Optimal recovery of twice differentiable functions based on symmetric splines

Transcription

1 Avalable onlne at Journal of Approxmaton Theory 164 (01) Full length artcle Optmal recovery of twce dfferentable functons based on symmetrc splnes Sergy Borodachov, Tatyana Sorokna Towson Unversty, 7800 York Road, Towson, MD 15, Unted States Receved 1 January 01; receved n revsed form 9 June 01; accepted July 01 Avalable onlne 31 July 01 Communcated by Günther Nürnberger Abstract Gven values and gradents of a functon at a fnte set of nodes n R d, we ntroduce symmetrc splne recovery methods based on local nformaton. We explctly construct bvarate symmetrc nterpolatng contnuous splnes on regular trangulatons that solve the optmal global recovery problem studed n Babenko et al. (010) [] for the class of functons whose second dervatves n any drecton are unformly bounded. We further prove that n contrast to the unvarate case, there are no smooth symmetrc splnes that solve ths recovery problem for d > 1. c 01 Elsever Inc. All rghts reserved. Keywords: Optmal recovery; Multvarate splnes; Trangulaton; Hermte nterpolaton 1. Introducton The problem of optmal recovery of functons based on ncomplete nformaton (see the second part of the ntroducton for rgorous defntons) s extensvely studed n approxmaton theory and related areas. In practcal applcatons, t s mportant to use recovery methods that are based on local nformaton. In partcular, every soluton of ths type s an optmal recovery fnte element. In the unvarate settng, the vast majorty of the recovery problems have been solved. However, n the multvarate settng, the stuaton s strkngly dfferent. In 1980 Babenko [1] proved that gven functon values at a fnte set of nodes n R d, the lnear nterpolatng splne solves an optmal recovery problem for Lpschtz and Hölder classes. In 1996 Correspondng author. E-mal addresses: sborodachov@towson.edu (S. Borodachov), tsorokna@towson.edu (T. Sorokna) /$ - see front matter c 01 Elsever Inc. All rghts reserved. do: /j.jat

2 1444 S. Borodachov, T. Sorokna / Journal of Approxmaton Theory 164 (01) Klzhekov [5] establshed the optmalty of the lnear nterpolatng polynomal for the class of twce dfferentable functons defned on a sngle smplex. The next step was made n 011. Gven values and gradents of a functon at a fnte set of nodes n R d, we were able to construct a quadratc contnuous splne on Delaunay trangulaton of the nodes (see [4]) that acheves exactly the optmal error bound found n [] for the class of twce dfferentable functons. In other words, ths splne solves exactly the correspondng optmal recovery problem. The splne has a symmetrc structure: the usage of nformaton at a node does not depend on the locaton of the node, and the splne s locally defned on each smplex. Its constructon fts the framework of tradtonal macro-elements (or fnte elements). The drawback s that t does not nterpolate the gradents at the nodes. The next step n our research was to fnd a smooth recovery macro-element that nterpolates both values and gradents at the nodes. It turns out that only half of ths problem can be solved n the case of symmetrc splnes. In Secton 6, we explctly construct two symmetrc splnes on a hexagonal mesh that do nterpolate both values and gradents but that are not globally smooth. The reason s that, n contrast to the unvarate case, smooth optmal symmetrc splnes do not exst, n general, n R d for d. The other goal of ths paper s to prove ths. Despte rather dscouragng news, we are hopeful that symmetrc splnes wll stll provde smooth solutons to optmal recovery problems when constructed on local refnements of Delaunay trangulatons. The settng consdered n ths paper can be stated as follows. Let F be a class of smooth functons f : Ω R d R, where Ω s a compact polytope n R d of full dmenson. Let the vector I X ( f ) := ( f (x 1 ),..., f (x n ), f (x 1 ),..., f (x n )) R l contan l = n(d + 1) unts of nformaton about f sampled at ponts of a set X = {x 1,..., x n } Ω. Then any operator Φ : R l C(Ω) can be consdered as a contnuous method σ [ f ] of recovery of f based on nformaton I X ( f ): σ [ f ] := Φ(I X ( f )). Consequently, the error of approxmaton of an ndvdual functon f by the method σ [ f ] s gven by R( f ; σ ) = f σ [ f ] Ω = f Φ(I X ( f )) Ω, where Ω s the unform norm on the doman Ω. To determne how well a method σ performs on the whole class F, we defne the quantty R(F; σ ) = R(F; X, Φ) = sup f Φ(I X ( f )) Ω () f F known as the global or the worst-case error of the algorthm σ over the class F. The frst problem s n fndng an optmal method of recovery σ wth gven nformaton I X and computng the global error of σ : R(F; X) := R(F; σ ) = nf R(F; X, Φ). (3) Φ We note that σ s not necessarly unque. In ths paper, we concentrate on fndng methods of optmal recovery defned locally. More precsely, we search for σ wth the followng property: f x s located n a smplex S R d, then for any f, the value σ [ f ](x) depends on I X ( f ) S only. The second problem s n fndng the best set X Ω of n samplng ponts and the global error of an optmal method Φ based on I X : (1)

3 S. Borodachov, T. Sorokna / Journal of Approxmaton Theory 164 (01) R n (F) := R(F; X ) = R(F; X, Φ ) = nf R(F; X). (4) X Ω #X=n More nformaton on the general theory of recovery problems can be found n [1,11,10]. The paper s organzed as follows. In Secton we gve rgorous defntons and menton known relevant results. We also ntroduce local symmetrc splne recovery methods n ths secton. We study propertes of symmetrc splnes n Secton 3. Secton 4 shows that under some addtonal non-restrctve assumptons there are no smooth optmal symmetrc splne methods. Secton 5 provdes a general error estmate for recovery methods of our nterest. Secton 6 contans explct constructons of optmal bvarate symmetrc nterpolatng quartc and quntc splnes over regular trangulatons. In Secton 7 we prove some techncal lemmas used n Secton 6.. Prelmnares We begn ths secton by ntroducng the classes of functons to be recovered. Gven a compact Jordan measurable regon Ω R d, d N, and a measurable functon f : Ω R, denote ts norm by f Ω := ess sup f (x). x Ω Let C 0 (Ω) := C(Ω) and let C 1 (Ω) be the subspace of contnuously dfferentable functons n C(Ω). Denote by W Ω the class of functons f C 1 (Ω) such that for every unt vector r R d, the drectonal dervatve f exsts nsde Ω at least n the generalzed sense and r f r 1. Ω We study problems (3) and (4) when Ω s a d-dmensonal polytope and F = W Ω. We also consder the perodc case of the optmal recovery problems (3) and (4). Let L be a full-rank lattce n R d, that s a lattce generated by d lnearly ndependent vectors {u 1,..., u d }. We denote the fundamental parallelepped of L by Π L := {α 1 u α d u d : α 1,..., α d [0, 1)}. We say that a functon f : R d R s L-perodc, f for every pont x R d and for every vector u L, there holds f (x + u) = f (x). Let C L be the space of contnuous L-perodc functons f : R d R and CL 1 be the subspace of contnuously dfferentable functons n C L. Denote by W L the class of functons f CL 1 such that for every unt vector r Rd, the drectonal dervatve f exsts at least n the generalzed sense n R d and r f r 1. ΠL We study problems (3) and (4) when Ω s the closure of Π L, F = W L, and recovery algorthms have form (1) wth Φ : R n(d+1) C L. When d = 1 and Ω = [a, b], problems (3) and (4) were ndependently solved n [3] and n [9] on the class W [a,b]. The case d > 1 of problem (3) under certan assumptons on the set of nodes X was solved n [] n the perodc case (for the class W L ) and n the non-perodc case (for the class W Ω defned on a convex body Ω). Problem (4) was solved n [] n the perodc bvarate

4 1446 S. Borodachov, T. Sorokna / Journal of Approxmaton Theory 164 (01) case when the perod s commensurable wth the regular hexagonal lattce. A non-constructve optmal method was presented n each of these cases. In [4] we constructed an optmal contnuous quadratc splne whch nterpolates only functon values at the nodes n the perodc case and n the non-perodc case when Ω s a convex polytope and X satsfes some addtonal assumptons. A more detaled revew s gven n [4]. In ths paper we construct two bvarate contnuous perodc splne methods that fully nterpolate the data and are optmal n the sense of problem (4) when the perod s commensurable wth the regular hexagonal lattce. Both methods are obtaned from a certan symmetrc form and we show that no smooth splne method of that form can be optmal (under certan addtonal assumptons). Let V = V T := {v 0, v 1,..., v d } be the set of d + 1 affnely ndependent ponts n R d, and let T = T (v 0,..., v d ) be the non-degenerate d-smplex wth vertces v V. For a pont v T, let b 0 = b 0 (v),..., b d = b d (v) be ts barycentrc coordnates wth respect to the smplex T,.e. non-negatve numbers such that b b d = 1 and v = d =0 b v. Let a b be the dot-product of vectors a, b R d. For brevty, we wrte nstead of v V T. We seek a splne s( f ) n d varables whose polynomal representaton over T can be wrtten as s( f, v) = s(b 0,..., b d ) = (α f + β f ), where f := f (v ), (5) f = f (v) = D v v f (v ) := f (v ) (v v ), α = α T (v), β = β T (v) (α T and β T are polynomals). We note that form (5) s equvalent to s( f, v) = α f + β b j f j, where (6) j f j := D v v j f (v ) = f (v ) (v j v ). We emphasze that symmetrc splnes are locally defned,.e., for any functon f and any pont v T the value s( f, v) depends only on the nformaton on f at V T. Thus, symmetrc splnes are fnte or macro-elements. A collecton = {T } of non-degenerate smplces n R d s called a trangulaton f (1) the nterors of the smplces are parwse dsjont; () each facet of a smplex s ether on the boundary of the unon of, or else s a common facet of exactly two smplces from ; (3) the unon of s closed and has path connected nteror. Followng [8], we ntroduce the followng defnton. Defnton.1. We say that a gven trangulaton s a trangulaton of X n Ω f X s the set of vertces of all smplces n and Ω s the unon of all smplces from. A smplex T s sad to be nteror, f every facet of T s shared wth some other smplex from. We wll denote by Int the collecton of all nteror smplces n. Denote by P d m the set of all polynomals of total degree m n d varables and for a gven trangulaton wth the unon Ω, let Sm r () := {s Cr (Ω) : s T Pm d, T }, r = 0, 1, be the spaces of contnuous and smooth splnes of degree m over ; see [7] for propertes of those spaces.

5 S. Borodachov, T. Sorokna / Journal of Approxmaton Theory 164 (01) Symmetrc splnes In ths secton we establsh several general results on nterpolatng propertes and smoothness of symmetrc splnes (6) as well as ther capabltes to reproduce low order polynomals. Our frst theorem provdes a necessary condton for s( f, v) to be optmal n the sense of (3). Throughout the rest of the paper we shall omt the frst or the second argument of s f there s no ambguty. We recall that W T s the class of functons defned on a smplex T n R d, whose second dervatves n any drecton are unformly bounded; see Secton. Let θ := 1 β. Theorem 3.1. If s of type (6) has a bounded error on W T, then α = β + b θ for all v V T. Proof. Snce W T contans every lnear polynomal, f s(p) p T 0 for some lnear polynomal p(v) then the error s(cp) cp T can be made arbtrarly large by varyng the constant c. Hence, s must reproduce all lnear polynomals. Consder a lnear polynomal f (v) = b k (v). Then f k = 1 and for all k, f = 0, f k = 1, f k = 1, whle f j = 0 for all k and j k. Consequently, the polynomal b k on T must concde wth the splne s(b k ) gven by s(b k ) = α k β k b j + b k β = α k β k (1 b k ) + b k and the result follows. j k k = α k β k + b k β = α k β k + b k (1 θ), β k Snce we are nterested n optmal methods of type (5) only we can restrct our attenton to the followng form of s( f ) on T : s( f ) = θ = θ b f + b f + β ( f + f ) β f + b j f j. (7) j Theorem 3.. Let T be a smplex n R d wth vertces {v 0,..., v d } and T be ts neghbor wth vertces {v 0,..., v d 1, ṽ d } sharng the facet F defned by b d = b d = 0. Let s and s be symmetrc splnes of type (7) over T and T, respectvely. Then s s C 0 (T T ) ff β d = b d γ d, β d = b d γ d and β F = β F for all d. Proof. Snce b = b on F for all d, we have the followng representatons s F = (θb + β ) f + β b j f j + β d f d + b j f d j, d j, j d j d s F = ( θb + β ) f + β b j f j + β d f d + b j f d j. d j, j d j d We see that s = s on F f and only f β d and β d vansh on F and β = β on F for all d.

6 1448 S. Borodachov, T. Sorokna / Journal of Approxmaton Theory 164 (01) Snce we are nterested n contnuous optmal methods of type (5) only, we update s( f ) on every nteror smplex T to the followng: s( f ) = = b [(θ + γ ) f + γ f ] where γ are some polynomals. b (θ + γ ) f + γ b j f j, (8) j Theorem 3.3. Let s be of type (8) on a gven smplex T. Then s nterpolates the values of f at the vertces of T. Proof. Snce at a gven vertex v, b = 1 and b j = 0 for all j, we have θ(v ) = 1 β = 1 γ. Thus, s(v ) = f. Theorem 3.4. Let s be of type (8) on a gven smplex T. Then s nterpolates the gradent of f at the vertces of T f and only f γ (v j ) = δ j for all, j, where δ j s the Kronecker symbol. Proof. Wthout loss of generalty we consder the dervatve D j s(v ) = s(v ) (v j v ) = D j b l (θ + γ l ) f l + γ l b k f lk l k l = D j b (θ + γ ) f + γ b k f k + b j k (θ + γ j ) f j + γ j b k f jk k j v = ([D j (θ + γ )] v (θ + γ ) v ) f + (θ + γ j ) v f j + γ (v ) f j + γ j (v ) f j = (1 γ (v ) + γ j (v ))( f j f ) + γ (v ) f j + γ j (v ) f j. Then D j s(v ) wll equal f j f and only f γ (v ) = 1 and γ j (v ) = 0. Theorem 3.5. Let T, T, s and s be as n Theorem 3., and F be defned by b d = b d = 0. If s s C 1 (T T ) then β d = b d λ d, β d = b d λ d and θ F = θ F = 0. Proof. By Theorem 3., we have β d = b d γ d and β d = b d γ d. Consder D d s and D d s, d, restrcted to F. We note that D d b = D d b = 1, D d b d = D d b d = 1, D d b j = D d b j = 0, for all d, j d,. In D d s F and D d s F the coeffcents n front of f d j and f d j, respectvely, for all j d, must vansh. Thus 0 = D d (γ d b d b j ) F = b j γ d F = D d ( γ d b d b j ) F = b j γ d F v

7 S. Borodachov, T. Sorokna / Journal of Approxmaton Theory 164 (01) mply that γ d F = γ d F = 0,.e. γ d = b d λ d and γ d = b d λ d. Addtonally, n D d s F and D d s F the coeffcents n front of f d and f d, respectvely, must vansh: 0 = D d [b d (θ + γ d )] F = θ F + γ d F = θ F, 0 = D d [ b d ( θ + γ d )] F = θ F + γ d F = θ F, d, and the proof s complete. If we consder only smooth optmal splnes of type (5) then on every nteror smplex T they can be wrtten as follows: s( f ) = = b [(φ Π j b j + λ ) f + λ f ] b φ f Π j b j + λ f + λ b j f j, where φπ b = 1 where λ and φ are some polynomals. j b λ, (9) Remark 3.1. In general, nether the converse of Theorem 3.5 s true, nor (9) mples Hermte nterpolaton at the vertces. We are not nterested n obtanng necessary and suffcent condtons here snce for d, under non-restrctve assumptons the necessary condtons of Theorem 3.5 turn out to be ncompatble wth optmal splnes of type (5) (or of type (7) n the perodc case); see Secton 4. We conclude the lst of propertes wth a result resolvng the ssue of reproducton of quadratc polynomals by symmetrc splnes. The followng theorem provdes the frst example of multvarate symmetrc splnes as well. Theorem 3.6. The only method of type (7) wth d that reproduces quadratc polynomals s gven by β = 1 b. Moreover, t s contnuous and optmal. Proof. Let s( f ) of type (7) reproduce all quadratc polynomals that can be wrtten as a product b b j, j. Then f = 0 for all, and f j = f j = 1, whle f kl = 0 for all (kl) {( j), ( j)}. Thus we have b b j = β b j + β j b, for all j. Consder the trple of those equatons nvolvng arbtrary b, b j, b k : β b j + β j b = b b j, β b k + β k b = b b k, β j b k + β k b j = b j b k. We multply the frst equaton by b k, the second by negatve b j, the thrd by b, and then add them to obtan β j = 1 b j. Ths proves that the choce β = 1 b for all, satsfes a necessary condton for a method of type (7) to reproduce quadratc polynomals. In [4], we proved that ths condton s suffcent, and moreover the method s optmal. Snce ths choce satsfes Theorem 3. wth γ = 1, the splne s( f ) s contnuous. We conclude ths secton wth examples of optmal unvarate symmetrc splnes.

8 1450 S. Borodachov, T. Sorokna / Journal of Approxmaton Theory 164 (01) Theorem 3.7. Let X be a fnte subset of [a, b] contanng the endponts. Then the contnuous quadratc unvarate splne of type (8) wth γ = 1 and the unvarate smooth Hermte nterpolatng cubc splne of type (9) wth λ = 1 are optmal for the recovery problem (3) on the class W [a,b] wth the gven nformaton set X. Proof. The optmalty of the quadratc splne follows from the results of [4]. If λ = 1, = 0, 1, then φ = n (9), and s( f ) = b 0 [(b 1 + 1) f 0 + b 1 f 01 ] + b 1 [(b 0 + 1) f 1 + b 0 f 10 ] = (b0 (b 0 + b 1 ) + b0 b 1) f 0 + b0 b 1 f 01 + (b1 (b 0 + b 1 ) + b1 b 0) f 1 + b1 b 0 f 10 = f 0 b f f 01 b0 b 1 + f 1 b f f 10 b1 b 0, whch s the nterpolatng smooth cubc splne of [6] known to be optmal. 4. Non-exstence of symmetrc smooth optmal splnes For the proofs of the man results of ths secton, we need to state two techncal lemmas. The frst lemma s well known and we omt ts proof. Lemma 4.1. Let T be a non-degenerate smplex n R d, d, whch contans ts crcumcenter, and let R be the crcumradus of T. Then there exsts a facet of T whose crcumradus r R 1 1/d. To state the second lemma we need some addtonal notaton. Gven a non-degenerate smplex T n R d, let q T be the crcumcenter of T and ρ(t ) be the Chebyshev radus of T : ρ(t ) = mn max x v. (10) x R d =0,d The pont x, where the mnmum on the rght-hand sde of (10) s attaned, s the Chebyshev center of T. Snce T s convex, x must be n T. If the crcumcenter q T belongs to T we denote by c(t ) the crcumcenter of a facet of T wth the largest crcumradus, and by r(t ) the largest crcumradus of a facet of T. Otherwse, we let c(t ) be the Chebyshev center of T and r(t ) := ρ(t ). We shall use a quadratc unvarate splne wth two fxed nodes r and r: r t /, t [0, r], ψ r (t) := (t r) /, t (r, r], 0, t (r, ), to defne a radal functon centered at c R d : f c,r (x) = ψ r ( x c ), x R d. Denote also by B[c, r] the closed ball of radus r n R d centered at c. Lemma 4.. Let T be a non-degenerate smplex n R d, d, and f C 1 (T ) be any functon such that f T B[c,r] = f c,r, where c = c(t ) and r = r(t ). Then for every polynomal s( f ) of form (9) defned on T, there holds f s( f ) T r (T ) > ρ (T ). (11) 4

9 S. Borodachov, T. Sorokna / Journal of Approxmaton Theory 164 (01) Proof. Denote by F a facet of T wth the crcumradus r(t ) that contans c(t ) f q T T. If q T T let F be the face of T whose vertces are all the vertces of T that are at a dstance ρ(t ) from c(t ). Then n both cases c(t ) wll be n F and wll be the crcumcenter of F. Denote I = { : b F 0}. Clearly, b = 0 everywhere on F. Moreover, at every pont of F, b λ = b λ = 1, and b j = 0, whenever I. j I Then snce c F, we have f (c) s( f )(c) = f c,r (c) b (c) φ(c) f b j (c) + λ (c)( f + f (c)) j = f c,r (c) b (c)λ (c)( f + f (c)). I For I, we have c v = r, f = f c,r (v ) = ψ r ( c v ) = r /, and f (c) = f c,r (v ) (c v ) = ψ r ( c v ) (v c) (c v ) = ψ r r (r)r = r. Then f s( f ) T f (c) s( f )(c) = r 3r b (c)λ (c) = r. In vew of Lemma 4.1, ether r ρ(t ) 1 1/d > ρ(t )/ or r = ρ(t ). In both cases, estmate (11) follows. Let σ (, ) be the splne on a trangulaton descrbed n Theorem 3.6. It was shown n [4, Theorem 4.1] that for every smplex T, sup f σ ( f, ) T ρ (T )/4. (1) f W T I Theorem 4.1. Let be a trangulaton of a fnte set X n P R d, d, (see Defnton.1) that has an nteror smplex T wth r(t ) > 1 max ρ(t ). T Then for any degree m, all symmetrc splne methods s Sm 1 () of the form (5), are not optmal n the sense of (3) for the recovery of the class W P wth the gven set of nodes X. Proof. Every symmetrc splne method s Sm 1 () of the form (5) that has a bounded error on W P reproduces lnear polynomals. The proof of Theorem 3.1 mples that s must have form (7). In vew of Theorem 3., relaton (8), and Theorem 3.5, any splne method s Sm 1 () of the form (7) can be wrtten as (9) on T. Let c = c(t ) and r = r(t ). Clearly, f c,r W P. From Lemma 4. and (1), we obtan R(W P ; X) R(W P ; σ (, )) 1 4 max T ρ (T ) < r (T ) f c,r s( f c,r ) T R(W P; s).

10 145 S. Borodachov, T. Sorokna / Journal of Approxmaton Theory 164 (01) Gven a full-rank lattce L n R d and a fnte subset X Π L, defne X + L = {x + m : x X, m L}. A trangulaton of X + L n R d s called L-perodc, f for every smplex T and every vector m L, the smplex T + m s also n. Denote J(L) = mn m. m L\{0} Theorem 4.. Let L be a full-rank lattce n R d, d, X Π L be a fnte subset and be an L-perodc trangulaton of X + L n R d such that max r(t ) J(L)/4. T Then for any degree m, all symmetrc L-perodc splne methods s Sm 1 () of the form (7), are not optmal n the sense of (3) for the recovery of the class W L wth the gven set of nodes X. Proof. Denote by T a smplex from wth the largest Chebyshev radus. In vew of Theorem 3., relaton (8), and Theorem 3.5, any L-perodc splne method s Sm 1 () of form (7) can be wrtten as (9) on T. Let c = c(t ), r = r(t ), and f (x) = f c,r (x + m). m L (13) In vew of condton (13), every x R d has some neghborhood, where ths sum contans at most one non-zero functon and, hence, f W L. Takng nto account (1) and Lemma 4., we obtan R( W L ; X) R( W L ; σ (, )) 1 4 max T ρ (T ) = ρ (T ) < r (T ) 4 f s( f ) T f s( f ) ΠL R( W L ; s). Theorem 4. s proved. Remark 4.1. When d = 1 (n contrast to the multvarate case) there exsts a smooth optmal recovery method of form (5) on the class W [a,b] and of form (7) on the class W Z ; see Theorem A general error estmate Theorem 5.1. Let T be a non-degenerate d-smplex n R d wth vertces v 0,..., v d, and s( f ) be of type (8). Then for every functon f W T, f (v) s( f, v) 1 where we set θ θ +γ γ θ+γ = γ f θ + γ = 0. b v v θ θ + γ γ θ + γ, v T \ V T, (14) Proof. Consder an arbtrary pont v n T, whch s dstnct from any vertex of T. Let τ = v v and g (t) := f v + t (v v ), t [0, τ ], = 0,..., d. τ

11 S. Borodachov, T. Sorokna / Journal of Approxmaton Theory 164 (01) Snce f W T, we have g W [0, τ ] := {g C 1 [0, τ ] : g AC, g [0,τ ] 1}, = 0,..., d, (see e.g. [, Lemma 5]). Let I τ (g ) := τ 0 (τ u)g (u)du, = 0,..., d. Some deas used n ths proof appear n [3]. By Taylor s formula, f (v) can be wrtten as follows f (v) = g (τ ) = g (0) + g (0)τ + I τ (g ) = f + f + I τ (g ) = f (v ) + f (v ) (v v ) + I τ (g ), = 0,..., d. (15) Note that every splne of form (8) satsfes α = 1. Then usng (15) and representaton (5) we have f (v) s( f, v) = α ( f + f + I τ (g )) s( f, v) = [(α β ) f + α I τ (g )]. Snce α = β + b θ, b (v v ) = v b v = 0, and D v v f (v) = f (v) (v v ) = g (τ )τ, = 0,..., d, we obtan f (v) s( f, v) = θ [b D v v f (v ) b D v v f (v)] + (β + θb )I τ (g ) = θ b τ [g (0) g (τ )] + (β + θb )I τ (g ) = θ = = = τ 0 b τ b τ f (v) s( f, v) = 1 b τ b τ τ 0 g (u)du + (β + θb ) [β τ (β + θb )u]g (u)du 0 1 [γ τ (γ + θ)u]g (u)du [γ (γ + θ)u]g (τ u)du. γ (γ + θ)u du where we set θ θ +γ γ θ+γ = γ f θ + γ = 0. τ 0 Then (τ u)g (u)du b τ θ θ + γ γ, v T \ V T, (16) θ + γ 6. Optmal bvarate symmetrc splnes over hexagonal mesh Throughout the rest of the paper, let be a regular hexagonal mesh n R wth sde length l, and L be any sublattce of the lattce consstng of the vertces of that satsfes J(L) l.

12 1454 S. Borodachov, T. Sorokna / Journal of Approxmaton Theory 164 (01) We recall that W L s the class of perodc functons f CL 1 whose second dervatves n any drecton are unformly bounded; see Secton. The followng result was proved n [4, Theorem 6.4], and equalty (17) was frst obtaned n []. Theorem 6.1. Let γ = 1/, = 0, 1,. Then the contnuous quadratc bvarate splne σ (, ) of type (8) on s optmal for the recovery problem (4) on the class W L wth n = Π L /( 3l ). Moreover, R n ( W L ) = l 1. (17) Remark 6.1. Ths theorem makes less restrctve assumptons on L than the ones n [4,]. The condton J(L) l ensures that the assumptons of Theorem 4 n [] hold. If Xn denotes the set of vertces of the regular hexagonal mesh that le n Π L then relaton () from Theorem 4 n [] holds. Ths yelds relaton (17) of Theorem 6.1 above, whle results of the paper [4] mply that R( W L ; σ (, )) ρ (T )/4 = l /1 for any T. The quadratc splne of Theorem 6.1 nterpolates the values of f at the vertces and does not nterpolate the gradents (see Theorems 3.3 and 3.4). In ths secton we fnd optmal bvarate Hermte nterpolatng splnes. If the ndex {0, 1, } s chosen, then j and k wll denote the other two elements from {0, 1, }. We do not need to specfy whch of the two values s k and whch s j due to the symmetry of the splnes. Throughout the rest of the paper we let a := b 0 b 1 b and c := b 0 b 1 + b 1 b + b b 0 be the elementary symmetrc polynomals. Theorem 6.. Let γ = b + 3 b jb k, = 0, 1,. Then the contnuous quartc bvarate splne of type (8) s optmal for the recovery problem (4) wth n = Π L /( 3l ) nodes on the class W L. Moreover, ths splne s Hermte nterpolatng and smooth at the vertces of. Proof. Let T be any trangle n. Snce v = b v, we obtan v v = b j v v j + b k v v k, v v v v = v v = l (b j + b jb k + bk ). (18) The followng denttes are easly verfed θ = 1 b γ = c 9 a, b v v = l c, b v v γ = l cθ. (19) Denote by s the splne of type (8) wth γ = b + 3 b jb k. From Theorem 5.1, takng nto account relatons (19), for every v T \ V T, we obtan f (v) s( f, v) 1 b v v θ + γ θ + γ = 1 b v v (γ θ) + b v v θ θ + γ = θ In vew of (17), to prove the optmalty of the splne s we need to show that θ b v v θ + γ l 1, v T \ V T. b v v θ + γ.

13 S. Borodachov, T. Sorokna / Journal of Approxmaton Theory 164 (01) Ths nequalty can be rewrtten as θ b v v (θ + γ j )(θ + γ k ) l 1 (θ + γ ). (0) From (19) t follows that θ c b v v (θ + γ j )(θ + γ k ) 1 b v v (1 + θ γ ) (θ + γ s ), 1 s b v v (θ + γ j )(θ + γ k )(1cθ (1 + θ γ )(θ + γ )) 0. It remans to show that for every = 0, 1,, there holds 1cθ (1 + θ γ )(θ + γ ) 0. (1) We start by provng the followng basc nequaltes: 0 c 1/3, and 0 c 9a/ 1/. () Clearly, c 0 and θ = 3a/ + b (b j + b k ) 0. Let m = b 1 + b and x = b 1 b. Then b 0 = 1 m, c = (1 m)m + x, θ = c 9a/ = (9m 5)x/ + (1 m)m. Snce 0 x = b 1 b m /4, we need to show that δ m (x) := c 1/3 0, ψ m (x) := θ 1/ 0, x [0, m /4], m [0, 1]. (3) Snce (1 m)m 1/4, m [0, 1], the followng nequaltes are true δ m (0) = (1 m)m 1/3 < 0, ψ m (0) = (1 m)m 1/ 0, m δ m = 3 m m 0, ψ m = (m 1) m 3 0. Snce δ m and ψ m are lnear n x, both () and (3) follow. Then (1) can be rewrtten as 6θ(cθ 1/6) + 3θ (c 1/3) + 3cθ γ + γ 0. Ths nequalty follows from () and Lemma 7.1 from the next secton. Thus (1) s true, and the proof of the optmalty of s on the class W L s complete. Interpolaton of the functon values at the vertces follows from Theorem 3.3. The fact that s nterpolates the gradent at every vertex, and, hence, s C 1 -contnuous at every vertex, mmedately follows from Theorem 3.4. The quartc splne of Theorem 6. s contnuous (see Theorem 3.) but not smooth. Accordng to Theorem 4., there s no smooth splne recovery method of the form (7) on the regular hexagonal mesh wth the same or lower error estmate on the class W L f J(L) l. Moreover, for any L-perodc trangulaton 1 n R that has n = Π L /( 3l ) vertces per perod and satsfes (13), no smooth splne on 1 of the form (7) s optmal for the recovery problem (4) on W L. Theorem 6.3. Let γ = b (1 + 9 b jb k ), = 0, 1,. Then the contnuous quntc bvarate splne of type (8) s optmal for the recovery problem (4) wth n = Π L /( 3l ) nodes on the class W L. Moreover, ths splne s Hermte nterpolatng and smooth at the vertces of.

14 1456 S. Borodachov, T. Sorokna / Journal of Approxmaton Theory 164 (01) Proof. Let T be any trangle n. We frst compute θ = 1 b (1 + 9b jb k /) = c 9a/, and γ + θ = b + c. (4) Denote h := (b + c) = a + 6c + 8c 3. By Theorem 5.1 takng nto account (18), (4), the second equalty n (19), and the fact that b = 1 c and b (b 3 j + b3 k ) = c a c, for every v T \ V T, we wll have f (v) s( f )(v) 1 b v v (γ θ) + b v v θ θ + γ = l b (b j + b jb k + bk )(b + 9a c) + l θ b (b j + b jb k + bk ) b + c = l (9ac 3a) + l θ b (b j h + b jb k + bk )(b j + c)(b k + c) = 3al (3c 1) + l θ h (a + 3ac + c ). By (), the frst term n ths sum s non-postve. Then n vew of (17) to prove the optmalty of the splne s we must show that l θ h (a + 3ac + c ) l 1. (5) Ths nequalty can be rewrtten as (c 9a/) (a + 3ac + c ) 1 1 (a + 6c + 8c 3 ), 9 4 a c(7a 1) + 7 a (3a c ) a ac 1 c (9a c) 81 +c 8c c 6 a 4ac 1 0. (6) In vew of () and the nequaltes a 1 7, and c 3a = 1 b (b j b k ) 0, the frst lne of (6) s non-postve. Lemma 7. n the next secton shows that the second lne of (6) s also non-postve. Ths completes the proof of (5), and mples the optmalty of s on the class W L. The nterpolaton of the functon values at the vertces follows from Theorem 3.3. The fact that s nterpolates the gradent at every vertex, and, hence, s C 1 -contnuous at every vertex, mmedately follows from Theorem 3.4.

15 S. Borodachov, T. Sorokna / Journal of Approxmaton Theory 164 (01) Remark 6.. We have a dfferent method of provng Theorem 6.3 that nvolves representng the left-hand sde of (6) as a sum of complete squares multpled by some negatve expressons. Ths technque may allow an extenson to the case of arbtrary trangulatons, and the work n ths drecton s currently n progress. 7. Auxlary nequaltes for regular trangles In ths secton we collect rather techncal proofs of the nequaltes on a regular trangle T used n the proofs of the man results. Lemma 7.1. Let γ = b + 3 b jb k. Then 3cθ γ + γ 0 for all v T, for every = 0, 1,. Proof. Denote m = b j + b k and x = b j b k. We note that θ = 1 b γ = c 9 a and γ = 1 m + 3x/. We need to show that r m (x) := 3cθ γ + γ 0, x [0, m /4], m [0, 1]. (7) The polynomal r m can be wrtten as r m (x) = u(m)x 3 + k(m)x + p(m)x + q(m), u(m) = 3 4 (9m 5), k(m) = 9 ((1 m)m(9m 5)(3m + 1) + 1), 4 p(m) = 18(1 m) m (3m 1) 3m + 3/, q(m) = (1 m)m(1(1 m) m 1). where Snce (1 m)m 1/4, m [0, 1], we have r m (0) = q(m) 0. Then m r m = m 4 56 ( 3m) (7(1 m) 4 (3m + 1) + 9(1 m) 3 (9m + 1) + 81m(1 m) + 4(1 m)(3m + 7) + 1m) 0, m [0, 1]. We next show that r m (x) attans ts maxmum on [0, m /4] at an endpont. Frst, we verfy that k(m) > 0, m [0, 1]. Ths s clear for m [5/9, 1]. Snce k(m) assumes the value 9/4 four tmes, ts dervatve k (m) has three zeros of multplcty one. Hence, k (m) changes ts sgn only once on [0, 5/9]. Snce k (0) < 0 and k (1/3) > 0, the dervatve k (m) changes ts sgn on [0, 1/3] and hence, k(m) s ncreasng on [1/3, 5/9]. Snce k(1/3) > 0, we have k(m) > 0, m [1/3, 5/9]. Fnally, assume that m [0, 1/3]. Then 0 (1 m)(3m + 1) 4/3 and 5/36 m(9m 5) 0. It follows that k(m) = 9 4 ((1 m)m(9m 5)(3m + 1) + 1) = 1 6 > 0. Thus, k(m) > 0 for every m [0, 1]. Let m = 5/9. Then r m (x) s a quadratc functon of x. Snce k(m) > 0, the dervatve r m (x) = k(m)x + p(m) ether preserves ts sgn on [0, m /4] or changes sgn from to + there. In both cases r m (x) attans ts maxmum on [0, m /4] at an endpont. Snce we know that both r m (0) and r m (m /4) are non-postve, we have r m (x) 0, x [0, m /4].

16 1458 S. Borodachov, T. Sorokna / Journal of Approxmaton Theory 164 (01) Assume now that m [0, 1] \ {5/9}. Consder the dervatve r m (x) = 3u(m)x + k(m)x + p(m). Note that 3u(m) > 0. If the quadratc polynomal r m (x) has no roots, we have r m (x) > 0, x [0, m /4]. If t has roots, snce k(m) > 0, the lesser of the two roots s negatve. Dependng on the poston of the larger root, we ether have r m (x) 0, x [0, m /4], or r m (x) changes ts sgn once on [0, m /4] (from to + ), or r m (x) 0, x [0, m /4]. In each of these cases the functon r m (x) attans ts maxmum on [0, m /4] at one of the endponts. Snce we know that both r m (0) and r m (m /4) are non-postve, we have r m (x) 0, x [0, m /4]. Thus, nequalty (7) holds. Lemma 7.. If v T, then 8c c 6 a 4ac 1 0. Proof. Let m = b 1 + b [0, 1] and x = b 1 b [0, m /4]. Then r m (x) := 8c c/6 a 4ac 1/ = k(m)x + p(m)x + q(m), k(m) = 4m 16, p(m) = 4m 3 + 3m 7m 7/6, q(m) = 8(1 m) m (1 m)m/6 1/. where We next show that r m (x) 0 on [0, m /4]. Snce 0 (1 m)m 1/4, for every m [0, 1], we have r m (0) = q(m) < 0 and r m (m /4) = ( 3m) (6m(1 m) + 3/ m)/1 0. When m (/3, 1], the parabola r m (x) s open upward, and t attans ts maxmum on [0, m /4] at one of the endponts. Hence, r m (x) 0 on [0, m /4]. Ths s also true when m = /3 snce r m (x) becomes a lnear functon n ths case. Consder m [11/1, /3). Let ρ(m) := r m (m /4) = 1m 3 + 4m 7m 7/6. Then ρ( 1) > 0, ρ(0) < 0, ρ(11/1) > 0, ρ(1) > 0, ρ() < 0. Hence, the cubc polynomal ρ(m) s postve for m [11/1, /3). Snce the parabola r m (x) s open downward and r m (m /4) > 0, the functon r m (x) s ncreasng on [0, m /4]. Snce r m (x) s non-postve at x = m /4, t wll be non-postve on [0, m /4]. Fnally, assume that m [0, 11/1]. Snce k(m) < 0, we obtan r m (x) p(m)x + q(m). We have q(m) < 0 and hence, the functon p(m)x + q(m) has a negatve value at x = 0. Let t(m) := p(m)m /4 + q(m) = 6m m 4 71m 3 /4 + 63m /8 m/6 1/. Snce t (m) = 360m + 384m 13/ < 0, the polynomal t(m) has at most three zeros. Snce t( 1) > 0, t(0) < 0, t(11/1) < 0, t(3/5) > 0, and t(1) < 0, we must have t(m) < 0, m [0, 11/1], whch s the value of the lnear functon p(m)x + q(m) at x = m /4. Then r m (x) p(m)x + q(m) < 0, x [0, m /4], and the proof s complete. Acknowledgment We would lke to thank undergraduate student Matthew Tger who used Mathematca computatons to assst our work on some parts of Secton 7. References [1] V.F. Babenko, On optmal pecewse lnear nterpolaton of contnuous mappngs, n: Investgatons n Current Problems n Summaton and Approxmaton of Functons and ther Applcatons, Dnepropetrovsk. Gos. Unv., Dnepropetrovsk, 1980, pp. 3 7 (n Russan).

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