �nu.Next, we extend b to Su2V Xnu for n = �+ 1; � � � ; 2r. This is done inductively on n asfollows. Let n be given with �+ 1 � n � 2r. Suppose that bb(x) has been determined forx 2 Sn�1j=0 Su2V Xju in such a way that h�;bbi = 0 for all � 2 Sn�1j=1 Su2V �ju. We wish todetermine the values of bb on Su2V Xnu such that for every u 2 V ,Xx2X �(x)bb(x) = 0; � 2 �nu:We claim that the value bb(x) has been determined whenever x 2 X nXnu and �(x) 6= 0 forsome � 2 �nu. To establish this claim, we consider � = �e;�, where e is an edge attachedto u and � = (�u; �v; �w) with �u = k � n. Without loss of generality, we assume thate = [u; v] is an oriented interior edge and both of u and v are nonsingular, for otherwisethe proof is analogous. Let � = [u; v; w] and ~� = [u; v; ~w] be the two triangles withcommon edge e. If �(x) 6= 0 and x =2 Xnu , then x = x�;� for some � 2 Z3+ with j�j = k,�u � �u; �v � �v. Thus, we have �u = k � m for some m � n. Since bb(x) has beendetermined for x 2 Sn�1j=0 Su2V Xju, we may assume that x =2 Sn�1j=0 Su2V Xju. It follows

STABILITY OF OPTIMAL-ORDER APPROXIMATON 13that �+ 1 � m � 2r. By the de�nition of Xmu , we have �w < 2m� 3r� 1. Consequently,x�;� =2 Xpv for any p � m. This shows thatx�;� =2 2r[n=�+1�Xnu;e [Xnv;e�:By the de�nition of Ye, we have x = x�;� 2 Ye, and therefore bb(x) = b(x) is alreadydetermined. This veri�es our claim.Suppose now that u is a nonsingular vertex. Remember that Xnu is the disjoint unionof Y nu and Znn and b(x) = b(x) for x 2 Y nu . Moreover,det��(x)��2�nu;x2Znu 6= 0:Thus, the values of bb on Znu can be determined by solving the systemXx2Znu �(x)b(x) = � Xx2XnZnu �(x)b(x); � 2 �nuof linear equations.It remains to deal with the case where u is a singular vertex. Suppose that �1, �2, �3and �4 are the four triangles attached to u and arranged in a consecutive way. We mayassume that Y nu = Xnu;�1 , n = �+ 1; : : : ; 2r. Let ej , j = 1; 2; 3; 4, be the common edge of�j and �j+1 with �5 := �1. We have bb(x) = b(x) for x 2 Y nu = Xnu;�1 . Note that the matrix��(x)��2�nu;ej ;x2Xnu;�j+1 is a nonsingular diagonal matrix if the �'s and the x's are arrangedin an appropriate way. Thus, we can determine the values of bb on Xnu;�j+1 , j = 1; 2; 3, bysolving the system Xx2Xnu;�j+1 �(x)b(x) = � Xx2XnXnu;�j+1 �(x)b(x); � 2 �nu;ejof linear equations.It is known (see, e.g., [7]) that the function b so obtained also satis�esXx2X �(x)b(x) = 0 for all � 2 �nu;e4 :Finally, we extend b to all the remaining points X; i.e., to the points in Se2E X�e . Thiscan be easily done by applying the smoothness conditions (4) across each interior edge.

14 TRANS. AMS, 347 (1995), 3301-3318To summarize, we have constructed a function bb on X such that bb 2 �? and bbjY = b.From this construction, we see that such a bb is unique. Indeed, if b = 0 on Y and bb 2 �?satis�es bbjY = b = 0, then bb must vanish on all of X. This completes the proof of thetheorem. �Theorem 1 suggests the following Approximation Scheme.Step 1. Given f 2 C(�) and a triangle � 2 �, �nd p� 2 �k such thatp� (x�;� ) = f(x�;� ) for all j�j = k:Step 2. Let s 2 S0k(�) be the spline function given bysj� = p�for each triangle � 2 �. Then �nd the B-net representation b of s.Step 3. Find bb in accordance with Theorem 1, such that bbjY = bjY and bb 2 �?. Let g bethe spline in Sr;�k (�) whose B-net representation bg agrees with bb on X.We denote by T the linear operator f 7! g, f 2 C(�).In the sequel, we will denote by a the smallest angle in �, and by Consta;k we meana constant depending only on a and k, which may vary from situation to situation. Weuse the notation Dj , j = 1; 2, to denote the partial derivative operators with respect tothe jth coordinates. Also, the closed star of v, denoted by St(v) =: St1(v), is the union ofall the triangles attached to v, and the m-star of v, denoted by Stm(v), is the union of alltriangles that intersect with Stm�1(v), m > 1.Lemma 3. The linear operator T satis�es the following conditions:(i) Tp = p for every polynomial p 2 �k.(ii) If � is a triangle attached to a vertex u, thenk(Tf)j�k1 � Consta;kkf jN(u)k1; (22)where N(u) denotes the star Stbr=2c+2(u).Proof. The �rst part of this lemma is a straightforward consequence of the constructionof T . The second part will be proved in the next section. �We are now in a position to establish the main result of this paper.

STABILITY OF OPTIMAL-ORDER APPROXIMATON 15Theorem 2. If k � 3r+2, then there exists a linear operator T from Ck+1(�) to Sr;�k (�)such that kf � Tfk1 � Consta;kj�jk+1jf jk+1;1; (23)where jf jk+1;1 :=P 1+ 2=k+1 kD 11 D 22 fk1.Proof. Let T be the operator described by the above approximation scheme. Let f 2Ck+1(�) be given. In order to estimate the error f � Tf , we consider f(x) � (Tf)(x),where x is a point in a triangle � of �. Then there exists a polynomial p 2 �k such thatp(x) = f(x) andjp(y)� f(y)j � Constk jf jk+1;1j�jk+1 for all y 2 N(u): (24)By Lemma 3, we deduce from (24) thatjf(x)� Tf(x)j = jT (f � p)(x)j� kT (f � p)j�k1� Consta;kk(f � p)jN(u)k1 � Consta;kjf jk+1;1j�jk+1:This estimate is valid for every x 2 �. Hence, the proof of the theorem is complete. �4. Stable Bases with Local SupportIn this section, by using the determining set as described in Section 3, we shall constructa basis for Sr;�k (�) which is both stable and local.To begin with, we establish the following result about the norm estimation of the B-netordinates of any function in Sr;�k (�).Theorem 3. Every b 2 RX \ �? satis�eskbk1 � Consta;kkbjY k1;where Y is the determining set for Sr;�k (�) as de�ned by (21).Proof. Let M := kbjY k1. First, we show that, for n = 0; 1; : : : ; �,jb(x)j � Consta;k M; x 2 [u2V Xnu : (25)

16 TRANS. AMS, 347 (1995), 3301-3318Let u be any vertex. Among the triangles attached to u, let � = [u; v; w] be the one thatcontains Y nu , and let ~� = [u; v; ~w] be the other triangle attached to the edge [u; v]. Sinceb 2 �?, we have b(x�;~�) = Xj�j=� ~w �� ~w� �S�uu S�vv S�wwS� ~w b(x(�u;�v;0)+�;� );where � = (�u; �v; � ~w) 2 Z3+ with j�j = k and �u � k � �. From (16) we see that��b(x(�u;�v;0)+�;�)�� � M; �u � k � �:Moreover, jS�uu S�vv S�ww =S� ~w j � Consta:This shows that jb(x�;~�)j � Consta;k M; �u � k � �:Repeating this process, we obtainjb(x)j � Consta;k M; x 2 Xnu ; n = 0; 1; : : : ; �:Next, we shall prove (25) for n = �+1; : : : ; 2r. If u is a singular interior vertex, this canbe done by the same argument as before. On the other hand, if u is a nonsingular vertex,we then prove (25) by induction on n as follows. Let n be an integer in f� + 1; : : : ; 2rgand assume that (25) holds for 0; 1; : : : ; n� 1. We wish to prove that (25) also holds forn. For this purpose, we shall employ the smoothness conditions given in Lemma 2. Fora triangle � and � 2 Z3+ with j�j = k, we let C�;� be de�ned as in (5). Let e = [u; v] bean oriented edge attached to u, and let � = [u; v; w] and ~� = [u; v; ~w] be the two triangleswith common edge e. It is assumed that the orientation of � agrees with that of e. ByLemma 2, we have C(k�n;n�m;m);~� = mX=0C(k�n;n�`;`);��m�Sm�`v SwSm ;for 0 � m � n � k. In order to estimate C�;� , we introduce A�;� , as follows. For eachtriangle � = [u; v; w] attached to u and � = (�u; �v; �w) 2 Z3+ with j�j = k, letA�;� := C�;� for 0 � �v; �w � r:

STABILITY OF OPTIMAL-ORDER APPROXIMATON 17Moreover, if � = (k � n; n� `; `) for some (n; `) with � + 1 � n � 2r and 2n� 3r � 1 �` � n� r � 1, we will considerA(k�n;n�`;`);� := C(k�n;n�`;`);� + 2n�3r�2Xj=0 a`jC(k�n;n�j;j);� ;where the coe�cients a`j are to be determined. Fix an integer j 2 f0; 1; : : : ; 2n� 3r� 2g.If Sv = 0, we set a`j := 0 for all ` = 2n � 3r � 1; : : : ; n� r � 1; otherwise, let a`j be thesolutions of the systemn�r�1X`=2n�3r�1 a`j�m�� SvSw�j�` = �mj �; m = n� r; : : : ; r;of linear equations. Since the matrix ��m��n�r�m�r;2n�3r�1�`�n�r�1 is invertible, thereexists a unique solution for (a`j). The choice of (a`j) was made in such a way that theequalities A(k�n;n�m;m);~� = mX`=2n�3r�1A(k�n;n�`;`);��m�Sm�`v SwSm (26)are valid for all (m;n) with �+ 1 � n � 2r and n� r � m � r. Also, we have���a`j (Sv=Sw)j�`��� � Constk:Since ` � 2n� 3r � 1 > j and jSv=Swj � Consta;k, we obtainja`j j � Constk jSv=Swj`�j � Consta;k:Next, we de�ne, for convenience, C(x�;� ) := C�;�and A(x�;� ) := A�;� for x�;� 2 Xnu ; �+ 1 � n � 2r:Then it follows from (26) thatXx2Xnu �(x)A(x) = 0 8� 2 �nu: (27)

18 TRANS. AMS, 347 (1995), 3301-3318Recall from Section 3 that Znu is a subset of Xnu with #Znu = #�nu, such that the inequality(19) holds for any subset Z of Xnu with #Z = #�nu. Rewrite (27) asXx2Znu �(x)A(x) = � Xx2Xnu nZnu �(x)A(x);and apply Cramer's rule to the above system of linear equations to yieldjA(x)j � #(Xnu n Znu ) maxy2XnunZnu jA(y)j; x 2 Znu ; �+ 1 � n � 2r: (28)Since a is the smallest angle in �, the number of triangles attached to the same vertex isbounded from above by a constant depending only on a; hence we have#(Xnu n Znu ) � Consta;k:If y = x�;� for some � with �u � k � n and y =2 Xnu , then from the proof of Theorem 1,we see that y 2 Sn�1m=0 �Xmu;e [Xmv;e� [ Ye; thus, by the induction hypothesis, we havejb(y)j � Consta;k M:This together with the construction of C(y) and A(y) implies thatjA(y)j � Consta;k M; y 2 Y nu = Xnu nZnu :Therefore, by (28), we obtainjA(x)j � Consta;k M; x 2 Znu :Again, by the construction of C(x) and A(x), and by the induction hypothesis, we havejb(x)j � Consta;k M; x 2 Znu :This establishes (25) for any nonsingular vertex u.Finally, for x 2 X�e , it is easily seen from the smoothness conditions across the edge ethat jb(x)j � Consta;k M:This completes the proof of the theorem. �Now let us establish an equivalence relation between the norm of a spline function andthat of its B-net representation.

STABILITY OF OPTIMAL-ORDER APPROXIMATON 19Lemma 4. Let f 2 S0k(�) and bf its B-net representation. Thenkfk1 � kbfk1 � Constk kfk1: (29)Proof. According to the de�nition of bf , we havef(x) = Xj�j=k b�B�;� (x); x 2 �; (30)where b� := bf (x�;� ). Since B�;� are nonnegative and Pj�j=k B�;� (x) = 1 for all x 2 � , itfollows that kfk1 � kbfk1.In order to prove the second inequality in (29), we consider the standard 2-simplex� := f(x1; x2) : x1; x2 � 0;x1 + x2 � 1g and a one-to-one a�ne mapping Q from � onto� . Since barycentric coordinates are invariant under a�ne transforms, we have B�;�(y) =B�;� (Qy) for all y 2 �. Thus, it follows from (30) thatf(Qy) = Xj�j=k b�B�;�(y):Since B�;�, j�j = k, constitute a basis of �k, we havejb�j � Constk supy2� fjf(Qy)jg � Constk kfk1:This completes the proof of the lemma. �We are now in a position to describe a procedure for constructing a stable basis ofSr;�k (�). For a given point x in Y , it follows from Theorem 1 that there is a uniqueBx 2 Sr;�k (�) whose B-net representation b satis�esb(y) = � 1 y = x,0 y 2 Y nfxg. (31)Theorem 1 also tells us that fBx : x 2 Y g constitutes a basis of Sr;�k (�).Theorem 4. The basis fBx : x 2 Y g of Sr;�k (�) is stable in the sense that there are twopositive constants K1 and K2 depending only on k and a such thatK1 supx2Y jcxj � Xx2Y cxBx 1 � K2 supx2Y jcxj: (32)

20 TRANS. AMS, 347 (1995), 3301-3318This basis is also local in the sense that for any x 2 Y there exists a vertex u such thatsuppBx � Stbr=2c+1(u): (33)Proof. We �rst prove (32). Let f = Px2Y cxBx. Then the B-net representation bf of fsatis�es bf (x) = cx for all x 2 Y . By Lemma 4 and Theorem 3, we havekfk1 � kbfk1 � Consta;k supx2Y jcxj:On the other hand, Lemma 4 implies thatsupx2Y jcxj � kbfk1 � Constk kfk1:The desired inequality (32) now follows at once from the above estimates.To prove (33), let x 2 Y be arbitrarily chosen. If x 2 Y� for some triangle � , thensuppBx � � . Generally, for a given x 2 Y , there exists a vertex u such that the barycentriccoordinate (�u; �v; �w) of x, with respect to any triangle [u; v; w] with u as a vertex,satis�es �u � k2 . For two vertices u and v in �, we denote by d(u; v) the smallest number ofedges among all paths joining u and v. We claim that for any positive integerm � 2r+1��,if d(u; v) � m, then the B-net representation b of Bx vanishes on S�+m�1n=0 Xnv . This willbe proved by induction on m. If m = 1, then for any vertex v 6= u, b vanishes on S�n=0 Y nv ;hence, by the smoothness conditions around v, we see that b vanishes on S�n=0Xnv . Let1 < m � 2r + 1� � and assume that our claim has been justi�ed for any positive integer` < m. We must verify it for m. Suppose that d(u; v) � m and d(v; w) = 1. Thend(u;w) � m�1. By the induction hypothesis, we see that b vanishes on S�+m�2n=0 (Xnv [Xnw).If y 2 X n Z�+m�1v and �(y) 6= 0 for some � 2 ��+m�1v , then we see from the proof ofTheorem 1 that b(y) = 0. Hence, b also vanishes on Z�+m�1v . This shows that b vanisheson X�+m�1v , and therefore completes the induction procedure. If d(v; u) � 2r + 2 � �and d(v; w) = 1, then b vanishes on S2rn=0Xnv and S2rn=0Xnw. Moreover, if one of u andv is an interior vertex, then b vanishes on X�e , where e is the oriented edge joining vand w. This shows that b vanishes on the star St(v) of the vertex v. Therefore, since2r + 1� � = 2r + 1� b 3r+12 c = b r+22 c, we have suppBx � Stbr=2c+1(u). �It only remains to prove part (ii) of Lemma 3. For f 2 C(�), let s 2 S0k(�) be thespline functions given in the approximation scheme as described in Section 3, and let b be

STABILITY OF OPTIMAL-ORDER APPROXIMATON 21the B-net representation of s. By the construction of Tf , we haveTf(x) = Xy2Y b(y)By(x); x 2 �: (34)Let � be a triangle of � with vertex u and x 2 � . Then By(x) 6= 0 only if d(y; u) � br=2c+2,or equivalently, y 2 Stbr=2c+2(u) = N(u). Hence, the number of nonzero terms in (34) isbounded above by Consta;k. Moreover, kByk1 � Consta;k by Theorem 4. Thus, it followsfrom (34) that jTf(x)j � Consta;k maxy2N(u)\Y fjb(y)jg:By Lemma 4, we may now conclude thatmaxy2N(u)\Y fjb(y)jg � ConstkksjN(u)k1 � Constkkf jN(u)k1:Combining the above estimates, we obtain the desired result (22). �Final Remarks1. Recently, de Boor and Jia [3] proved that the order of approximation of Srk( ~�) fork � 3r+ 1 and the three-direction mesh ~� is at most k. Hence, k = 3r+ 2 is the smallestdegree for which Srk(�) achieves the optimal approximation order of k + 1.2. The main di�erence between our approach and the previous attempts in [5] and[6] is that the set Znu for �nu with the property that assertion (28) holds for all x 2Znu ; n = �+ 1; : : : ; 2r, is obtained by applying (19). Consequently, the dependence of theapproximation error on the near-singularity of the triangulation � is eliminated. The priceto pay is that the supports of the basis functions, as given in Theorem 4, are necessarilylarger than those of the vertex splines in [5].References1. C. de Boor, A local basis for certain smooth bivariate pp spaces, Multivariate Approximation IV (C.K. Chui, W. Schempp, and K. Zeller, eds.), Birkh�auser, Basel, 1989, pp. 25{30.2. C. de Boor and K. H�ollig, Approximation power of smooth bivariate pp functions, Math. Z. 197 (1988),343{363.3. C. de Boor and R. Q. Jia, A sharp upper bound on the approximation order of smooth bivariate ppfunctions, J. Approx. Theory 72 (1993), 24{33.4. C. K. Chui, Multivariate Splines, CBMS Series in Applied Mathematics, vol. 54, SIAM, Philadelphia,1988.5. C. K. Chui and M. J. Lai, On bivariate super vertex splines, Constr. Approx. 6 (1990), 399{419.

22 TRANS. AMS, 347 (1995), 3301-33186. D. Hong, On bivariate spline spaces over arbitrary triangulation, Master Thesis, Zhejiang University,1987.7. D. Hong, Spaces of bivariate spline functions over triangulation, Approx. Theory and Appl. 7 (1991),56{75.8. R. Q. Jia, B-net representation of multivariate splines, Ke Xue Tong Bao (A Monthly Journal ofScience) 11 (1987), 804{807.9. J. J. Rotman, An introduction to algebraic topology, Graduate Texts in Math., vol. 119, Springer-Verlag, New York, 1988.10. L. L. Schumaker, Recent progress on multivariate splines, Mathematics of Finite Elements and Appli-cations VI (J. Whiteman, ed.), Academic Press, London, 1991, pp. 535-562.11. A. �Zeni�sek, Interpolation polynomials on the triangle, Numer. Math. 15 (1970), 283{296.C.K. Chui, Center for Approximation Theory, Texas A&M University, College Station,TX 77843D. Hong, Department of Mathematics, The University of Texas at Austin, Austin, TX78712R.Q. Jia, Department of Mathematics, University of Alberta, Edmonton, Canada T6G2G1