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1 Journal of Computatonal Mathematcs, Vol., No.6, 4, 87{86. THE STRUCTURAL CHARACTERIZATION AND LOCALLY SUPPORTED BASES FOR BIVARIATE SUPER SPLINES ) Zh-qang u (Insttute of Mathematcal Scences, Dalan Unversty of Technology, Dalan 64, Chna) (Department of Computer Scence, Tsnghua Unversty, Bejng 84, Chna) Ren-hong Wang (Insttute of Mathematcal Scences, Dalan Unversty of Technology, Dalan 64, Chna) Abstract Super splnes are bvarate splnes dened on trangulatons, where the smoothness enforced at the vertces s larger than the smoothness enforced across the edges. In ths paper, the smoothness condtons and conformalty condtons for super splnes are presented. Three locally supported super splnes on type- trangulaton are presented. Moreover, the crtera to select local bases s also gven. By usng local supported super splne functon, a varaton-dmnshng operator s bult. The approxmaton propertes of the operator are also presented. Mathematcs subject classcaton: 4A, 4A, 6D7, 6D. Key words: Splne, Local Bases, Super Splne.. Introducton Let D be a polygonal doman n R and a trangulaton of D consstng of ntely straght lnes or lne segments dened by ; : y ; a x ; b = = N: Denote by v = V I all the vertces of. Denote by D = T, all the cells of. For ntegers r, we say that S r () = fs Sr () : s C (v ) = V I g s a super splne space of degree and smoothness r (cf.[,,]), where C (v) denotes the set of functons dened on D whch are tmes contnuously derentable at the pont v and S r () s an ordnary splne space dened as S r () = fs C r () : sj D P (x y)8g: Throughout the paper, P (x y) andp (x) denote the collecton of polynomals P (x y) :=f ; = j= c j x y j jc j Rg P (x) :=f = c x jc Rg respectvely. Moreover, f < P (x y) and P (x) are both equal to zero. If S r () 6= S () the super splne space Sr () s called a nontrval super splne space ofdegree and smoothness r. Super splnes have strongly appled bacground n nte elements, vertex splne and Hermte nterpolaton. In [], the relaton between super splne theory and nte element theory was ntroduced. In [,,], by usng super splne, bvarate macro element was bult. In [9], based on super splne spaces, Hermte nterpolaton was dscussed. In [,4,,8,], the Receved September 8, nal revsed June 9, 4. ) Project Supported by The Natonal Natural Scence Foundaton of Chna and Chna Postdoctoral Scence Foundaton.

2 88 Z.Q. U AND R.H. WANG dmensons of S r () were gven. In these papers, super splnes were dscussed by a B net method. In ths paper, the smooth cofactors method for studyng super splnes s presented. The method s more eectve for solvng some problems about super splnes. By usng Bezout's theorem from algebrac geometry, Wang dscovered the followng smooth condtons and the conformalty condtons for bvarate splnes (cf.[]). Theorem []. The functon s(x y) s a bvarate splne belongng to S () f and only f the followng condtons are satsed. () For any grd-segment ; dened by l (x y) = there exsts the so-called smoothng cofactor q (x y) such that p (x y) ; p (x y) =l + (x y)q (x y) () where the polynomals p and p are determned by the restrcton of s(x y) to the two cells D and D wth ; as common edge and q P ;(+) (x y): () For any nteror vertex v j of, the followng conformalty condtons are satsed (l (j) (x y)) + q (j) (x y) () where the summaton s taen over all the nteror edges ; (j) passng through v j and the sgns of the smoothng cofactors q (j) are rexed n such a way that when a pont crosses ; (j) from D to D t goes around v j n a counter-clocwse manner. Smooth condtons and the conformalty condtons are very eectve tools for studyng bvarate splnes (cf.[6]). The purpose of ths paper s to present the smooth condtons and the conformalty condtons for super splnes. Usng the smooth condtons and conformalty condtons, the locally supported bases of super splnes on type- trangulaton are also dscussed. The local supported bases of super splnes have a wde range of applcatons n approxmaton, nterpolaton, numercal analyss and nte element methods. We shall only dscuss some approxmaton propertes arsng from the varaton-dmnshng super splne seres.. The Smooth Condtons and the Conformalty Condtons for Super Splne To obtan the smooth condtons and the conformalty condtons for super splnes, we rstly ntroduce a lemma. One can nd a smlar result n [4]. Lemma. Denote by l(x y) the straght lne y ; ax ; b = : Let p(x y) P (x y) and (x y ) (x y ) be two dstnct ponts lyng on l. p(x j x@ n;j y j (x y ) p(x j x@ n;j y j (x y ) = j n f and only f there exst q(x y) P ;; (x y) and c m (x) P ;;m; (x) such that p(x y) =(y ; ax ; b) + q(x y)+ + m= (y ; ax ; b) +;m ( where c m (x) provded ; ; m ; <. Proof. There exst q(x y) P ; (x y) and c(x) P (x) such that Y = (x ; x )) m c m (x) () p(x y) =(y ; ax ; b)q(x y)+c(x): (4) When = then c(x )= c(x )=.e., there exsts c (x) suchthatc(x) =(x ; x )(x ; x )c (x): So, the theorem holds for =: Suppose that the lemma holds for = g ;,.e. there exst q (x y) andc () m (x) suchthat p(x y) =(y ; ax ; b) g q (x y)+ g m= (y ; ax ; b) g;m ( Y = (x ; x )) m c () m (x): ()

3 The Structural Characterzaton and Locally Supported Bases for Bvarate Super Splnes 89 To prove thelemma for = g: p(x g j (x y) p(x g j (x y) = g!q (x y ) we have q (x y ) = = : By q (x y ) = = and the result for = there exst q(x y) andc (x) such that Substtutng (6) nto (), we have q (x y) =(y ; ax ; b)q(x y)+(x ; x )(x ; x )c (x): (6) p(x y) = (y ; ax ; b) g+ q(x y)+(y ; ax ; b) g (x ; x )(x ; x )c (x) + g m= (y ; ax ; b) g;m ( Y = (x ; x )) m c () m (x): (7) Q Denote by S m (x y) the (y ; ax ; b) g;m ( = (x ; x )) m c () m (x) m g: g S m(x t x@ g;t y j (x y ) 6= t g = f and only f m t: When t S m(x t x@ g;t y j (x y ) 6= = f and only f m = : When t g p(x g p(x t x@ g;t y j (x y ) S (x t x@ g;t y j (x y ) : t x@ g;t y j (x y ) = we have c () (x )= =.e., there exsts c (x) such thatc () (x) = Q = (x ; x )c (x): Suppose c () (x ) = j h ; = h Z: Then when t = g S m(x y) t x@ g;t y j (x y ) 6= = f and only Q f m = h whch mples c () h (x ) = =.e. there exsts c h+ (x) such that c () h (x) = = (x ; x )c h+ Q (x): We now contnue ths process and obtan that there exsts c m+ (x) suchthatc () m (x) = = (x ; x )c m+ (x) m g: Substtutng q (x y) andc () m (x) nto (6), we have p(x y) =(y ; ax ; b) g+ q(x y)+ g+ m= (y ; ax ; b) g+;m ( Y = (x ; x )) m c m (x): (8) Hence, the lemma holds. Theorem. s(x y) S r () f and only f the followng condtons are satsed: () For each nteror edge of dened by ; : l(x y) =y ; a x ; b = there exst q (x y) and c m (x) m ; r such that p (x y) ; p (x y) =l + (x y)q (x y)+ ;r m= l ;m+ (x y)((x ; x )(x ; x )) m c m (x) (9) where (x y ) and (x y ) are two vertces lyng on ;, the polynomals p and p are determned by the restrcton of s(x y) to the two cells D and D wth ; as the common edge and q(x y) P ;; (x y) c m (x) P h (x) h = ; ; m ;. c m (x) provded h<. ()For any nteror vertex v j :(x j y j ) of,the followng conformalty condtons are satsed: ((l (j) ) + q (j) ;r (x y)+ (l (j) m= ) ;m+ ((x ; x )(x ; x j )) m c (j) (x)) () where the summaton s taen over all the nteror edges ; (j) passng through v j and (x y ) and (x j y j ) are two vertces lyng on ; (j) and the sgns of the q (j) and c (j) m are rexed nsuch a way that when a pont crosses ; (j) from D to D t goes around v j n a counter-clocwse manner. Proof. By usng Theorem, we have p (x y) ; p (x y) =l r+ (x y)h (x y) () where h (x y) P ;r; (x y): Accordng to the denton of a super splne, we n h (x j x@ n;j y j (x y ) h (x j x@ n;j y j (x y ) = where j n n ; r ; : Usng the Lemma m

4 8 Z.Q. U AND R.H. WANG, we have h (x y) =(y ; ax ; b) ;r q(x y)+ ;r m= (y ; ax ; b) ;r;m ( Y = (x ; x )) m c m (x): Substtutng h (x y) nto (), we can prove (). The proof of () s smlar to the proof of () n Theorem. Hence,we omt t. In general, (9) and () are called smooth condtons and conformalty condtons of super splne space S r () respectvely. Usng the smooth condtons, we obtan the followng theorems. Theorem. When r< S r () = S () fandonlyf ; : proof. We use the same notatons as n Theorem. Usng the smooth condton (9), we have c m (x) P ;;m; (x y) m ; r: Obvously, ; ; m ; ; ; : ; ; < f and only f ; : So, f and only f ; ; ; m ; < where m ; r: Hence, f and only f ; c m (x) where m ;r.e. S r () S (): Because of S () Sr () we have S () = Sr () f and on f ; : Theorem 4. For any trangulaton f + then S ; () = S +; () where S ; () = fs C (v j ) : sj D P (x y) 8 jg: Proof. We use the same notatons wth Theorem. Let s(x y) S ; (): Usng the smooth condtons p (x y) ; p (x y) =l + (x y)q (x y)+ + m= we have whenm>; ; c m (x) : Hence, So, l ;m+ (x y)((x ; x )(x ; x )) m c m (x) ;; p (x y) ; p (x y) =l + (x y)q (x y)+ l ;m+ (x y)((x ; x )(x ; x )) m c m (x): m= p (x y) ; p (x y) = l +; (x y)(l ;; (x y)q (x y) + By Theorem, s(x y) S ;+ () S ; ;; m= l ;;;m (x y)((x ; x )(x ; x )) m c m (x)): (): Hence S ; () S ;+ (): Because of S ;+ () we have S ;+ () = S ; (): Remar.. If the grd lne l s dened by x ; a y ; b = then (9) can be replaced by the followng smooth condton : p (x y) ; p (x y) =l + (x y)q (x y)+ ;r m= l ;m+ (x y)((y ; y )(y ; y )) m c m (y): () The proof s smlar to that of Lemma.. If the degrees of super-smooth at the vertex (x y ) and (x y ) are and respectvely then (9) can be replaced by the followng smooth condton: + p (x y) ; p (x y) =l + (x y)q (x y) () ;r m= l ;m+ (x y)(x ; x ) (m+;)+ (x ; x ) (m+;)+ c m (x)

5 The Structural Characterzaton and Locally Supported Bases for Bvarate Super Splnes 8 where = maxf g and () + = maxf g: The proof s also smlar as Lemma.. The Spaces of Super Splne on Type- Trangulaton We begn wth the necessary notatons. Let D mn =[ m+] [ n+] where m and n are postve ntegers. Partton D mn rst by drawng n the vertcal lnes x ; = and horzontal lnes y ; j = = m and j = n: Then by drawng n the dagonals wth postve slopes to the rectangles [ +] [j j +] we obtanatype- trangulaton () mn of D mn. By Theorem, when r< the necessary condton for the super splne space S r (() mn) beng nontrval s +: In practce, super splne spaces wth the lowest possble degree and the hghest possble smooth degree r are the most useful. Hence, the mportant spaces to study are S (() mn) S 4 (() mn) S (() mn) : The space S (() mn) strval from the mathematcal pont ofvew. In ths secton, we wll dscuss varous local support bases of the bvarate super splne space S 4 (() mn). By usng the smooth condtons and conformalty condtons, we obtan three locally supported super splne functons, wrte as B () = respectvely. A () A () A () A () A () A () A () A () : ( ; ; ; 6; ; 6) B () A () A () A () : ( ; ; ; ; ; ) : ( ; ; ; ; ; 6) : ( ; ; ; 6; ; ) B () Fg. A () A () A () : ( ; ; ; 6; ; ) : ( ; ; ; ; ; ) : ( ; ; ; ; ; 6) B () In Fg., the supports of B () are shown. The vertces A () A () A() A() = nsde the support of B () are labelled and the values of B () D x B () D y B () D x B() D xy B() Dy B(), respectvely, at these vertces are also gven. These values completely determne B () wth the excepton of a translaton. To determne B () = unquely, we place the vertex A () A () A() n Fg. at the orgn respectvely. /4 /4 / / / / /4 /4 /4 /4 /4 /4 /4 / /4 /4 / /4 / / / /4 / / /4 / / / / /4 /4 /4 /4 / / /4 / / /4 /4 / / /4 /4 / / /4 / / /4 /4 /4 /4 / /4 / / /4 / / / /4 / /4 /4 / /4 /4 /4 /4 /4 /4 /4 / / / / /4 /4 Fg.

6 8 Z.Q. U AND R.H. WANG In Fg., the B net coordnates of B () = are also presented Fg. In Fg., the graphs of the three locally supported super splnes are shown. By usng the conformalty condtons of bvarate super splnes, t can be shown that the supports of B () are mnmal. We now translate B () = to obtan bases of S 4 (() mn). That s, we consder B (p) j (x y) =B(p) (x ; y ; j) p= : To facltate our presentaton, we ntroduce the ndex sets p = f( j) :B (p) j does not vansh dentcally on D mng: S It s clear that the cardnalty of p= p s mn +8(m + n)+: From [4] we also now that the dmenson of S 4 (() mn) s Hence the collecton dms 4 (() mn) =mn +8(m + n)+9: B = [ fb (p) j p= :( j) p g must be lnearly dependent ond mn. We wllgve crtera to determne whch element canbe deleted from B to gve a local S bass of S 4 (() mn): Theorem. For any f p= fb(p) j :( j) p g,theelements of Bnf are lnear ndependent. N Proof. Let D =[ +] [j j + ] and For any ( j ) [ wrte F (x y) = p = f( j) :B (p) j does not vansh dentcally on D g: ( j) c j B () j (x y)+ ( j) d j B () j (x y)+ ( j) e j B () j (x y) where d j or e j s equal to zero. We have toshowthatff (x y) = for all (x y) D then all the other c j d j and e j are equal to zero. We assume F (x y) ond : Then usng the equatons F ( + j + v) = D x F ( + j + v) = D y F ( + j + v) = D x F ( + j + v) = D xy F ( + j + v) = D y F ( + j + v) = v f g

7 The Structural Characterzaton and Locally Supported Bases for Bvarate Super Splnes 8 and the values n Fg., we can arrve at the followng lnear systems: ; ; ; ; ;6 ;6 ;6 ; ; ;6 ;6 ;6 C A v f g: c + j +v d + j +v d ;+ j ;+v d + j ;+v e + j +v e ;+ j +v e ;+ j ;+v C A = C A Usng these lnear systems, we can show fd j or e j s equal to zero, all the other c j d j and e j are equal to zero. Suppose f = B () j ( j ) : Wrte () G(x y) = c j Bj (x y)+ d j B ( j) ( j) () j () (x y)+ e j Bj (x y) ( j) where, N d j = : Hence, the coecents of B () B () and B () on sub-rectangle [ + ] [j j +] are also : By Fg., t s easy to prove the coecent of local support super splne on adjacent sub-rectangle s also : We nowcontnue ths process and obtan that all the c j d j and e j S are equal to zero. Smlarty ff = B () j ( j ) the result also holds. Hence for any f g, the elements of B;f are lnear ndependent. p= fb(p) j By the above result and the S dmenson of S 4 (() mn), the followng theorem can be obtaned: Theorem 6. For any f p= fb(p) j : ( j) p g, the elements of Bnf form a locally supportedd bass of S 4 (() mn). Let H () j (x y) =B() j (x y) ; B() j (x y) H() j (x y) =B() j (x y) ; B() j (x y): Usng the values shown n Fg. or B net coordnates n Fg., we have Theorem 7. For all (x y) R H () j The H () j j H () j (x y) j H () j (x y) : () () (x y) andh (x y) have a partton of unty, but the functon values of H (x y) j j (x y) are not nonnegatve. To buld a "varaton dmnshng" operator, we ntroduce another local supported super splne functon, denote B. In Fg.4, the support of B s shown. For B () we need explct values of B: In Fg., the vertces nsde the support of B are labelled A A A A 4 and the values of B D x B D y B Dx B D xy B D yb respectvely, at these vertces are also gven as 6-tuples. We also assume that A s located at the orgn. It s clear that the locaton of A and the gven values n Fg.4 unquely determne B: In Fg., the B net coordnates of B are presented. In Fg.6, the graph of B s shown. Let B j (x y) = B(x ; y ; j): Usng the values shown n Fg.4 or B net coordnates n Fg., we have Theorem 8. () For any (x y) R j B j (x y) =: () Insde the support of B the functon values of B(x y) are strctlypostve. Let L be the "varaton-dmnshng" operator that map C(D mn )nto S 4 (() mn) dened by (Lf)(x y) = j f( + j+ )B j(x y):

8 84 Z.Q. U AND R.H. WANG A A A A 4 A : ( 4 ; ; ; ; ; ) A : ( 4 ; ; ; ; ; ) A : ( 4 ; ; ; ; ; ) A 4 : ( 4 ; ; ; ; ; ) Fg.4 /8 /8 /4 /4 /8 /8 /8 /4 /8 / /8 /4 /8 /4 /8 /8 /4 / /8 /8 /4 / /4 /4 / /4 /8 /8 / /4 /8 /8 /4 /8 /4 /8 / /8 /4 /8 /8 /8 /4 /4 /8 /8 Fg. Now, we dscuss some approxmaton propertes of the varaton dmnshng operator. We rst ntroduce a lemma. Lemma. L(f)=f for all f P (x y): We remar that the above theorem does not hold for f(x y) =x xy and y and that for f(x y) =twas already shown n Theorem 8. Snce a polynomal n P (x y) on a trangle wth vertces A B C vanshes dentcally f ts values at A B C and the values of ts two rstpartal dervatves, three second partal dervatves at A B and C are all equal to zero, the result follows by verfyng that L(f);f f P (x y) satses these condtons on each trangular cell of the partton () mn: Ths can be shown by usng the values gven n Fg Fg.6

9 The Structural Characterzaton and Locally Supported Bases for Bvarate Super Splnes 8 Suppose K s a closed set n R and f C(K): Let! K (f ) =supfjf(x y) ; f(u v)j :(x y) (u v) K j(x y) ; (u v)j <g and s the radus of a cell n () mn and s the radus of the support of B: Suppose D mn K and the centers of the support of B j le n the nteror of K: Let Dmn denote the maxm value on D mn : We have Theorem 9. If f C(K) then If f C (K) then If f C (K) then f ; V (f) Dmn! K (f ): (4) f ; V (f) Dmn max(! K (D x f =)! K (D y f =)): () f ; V (f) Dmn D f (6) where, the lnear operator D f(x y)( ) :R R! R s dened as D f(x y)((u u ) (v v )) = D x f(x y)u v + D xy f(x y)u v + D yx f(x y)u v + D y f(x y)u v : Proof. Snce the property of partton of unty, (4)obvously holds. For f C (K) suppose F be such a closure of a trangular cell that f ; L(f) Dmn = f ; L(f) F : Suppose (x y ) be equal to ( j ; )or(; j): By the mean-value theorem, we have f(x y) =p (x y)+(d x f(u v) ; D x f(x y ))(x ; x )+(D y f(u v) ; D y f(x y ))(y ; y ) (7) where (u v) =t(x y)+(; t)(x y ) t p (x y) =f(x y )+D x f(x y )(x ; x )+D y f(x y )(y ; y ): (8) By Lemma and L = we have f ; L(f) F f ; p F + L(f ; p ) F f ; p F : Hence, by (7), equaton () can be obtaned. For f C (K) by Taylor's formula f(x y) =p (x y)+ D f(u v)(x ; x y; y ) (9) where (u v) =t(x y)+(; t)(x y ) t [ ] and (x ; x y; y ) =((x ; x y; y ) (x ; x y; y )): By (9), (6) can be proved easly. Remar.. Usng the smooth condtons and conformalty condtons, smlar as the ordnary multvarate splne spaces, the dmensons of bvarate super splne spaces can be constructed. Moreover, some other problems, such as constructng a bvarate macro-element [], can also be nvestgated by the smooth condtons and the conformalty condtons.. For nonunform trangulated rectangles, snce the grd lnes determned by =x < < x m+ = m + and =y < <y n+ = n + are arbtrary, theycanbemoved approprately to t the gven data. In fact, adaptve schemes can be developed and the problems of approxmaton by super splne spaces of degree 4 and smoothness wth varable grd parttons can be nvestgated by usng the bvarate local supported super splne functons B j (x y): The study of these problems wll be delayed to a later date.

10 86 Z.Q. U AND R.H. WANG References [] P. Alfeld, L.L. Schumaer, Upper and lower bounds on the dmenson of super splne spaces, Constructve Approx, 9 (), 4-6. [] P. Alfeld, L.L. Schumaer, Smooth Macro-elements based on Powell-Sabn trangle splts, Advances n Comp. Math., 6 (), [] C. de Boor, K. Ho llg, S. Remenschneder, Boxs Splnes, Sprnger-Verlag, New Yor, (99). [4] C.K. Chu, T.. He, On the dmenson of bvarate super splne spaces, Math. Comp., :87 (989), 9-4. [] C.K. Chu and M.J. La, On bvarate super vertex splnes, Constr. Approx., 6 (99), [6] G. Nurnberger, Approxmaton by splne functons, Sprnger-Verleg, Berln, (989). [7] G. Nurnberger and F. Zelfelder, Developments n bvarate splne nterpolaton, J. Comp. Appl. Math., (), -. [8] Ibrahm, A. and L.L. Schumaer, Super splne spaces of smoothness r and degree d r +, Constr. Approx., 7 (99), 4-4. [9] La, M.-J. and L.L. Schumaer, Scattered data nterpolaton usng C super splnes of degree sx, SIAM J. Numer. Anal., 4 (997), 9{9. [] La, M.-J. and L.L. Schumaer, Macro-elements and stable local bases for splnes on Clough-Tocher trangulatons, Numer. Mathe., 88 (), -9. [] La, M.-J. and L.L. Schumaer, Macro-elements and stable local bases for splnes on Powell-Sabn trangulatons, Math. Comp., 7 (), {4. [] L.L. Schumaer, Dual bases for splne spaces on cells, CAGD, (988), [] L.L. Schumaer, On Super splnes and nte elements, J. Numer. Anal., 6:4 (989), [4] Ren-Hong Wang, Multvarate Wea Splne, The Natonal Conference on Numercal Analyss. GuangZhou, 979. [] Ren-Hong Wang, The structural characterzaton and nterpolaton for multvarate splnes, Acta Math. Snca, 8 (97), 9-6 (Englsh transl.bd. 8 (97), -9). [6] Ren-Hong Wang etc., Multvarate Splnes Functons and Ther Applcaton, Scence Press, Bejng, (994).