Approximation of Univariate Set-Valued Functionsan Overview

Transcription

1 Approxmaton of Unvarate Set-Valued Functonsan Overvew ra Dyn, Elza Farkh, Alona Mokhov School of Mathematcal Scences Tel-Avv Unversty, Israel Abstract The paper s an updated survey of our work on the approxmaton of unvarate setvalued functons by samples-based lnear approxmaton operators, beyond the results reported n our prevous overvew. Our approach s to adapt operators for real-valued functons to set-valued functons, by replacng operatons between numbers by operatons between sets. For set-valued functons wth compact convex mages we use Mnkowsk convex combnatons of sets, whle for those wth general compact mages metrc averages and metrc lnear combnatons of sets are used. We obtan general approxmaton results and apply them to Bernsten polynomal operators, Schoenberg splne operators and polynomal nterpolaton operators. Key words: compact sets, set-valued functons, lnear approxmaton operators, Mnkowsk sum of sets, metrc average, metrc lnear combnatons 1 Introducton In ths paper we present the progress of our work on the approxmaton of unvarate setvalued functons (multfunctons) by lnear approxmaton operators, beyond the results reported n [11]. We adapt lnear samples-based approxmaton operators for real-valued functons to set-valued functons (SVFs) wth compact mages n R n, by replacng operatons between numbers by operatons between sets. For ths purpose, the well-known Mnkowsk sum of sets s a proper substtute for addton of numbers, only n case of SVFs wth convex mages. For such multfunctons, the representaton of convex compact sets n terms of ther support functons allows to reduce approxmaton of SVFs by lnear postve operators to the approxmaton of the correspondng support functons. The applcaton of known approxmaton results from the case of real-valued functons to the case of SVFs wth convex compact mages s studed n [19, 6, 2, 8]. The postvty of the operators s necessary for the approxmants to be well defned. It was notced by Vtale [19] that postve approxmaton operators wth Mnkowsk sums of sets fal to approxmate multfunctons wth general compact mages (not necessarly convex). Vtale also observed that the mages of the Bernsten approxmants of ncreasng degree tend to convex sets. Smlarly, lmts of splne subdvson schemes are convex-valued 1

2 SVFs for any ntal data [10]. The obvous concluson s that operators wth Mnkowsk sums of sets are not approprate for the approxmaton of SVFs wth general compact mages. In [1] a bnary operaton between sets, the metrc average, s ntroduced, and the pecewse lnear nterpolant based on t s shown to approxmate contnuous SVFs wth general mages. The use of ths operaton n the adaptaton of known postve approxmaton operators to SVFs, requres a representaton of the approxmaton operators by repeated bnary averages. Such a representaton exsts for any samples-based lnear operator, whch reproduces constants, but s not unque [20]. Ths non-unqueness leads to dfferent operators for SVFs whch are not necessarly approxmatng. Yet, splne subdvson schemes represented by repeated bnary averages [9], and the Schoenberg operators defned n terms of the de Boor algorthm [13], approxmate SVFs wth general compact mages. On the other hand, for the adaptaton of the Bernsten operators based on the de Casteljau algorthm, an approxmaton result was obtaned only for a certan clas of SFVs wth mages n R [13]. The lack of assocatvty of the metrc average s the reason why t s hard to extend ths bnary operaton to an average of three or more sets. Yet, n [12] a set-operaton on a fnte sequence of compact sets, termed metrc lnear combnaton, whch extends the metrc average, s devsed. Wth ths operaton, lnear approxmaton operators are successfully adapted to unvarate SVFs. It should be emphaszed that ths adaptaton method s not restrcted to postve operators. To the best of our knowledge so far only postve operators were appled to SVFs. We apply the dfferent adaptatons to two classes of postve operators, Bernsten operators and Schoenberg splne operators. Adaptaton of polynomal nterpolaton operators s constructed only wth metrc lnear combnatons of sets. Such nterpolaton operators at the zeros of the Tchebyshev polynomals of growng degree are shown to converge to Lpschtz contnuous SFVs [12]. The outlne of the paper s as follows. Secton 2 contans defntons and notaton. Secton 3 dscusses operators based on Mnkowsk averages of sets; ther applcablty for the approxmaton of convex-valued multfunctons, and ther falure n the case of SVFs wth general compact mages. In Secton 4 two metrc operatons on sets are presented, and used n Secton 5 to construct approxmatng operators for multfunctons wth general compact mages. In Secton 6 error estmates for specfc approxmaton operators are presented. 2 Prelmnares Frst we ntroduce some notaton. The collecton of all nonempty compact sets n R n s denoted by K n, C n denotes the collecton of convex sets n K n., s the nner product, s the Eucldean norm and S n 1 s the unt sphere n R n. We use coa for the convex hull of A K n and dst(x, A) = nf a A x a for the dstance from a pont x R n to A. We defne the set of metrc pars of A, B K n by Π(A, B) = { (a, b) A B : a b = dst(a, B) or a b = dst(b, A)}. For A, B K n the Hausdorff metrc s haus(a, B) = sup{ a b : (a, b) Π(A, B) }. 2

3 The space K n s a complete metrc space wth respect to ths metrc. [17]. The support functon δ (A, ) : R n R s defned for A K n as δ (A, l) = max l, a, l R n. a A A lnear Mnkowsk combnaton of sets s k λ A = { =1 k λ a : a A, = 1,..., k}, A K n, λ R. =1 In partcular, A+B = {a+b, a A, b B} s the Mnkowsk sum of two sets. A Mnkowsk average (a Mnkowsk convex combnaton) of sets s a lnear Mnkowsk combnaton wth λ non-negatve, summng up to 1. We consder functons defned on [0, 1] wth mages n a metrc space (X, ρ), wth X ether R n or K n, and ρ ether the Eucldean metrc or the Hausdorff metrc respectvely. The notons of convergence, contnuty, Hölder/Lpschtz contnuty are to be understood wth respect to the approprate metrc, e.g. f( ) s Hölder contnuous wth exponent α f ρ(f(x), f(y)) L x y α, x, y [0, 1], where the constant L depends on f. The collecton of Hölder contnuous multfunctons wth exponent α and constant L s denoted by H α (L). For α = 1 the notaton s Lp(L). We recall that the modulus of contnuty (see e.g. [5], Chapter 2) of a functon f : [0, 1] X wth a step δ 0 s ω(f, δ) = sup h (f, ), 0<h δ where h (f, x) = { ρ(f(x + h), f(x)) for x, x + h [0, 1], 0 otherwse. ω(f, δ) s also known as the frst modulus of smoothness. The k th modulus of smoothness s defned by ω k (f, δ) = sup{ k h (f, ) : 0 < h δ}, wth 1 h = h and k h (f, x) = ( h k 1 h (f, x), x ). ote that for f H α (L), ω k (f, δ) = O(δ α ), k 1 and for f k tmes contnuously dfferentable ω k (f, δ) = O(δ k ). In ths paper we dscuss the adaptaton to unvarate SVFs of certan lnear operators approxmatng real-valued functons. We consder lnear operators based on samples at a set of ponts χ = {x 0,..., x }, 0 x 0 < x 1 <... < x 1 of the form Aχ(f, x) = c (x)f(x ). (1) We restrct ths class to operators whch approxmate contnuous functons n [0, 1] or n most of t. Thus, we requre that c (x) = 1 ether n [0, 1], or n most of t. We 3

4 denote by χ = max{x +1 x : = 0,..., 1}. χ denotes the set of equdstant ponts {/ : 0 }, wth χ = 1/. An operator based on χ we denote by A. We recall that a lnear operator L(f, x) s called postve f for a non-negatve f, L(f, x) s non-negatve. Obvously, Aχ s a postve lnear operator f c (x) 0, = 0,...,. It reproduces the constant functons, namely Aχ(f, x) = f(x) for f(x) = Const at all x such that c (x) = 1. At all such ponts, Aχ(f, x) s a weghted average of the functon values f(x ), = 0, 1,...,. 3 Approxmaton based on Mnkowsk averages The frst adaptatons of operators of type (1) to SFVs were done wth the help of Mnkowsk sum of sets. In ths secton we survey some general results for such adaptatons [19, 2, 6, 8]. For gven data ponts χ, a postve operator of the form (1) wth Mnkowsk sums of sets replacng addton of numbers s Aχ(F, x) = c (x)f(x ), x [0, 1], c (x) 0. (2) 3.1 The case of convex-valued multfunctons Here we consder SVFs wth mages n C n, and the operaton of Aχ defned by (2) on such multfunctons. It s clear, that Aχ(λF + µg, ) = λaχ(f, ) + µaχ(g, ), λ, µ 0. (3) Moreover, by the postvty of c (x), the mages of Aχ(F, ) reman n the cone C n. The approxmaton and the shape-preservaton propertes of such operators follow from the parametrzaton of convex compact sets by ther support functons. The well known propertes of the support functons δ, relevant to our nvestgaton, are [16]: for A, B C n, 1. δ (A + B, ) = δ (A, ) + δ (B, ), 2. δ (λa, ) = λδ (A, ), λ 0, 3. A B δ (A, l) δ (B, l) for each l R n, 4. haus(a, B) = max l S n 1 δ (A, l) δ (B, l). Thus, the operator Aχ n (2) s related to the operator Aχ n (1) by, δ (Aχ(F, t), l) = Aχ(δ (F, l), t), l R n. (4) Also, F H α (L) ff δ (F( ), l) H α (L), unformly n l S n 1. By the above two observatons, approxmaton results for postve operators can be extended from the case of real-valued functons to the case of set-valued functons wth compact convex mages. Here we formulate a general result of ths type. 4

5 Theorem 3.1. Let Aχ approxmate contnuous real-valued functons wth the error estmate Aχ(f, x) f(x) Cω k (f, ψ(x, χ )), where ψ : [0, 1] R + R + s a contnuous real-valued functon, non-decreasng n ts second argument, satsfyng ψ(x, 0) = 0. Then for a contnuous convex-valued multfuncton F : [0, 1] C n haus(aχ(f, x), F(x)) C sup l S n 1 ω k (δ (F, l), ψ(x, χ )). As n the real-valued case, the adapted postve operators (2) have shape preservaton propertes n the convex-valued case. The postvty of Aχ preserves the order between two multfunctons n the sense of set-ncluson: F(x) G(x) Aχ(F, x) Aχ(G, x). Moreover, by Property 3 of support functons, f Aχ preserves monotoncty of realvalued functons, then Aχ preserves monotoncty of multfunctons. Here monotoncty of SVFs s n the sense of set-ncluson, namely f F(x) F(x + h) for all h > 0, then Aχ(F, x) Aχ(F, x + h) for all h > 0 (see [8] for more detals). 3.2 The general case convexfcaton Vtale [19] notced that for the constant multfuncton F(x) = {0, 1}, the pecewse-lnear approxmaton constructed wth Mnkowsk sums does not converge to F(x) when χ 0. He also observed that the Bernsten approxmants of a multfuncton wth general compact mages converge, when ncreasng ther degree, to a convex-valued multfuncton. More generally, f the number of summands n (2) grows wth, as for the Bernsten operators, the Shapley-Folkman-Starr Theorem (see Appendx 2 n [18] and Theorem 2 n [4]) yelds haus(aχ(f, x), coaχ(f, x)) n max c (x) max sup{ y : y F(s)}, 0 s [0,1] for any multfuncton wth compact mages n R n. Snce coaχ(f, x) = Aχ(coF, x), by Theorem 3.1, lm Aχ(coF, x) = cof(x). Thus, f lm max c (x) = 0, as n the case of Bernsten operators, then lm Aχ(F, x) = cof(x) (see [10] for other operators wth ths property). Another type of operators for whch convexfcaton occurs are splne subdvson schemes. For these operators the Shapley-Folkman-Starr Theorem s not applcable. Subdvson schemes are recursve averagng procedures wth a fxed fnte number of summands and fxed weghts. For ths case an nequalty, nvolvng a measure of non-convexty of sets, ntroduced n [4], s used to prove that splne subdvson schemes wth Mnkowsk averages appled to arbtrary ntal compact sets n R n converge to a multfuncton wth convex mages [10]. The convexfcaton occurng wth Mnkowsk averages, motvated the search for alternatve operatons on sets. 5

6 4 Metrc operatons on general sets The lack of approxmaton by operators of type (2) n case of SVFs wth general mages, s due to the fact that the Mnkowsk averages of non-convex sets are too bg. For example, the convex combnaton λa + (1 λ)a, λ [0, 1] equals A f A s convex, but s a superset of A for general A. Two operatons on sets are ntroduced n [1] and [12] whch produce subsets of the Mnkowsk average or lnear Mnkowsk combnaton respectvely. Wth these operatons t s possble to avod convexfcaton and to acheve approxmaton for SVFs wth general mages. 4.1 The metrc average of two sets A bnary operaton between sets was constructed n [1] and used for pecewse-lnear approxmaton of SVFs wth compact (not necessarly convex) mages. Ths bnary operaton s termed n [9] metrc average. Defnton 4.1. Let A, B K n, t [0, 1]. The t-weghted metrc average of A and B s A t B = {ta + (1 t)b : (a, b) Π(A, B)}. The followng propertes of the metrc average are mportant for our applcatons. The frst three are easy to observe [9], and the fourth s the metrc property proved n [1]. Let A, B, C K n and 0 t 1, 0 s 1. Then 1. A 0 B = B, A 1 B = A, A t B = B 1 t A. 2. A t A = A. 3. A B A t B ta + (1 t)b. 4. haus(a t B, A s B) = t s haus(a, B). ote that the analogues of propertes 2 and 4 are true n the case of Mnkowsk averages only for convex sets, whle wth the metrc average these essental propertes are vald for general compact sets. Although the metrc average s a non assocatve bnary operaton, there exsts an extenson of ths operaton to a fnte number of ordered sets. 4.2 The metrc lnear combnaton of sets In [12] a new operaton on a fnte sequence of sets s ntroduced. It s based on the noton of a metrc chan, whch s an extenson of a metrc par. Defnton 4.2. For {A 0,..., A } wth A K n, a vector (a 0,..., a ) s called a metrc chan of {A 0,..., A }, f a A, = 0,...,, and there exsts j, 0 j such that a 1 Π A 1 (a ), 1 j and a +1 Π A+1 (a ), j 1. Here Π A (b) = { a A : a b = dst(b, A) } for b R n. 6

7 An llustraton of such a metrc chan s gven n Fgure 4.1. a 0 Π A0 (a 1 ) a j 1 Π Aj 1 (a j ) a j A j a j+1 Π Aj+1 (a j ) a Π A (a 1 ) Fgure 4.1. Thus each element of each set A, = 0,..., generates at least one metrc chan. We denote by CH(A 0,..., A ) the collecton of all metrc chans of {A 0,..., A }. The set CH(A 0,..., A ) depends on the order of the sets A, = 0,...,. Wth ths noton of metrc chans we can defne, Defnton 4.3. A metrc lnear combnaton of a sequence of sets A 0,..., A wth coeffcents λ 0,..., λ R, s { } λ A = λ a : (a 0,..., a ) CH(A 0,..., A ). (5) The followng dstrbutve laws are easly derved from the defnton, ( ( ) () λ A = λ )A, () λa = λ 1 A. ote that λ 0,..., λ can be any real numbers, and that f λ = 1, then by (), 5 Metrc approxmaton operators λ A = A. In ths secton we descrbe our general approach to the adaptaton of operators of type (1) to the set-valued settng, based on the metrc operatons of Secton 4. The dscusson of the adaptaton of specfc operators s postponed to Secton Operators based on the metrc average The metrc average was successfully used n [9] for the constructon of set-valued subdvson schemes and n [13] for the adaptaton of the Schoenberg splne operators to multfunctons. Also n [13] the Bernsten operators based on the metrc average are shown to approxmate a certan class of SVFs wth mages n R. The man dsadvantage of the metrc average, as an operaton on sets, s the lack of assocatvty. Hence t s not drectly extendable to several sets. Ths s the reason why the adaptaton of (1) based on the metrc average requres to represent t n terms of repeated bnary averages. Let us note that a representaton by repeated bnary averages exsts for any samples-based lnear operator, whch reproduces constants, but t s not unque [20]. The non-unqueness leads to a varety of operators, whch are not necessarly approxmatng. Therefore general approxmaton results are not avalable. Yet, the representatons chosen n [9, 13], for concrete approxmaton operators, are proved to be adequate theoretcally and expermentally. 7

8 5.2 Operators based on the metrc lnear combnatons All the results of ths subsecton are cted from [12]. We use the metrc lnear combnaton (5) to defne the metrc analogue of the lnear operator (1). Defnton 5.1. For F : [0, 1] K n, we defne a metrc lnear operator A M χ by A M χ (F, x) = c (x)f(x ). (6) In contrast to the adaptatons of postve operators based on the metrc average, the metrc analogues (6) of two lnear operators of the form (1), whch are dentcal on snglevalued functons, are dentcal on SVFs. Here we formulate a general error estmate for these operators. Theorem 5.2. Let A χ be of the form (1), then for a contnuous F : [0, 1] K n haus(a M χ (F, x), F(x)) 2 ω(f, χ ) + sup A χ (s(χ, ϕ), x) s(χ, ϕ)(x), (7) ϕ CH where s(χ, ϕ) s a pecewse-lnear sngle-valued functon nterpolatng the data (x, f ), = 0,...,, wth ϕ = (f 0,..., f ) CH(F(x 0 ),..., F(x )). In case F Lp(L), then also s(χ, ϕ) Lp(L), and we have Corollary 5.3. Let F Lp(L) and let A χ be of the form (1), satsfyng where ψ s as n Theorem 3.1. Then A χ (f, x) f(x) C Lψ(x, χ ), f Lp(L), haus(a M χ (F, x), F(x)) 2 L χ + CLψ(x, χ ), (8) 6 Adaptaton of specfc approxmaton operators The approxmaton results from Sectons 3,5 are specalzed here to two classes of postve operators: the Schoenberg splne operators and the Bernsten polynomal operators. We also present the adaptaton of polynomal nterpolaton operators to SVFs as examples of non-postve operators. Error estmates for the varous types of adapted approxmaton operators are provded, usng C as a generc constant. 8

9 6.1 Bernsten operators The Bernsten operator B (f, x) for a real-valued functon f : [0, 1] R s ( ) ( ) B (f, x) = x (1 x) f. (9) It s known (see [5], Chapter 10) that there exsts a constant C ndependent of f such that for a contnuous f f(x) B (f, x) Cω [0,1] (f, x(1 x)/). The value B (f, x) can be calculated recursvely by repeated bnary averages, usng the de Casteljau algorthm [15], f k = (1 x)f k 1 f 0 = f(/), = 0,...,, (10) + xf k 1 +1, = 0, 1,..., k, k = 1,...,, B (f, x) = f0. Ths algorthm s commonly used n CAGD. ext we present three dfferent adaptatons of the Bernsten operators to SVFs. The adapted Bernsten operator of the form (2) s and by Theorem 3.1 we have, B Mn (F, x) = ( ) x (1 x) F Theorem 6.1. For a convex-valued multfuncton F H α (L) haus ( F(x), B Mn (F, x) ) ( ) α x(1 x) 2 C. ( ), (11) In the adaptaton of (9), based on the metrc average wth the de Casteljau algorthm, startng wth F 0 = F(/), we replace n (10) the average f k = (1 x)f k 1 + xf k 1 +1 by the metrc average F k = F k 1 1 x F k 1 +1 and obtan the approxmant B MA(F, x) = F 0, x [0, 1]. It s not known whether these operators approxmate multfunctons wth general compact mages n R n, yet for a certan class of SVFs wth compact mages n R, the followng approxmaton result holds [13], Theorem 6.2. Let F Lp(L) be such that for x [0, 1], F(x) = J j=1 F j(x), where F j (x) are dsjont compact ntervals. Then for suffcently large haus ( B MA (F, x), F(x) ) C/, x [0, 1]. The metrc analogue of the Bernsten operator for set-valued functons s [12], B M (F, x) = ( ) ( ) x (1 x) F { ( ) } = x (1 x) f : (f 0,..., f ) CH, 9

10 where CH = CH(F(0), F(1/),..., F(1)). It follows from Corollary 5.3 that Corollary 6.3. Let F Lp(L), then 6.2 Schoenberg operators haus ( B M (F, x), F(x)) 2L/ + CL x(1 x)/. For the Schoenberg operators we have four successful adaptatons to SVFs. The approxmaton results n ths case are numerous The real-valued case The Schoenberg splne operator of order m wth unform samplng ponts χ, for a realvalued functon f, s S m, (f, x) = f(/)b m (x ), x [0, 1], (12) where b m (x) s the B-splne of order m (degree m 1) wth nteger knots and support [0, m]. By the known approxmaton result (see [3], Chapter XII), [ ] m + 1 m 1 S m, (f, x) f(x) ω [0,1] (f, 1/), x 2, 1, (13) where x s the maxmal nteger not greater than x. ote, that the rate of approxmaton of the Schoenberg operators can be mproved f b m n (12) s replaced by the centered B-splne b m = b m ( + m/2). We omt the detals here. In [3], Chapter X t s shown that (12) can be evaluated recursvely n terms of repeated bnary averages. For x [j, j + 1] let f 1 = f (/), = j m + 1,..., j, (14) f k = λ k fk ( ) 1 λ k f k 1, = j m + k,..., j, k = 2,..., m, S m, (f, x) = fj m. wth λ k = + m + 1 k t, = j m + k,..., j, k = 2,..., m. m + 1 k For real-valued functons the Schoenberg operators can be also evaluated by subdvson schemes (see e.g. [7]). Gven the ntal sequence f 0 = f( ), = 0,..., of values n R, wth f 0 = 0 for Z \ {0, 1,..., }, the splne subdvson scheme for the evaluaton of S m, (f, ) s gven by the refnement steps f k+1 = j Z a [m] 2j fk j, Z, k = 0, 1, 2,... (15) 10

11 where a [m] = ( ) m+1 /2 m, = 0, 1,..., m + 1 and a [m] = 0 for Z \ {0, 1,..., m + 1}. At the k th refnement level one defnes the pecewse-lnear functon f [k] (x) = Z f k b 2 (2 k x ), x R, (16) where {f k, Z} are the values generated by the subdvson scheme at refnement level k. The scheme (15) s unformly convergent, namely the sequence {f [k] ( )} k 0 s a Cauchy sequence, and ts lmt functon s of the form (see e.g. [7]) f (x) = f 0 b m(x ), x R. Therefore S m, (f, x) = f (x), x [0, 1]. (17) The refnement step (15) can be computed by repeated bnary averages as follows: f k+1,0 2 = f k, fk+1,0 2 1 = (1/2)f 1 k + (1/2)fk, Z, (18) f k+1,j f k+1 = (1/2)f k+1,j The convex-valued case = f k+1,m 1 + m 1 2, Z. + (1/2)f k+1,j 1 +1, j = 1,..., m 1 To defne Schoenberg operators for a multfuncton wth convex mages F, one can use the drect formula (12), the evaluaton procedure (14) or splne subdvson schemes wth a refnement step gven by (15) or by (18), replacng f by F and sums of numbers by Mnkowsk sums of sets. By the results obtaned for the real-valued case and by (4), all methods of computaton lead to the same SVF, denoted by S m, (F, ) [8],[11]. By Theorem 3.1 we have for F H α (L) m haus (S m, (F, x), F(x)) 2 α, x Schoenberg operators based on the metrc average [ ] m 1, 1. (19) In [13] the Schoenberg operator for a multfuncton F, Sm, MA (F, ), s defned n terms of algorthm (14) wth the bnary averages between numbers replaced by the correspondng metrc averages between sets. It s shown there that Theorem 6.4. For a set-valued functon F : [0, 1] K n, F H α (L), the Schoenberg operator Sm, MA (F, x) satsfes haus ( Sm, MA (F, x), F(x)) C α. (20) 11

12 Another way to adapt the Schoenberg operators usng the metrc average as a basc operaton s to adapt the m-th degree splne subdvson scheme, represented by the sequence of repeated bnary averages (18). Startng wth F 0 = F(/), = 0,..., and F 0 = {0} otherwse, we replace n (18) the averages of numbers by correspondng metrc averages of sets to obtan {F k+1 : Z} from {F k : Z} At the (k +1) th refnement level, a metrc pecewse-lnear SVF, F [k+1] (t) s defned by F [k+1] (t) = F k+1 λ(t) F k+1 +1, 2 (k+1) t ( + 1)2 (k+1), Z (21) wth λ(t) = ( + 1) t2 k+1. The followng results are proved n [9]. Theorem 6.5. Let {F 0 : Z} be compact sets wth L = sup{haus(f 0, F +1 0 ) : Z} <. Then the sequence {F [k] ( )} k Z+ n (21) converges unformly on R to a set-valued functon F ( ) Lp(L). Theorem 6.6. Let the ntal sets for the subdvson be the samples F 0 = F(), Z, wth F Lp(L) on R, and let F ( ) be as n Theorem 6.5. Then max x R haus(f (x), F(x)) L(7 + m)/2. Applyng these results to the ntal data relevant to the evaluaton of the Schoenberg operator of functons defned on [0, 1] we obtan, Corollary 6.7. Let F Lp(L) on [0, 1], and let Then F Lp(L/) on R, and F 0 = haus ( F (x), F(x) ) { F(/) 0, {0} otherwse. [ ] L(7 + m) m 1 2, x, 1. Corollary 6.7 can be extended to F H α (L) to obtan error of order O( α ) Metrc analogues of Schoenberg operators The metrc analogue of the Schoenberg operator of order m for a multfuncton F and a set of equdstant ponts χ s Sm, M (F, x) = ( ) { } b m (x ) F = b m (x ) f : (f 0,..., f ) CH, where CH = CH(F(0), F(1/),..., F(1)). By Corollary 5.3 and the known approxmaton result (13), we have for Lpschtz contnuous SVFs Corollary 6.8. Let F Lp(L) on [0, 1]. Then haus ( ( ) m + 1 L Sm, M (F, x), F(x)) = 2 + 2, 12 x [ ] m 1, 1.

13 6.3 Polynomal Interpolants For a real-valued functon f the polynomal nterpolaton operator at the set of ponts χ s P χ (f, x) = l (x)f(x ), wth l (x) = j=0,j x x j x x j, = 0,...,. For > 1, P χ s not a postve operator. Thus the only possble adaptaton of P χ to SVFs s the metrc analogue of Defnton 5.1. For a multfuncton F, the metrc polynomal nterpolaton operator at χ, s gven by { } Pχ M (F, x) = l (x)f(x ) = l (x)f : (f 0,..., f ) CH, wth CH = CH(F(x 0 ),..., F(x )), = 0, 1,...,. To llustrate the metrc set-valued polynomal nterpolants, and to see the geometry of metrc lnear combnatons of sets wth negatve coeffcents, we present n Fgure 6.1 an example of a metrc parabolc nterpolant to three sets n R. The parabolc nterpolant nterpolates the data (x, A ), = 0, 1, 2 wth x 0 = 0, x 1 = 1/2, x 2 = 1 and A 0 = [1/4, 1/2] [3/4, 1], A 1 = [9/16, 11/16], A 2 = A 0. Fgure 6.1. Metrc parabolc nterpolant In the above fgure the nterpolated sets are depcted n black. The gray curves n the left fgure are parabolc nterpolants to the data (x, a ), = 0, 1, 2 for some metrc chans (a 0, a 1, a 2 ) CH(A 0, A 1, A 2 ). The rght fgure s the graph of the set-valued nterpolant. ext we consder a specfc sequence of nterpolaton operators whch, when operatng on F Lp(L), converges to F. Let the nterpolaton ponts χ be the roots of the Tchebyshev polynomal of degree + 1 on [0, 1]. It s known (see e.g. [14]) that for these ponts For a real-valued functon f, f(x) l (x) C log. l (x)f(x ) ( l (x) )E (f),

14 wth E (f) the error of the best approxmaton by polynomals of degree on [0, 1]. Snce E (f) Cω(f, 1/) (see [5], Chapter 7), we obtan for a Lpschtz contnuous functon f f(x) l (x)f(x ) C log, x [0, 1], and the error n the nterpolaton of such a functon at the roots of the Tchebyshev polynomals tends to zero as. When adaptng these nterpolaton operators to Lpschtz contnuous SVFs, we get by Corollary 5.3 and by the observaton that χ π/(2), Corollary 6.9. Let F : [0, 1] K n, F Lp(L), and let the ponts χ be the roots of the Tchebyshev polynomal of degree + 1 on [0, 1], then ( ) haus(pχ M C log log (F, x), F(x)) 2L χ + = O. To the best of our knowledge ths result s the frst convergence result of non-postve operators to the approxmated set-valued functons. Although we get approxmaton results for adapted operators based on metrc lnear combnatons, the drect computaton of the approxmants accordng to defntons (5), (6) s of hgh complexty. From Fgure 6.1 t s clear that such a computaton s redundant. Our am s to devse effcent algorthms for the computaton of these operators. Acknowledgements The authors acknowledge the support of the Hermann Mnkowsk Center for Geometry and the Internal Research Foundaton at Tel-Avv Unversty. References [1] Z. Artsten. Pecewse lnear approxmatons of set-valued maps. Journal of Approx. Theory 56 (1989), [2] R. Baer. Mengenwertge Integraton und de dskrete Approxmaton errechbarer Mengen. Bayreuth. Math. Schr. 50 (1995), [3] C. de Boor. A practcal gude to splne. Sprnger-Verlag, [4] J. W. S. Cassels. Measures of the non-convexty of sets and the Shapley Folkman Starr theorem. Math. Proc. Camb. Phl. Soc. 78 (1975), [5] R. DeVore and G. Lorentz. Constructve Approxmaton. Sprnger-Verlag, Berln, [6] T. Donchev and E. Farkhr. Modul of smoothness of vector-valued functons of a real varable and applcatons. umer. Funct. Anal. Optmz. 11(586) (1990), [7]. Dyn. Subdvson schemes n computer-aded geometrc desgn. In: Advances n umercal Analyss, Vol. II, Wavelets, Subdvson Algorthms and Radal Bass Functons, (Ed W. Lght), Clarendon Press, Oxford, 1992,

15 [8]. Dyn and E. Farkh. Splne subdvson schemes for convex compact sets. Journal of Comput. Appl. Mathematcs 119 (2000), [9]. Dyn and E. Farkh. Splne subdvson schemes for compact sets wth metrc averages. In: Trends n Approxmaton Theory, (Eds K. Kopotun, T. Lyche and M. eamtu), Vanderblt Unv. Press, ashvlle, T, 2001, [10]. Dyn and E. Farkh. Set-valued approxmatons wth Mnkowsk averages - convergence and convexfcaton rates. umer. Funct. Anal. Appl. 25 (2004), [11]. Dyn and E. Farkh. Approxmaton of set-valued functons wth compact mages - an overvew. In: Approxmaton and Probablty, Banach Center Publc., Vol 72, 2006, [12]. Dyn, E. Farkh and A. Mokhov. Approxmatons of set-valued functons by metrc lnear operators. Constr. Approx. 25 (2007), [13]. Dyn and A. Mokhov. Approxmatons of set-valued functons based on the metrc average. Rendc. d Matem. S.VII 26 (2006), [14] I.P. atanson. Constructve Functon Theory. Vol. III, Frederck Ungar, ew York, [15] H. Prautzsch, W. Boehm and M. Paluszny. Bezer and B-Splne Technques. Sprnger, [16] R. T. Rockafellar. Convex Analyss. Prnceton Unversty Press, Prnceton, [17] R. Schneder. Convex Bodes: the Brunn Mnkowsk Theory. Cambrdge Unversty Press, Cambrdge, [18] R. Starr. Quas-equlbra n markets wth non-convex preferences. Econometrca 37 (1969), [19] R. Vtale. Approxmaton of convex set-valued functons. J. Approxmaton Theory 26 (1979), [20] J. Wallner and. Dyn. Convergence and C 1 analyss of subdvdson schemes on manfolds by proxmty. CAGD 22 (2005),