Approximations of Set-Valued Functions Based on the Metric Average

Size: px

Start display at page:

Download "Approximations of Set-Valued Functions Based on the Metric Average"

Gerard Edward Doyle
5 years ago
Views:

1 Approxmatons of Set-Valued Functons Based on the Metrc Average Nra Dyn, Alona Mokhov School of Mathematcal Scences Tel-Avv Unversty, Israel Abstract. Ths paper nvestgates the approxmaton of set-valued functons wth compact mages (not necessarly convex), by adaptatons of the Schoenberg splne operators and the Bernsten polynomal operators. When replacng the sum between numbers n these operators, by the Mnkowsk sum between sets, the resultng operators approxmate only set valued functons wth compact-convex mages [10]. To obtan operators whch approxmate set-valued functons wth compact mages, we use the well known fact that both types of operators for realvalued functons can be evaluated by repeated bnary weghted averages, startng from pars of functon values. Replacng the bnary weghted averages between numbers by a bnary operaton between compact sets, ntroduced n [1] and termed n [4] the metrc average, we obtan operators whch are defned for set-valued functons. We prove that the Schoenberg operators so defned approxmate setvalued functons whch are Hölder contnuous, whle for the Bernsten operators we prove approxmaton only for Lpschtz contnuous set-valued functons wth mages n R all of the same topology. Examples llustratng the approxmaton results are presented. Key words: Mnkowsk sum, metrc average, set-valued functons, compact sets, Schoenberg splne operators, Bernsten polynomal operators. 1 Introducton We present n ths paper a method for adaptng to set-valued functons (multfunctons) certan well known lnear postve approxmaton operators for real-valued functons. We study two types of lnear operators, the Schoenberg splne operators and the Bernsten polynomal operators. Both types of operators, when adapted by the usual method of replacng sums between numbers by Mnkowsk sums of sets, approxmate n the Hausdorff metrc 1

2 only multfunctons wth compact-convex mages [10]. It s shown n [5] that such Bernsten multpolynomals of a set-valued functon F wth compact mages, converge n the Hausdorff metrc, wth growng degree, to the setvalued functon whose mages are the convex hulls of the mages of F. Our adaptaton method s taken from [4], where the approxmaton operators were lmts of splne subdvson schemes. Here we apply the method successfully to the Schoenberg operators. We use the de Boor algorthm for the evaluaton of the Schoenberg operators n terms of repeated bnary weghted averages, and replace the bnary weghted average between two numbers by a bnary operaton between sets, ntroduced n [1], and termed n [4] the metrc average. We prove that wth ths procedural defnton of the Schoenberg operators for multfunctons, the Schoenberg operators approxmate a Hölder contnuous set-valued functon n a rate whch equals the Hölder exponent of the multfuncton. For the Bernsten operators we use the de Casteljau algorthm for the evaluaton of a Bernsten polynomal n terms of repeated bnary weghted averages, and replace the average between two numbers by the metrc average of two sets. We prove for F Lpschtz contnuous wth mages n R all of the same topology, that ts Bernsten multpolynomal of large enough degree m approxmates F wth an error bound proportonal to m 1/2. The approxmaton results for both types of operators are llustrated by examples. We conclude the Introducton by an outlne of the paper. In Secton 2 we gve basc defntons and notatons. In partcular we dscuss the metrc average and ts relevant propertes. In Secton 3 the Schoenberg splne operators for real-valued functons are defned, and ther evaluaton n terms of the de Boor algorthm s brefly revewed. The procedural defnton of the Schoenberg operators for set-valued functons s gven n Secton 4, together wth the approxmaton results, ther proofs and examples. Secton 5 dscusses the Bernsten polynomals of real-valued functons and ther evaluaton n terms of the de Casteljau algorthm. In Secton 6 the Bernsten operators for set-valued functons are defned, and the proof of the approxmaton result together wth an example are gven. 2 Prelmnares In ths secton we ntroduce some defntons and notaton. The collecton of all nonempty compact subsets of R n s denoted by K(R n ). By Co(R n ) we denote the collecton of all convex sets n K(R n ), and by coa we denote the convex hull of A. The Eucldean dstance from a pont a to a set 2

3 B K(R n ) s defned as dst(a, B) = nf a b, b B where s the Eucldean norm n R n. The Hausdorff dstance between two sets A, B K(R n ) s defned by { } haus(a, B) = max sup dst(a, B), sup dst(b, A). a A b B The set of all projectons of a R n nto a set B K(R n ) s Π B (a) = {b B : a b = dst(a, B)}. For A, B K(R n ) the projecton of A on B s the set Π B (A) = {Π B (a) : a A}. A lnear Mnkowsk combnaton of two sets A and B from K(R n ) s λa + µb = {λa + µb, a A, b B}, wth λ, µ R. The Mnkowsk sum corresponds to a lnear Mnkowsk combnaton wth λ = µ = 1. Defnton 2.1. Let A, B K(R n ) and 0 t 1. The t-weghted metrc average of A and B s A t B = {ta + (1 t)π B (a) : a A} {t Π A (b) + (1 t)b : b B} (1) The most mportant propertes of the metrc average are presented below [4] : For A, B K(R n ) and 0 t 1, 0 s 1 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 co(a B) 4. haus(a t B, A s B) = t s haus(a, B) 5. A t B = ta + (1 t)b, A, B Co(R) It follows from propertes 1 and 4 that haus(a t B, A) = (1 t)haus(a, B), haus(a t B, B) = t haus(a, B) (2) 3

4 3 Schoenberg operators for real-valued functons, and ther evaluaton by repeated bnary averages The m -th order Schoenberg splne operator (Schoenberg s varaton dmnshng splne approxmaton) S m f to a contnuous functon f on R s gven by S m f = Z f()b m ( ), where B m (t) s the B-splne of order m wth nteger knots and support [0, m] [3]. For the knot sequence hz, wth small h, we consder the operator S m, h f = Z f(h)b m ( h ). (3) For f C(R) lm h 0 S m, h f(t) = f(t) t R [3]. S m, h f can be evaluated by an algorthm (known as the de Boor algorthm) for the computaton of a splne functon gven n terms of the B-splne bass, based on the recurrence formula for B-splnes. For j t < j + 1, (3) can be wrtten as S m, h f(th) = wth 0 k m 1 and j =j m+k+1 a k B m k (t ), (4) a 0 = f(h), = j m + 1,..., j a k = + t Introducng the notaton a k t ak 1, = j m + k + 1,..., j. (5) λ k = + t, = j m + k + 1,..., j, k = 1,..., m 1, (6) wth coeff- we observe that a k s a convex combnaton of a k 1 1 cents λ k, 1 λ k. The case k = m 1 yelds and a k 1 S m, h f(th) = a m 1 j. (7) 4

5 Remark 3.1. It follows from (4) wth k = 0 that S m, h f(th) at t [j, j+1) depends only on f(h) = j m + 1,..., j. A better approxmaton s the symmetrc Schoenberg operator: S m, h f = f(h) B ( ) m h, where B ( m (t) = B m t m ) (8) 2 Z For t [j, j + 1) Sm, h f(th) s a convex combnaton of values of f at a set of symmetrc ponts relatve to (jh, (j +1)h). For even m the evaluaton of S m, h f s smlar to that of S m, h f. In ths work we study the operator S m,h for set-valued functons. 4 Schoenberg operators for set-valued functons Let F : R K(R n ) be a set-valued functon. We defne the set-valued Schoenberg operator of order m n terms of ts evaluaton accordng to the de Boor algorthm, usng the metrc average as the basc bnary operaton and the ntal sets {F 0 = F(), Z}. To calculate the splne operator S m, h F(th) at t [j, j + 1) we use an extenson of (5) and (7) wth the average between two numbers replaced by the metrc average of two sets. Thus for k = 1,..., m 1 we defne recursvely the sets F k = F k 1 1 λ k F k 1, (9) wth λ k gven by (6) and as n (7), determne S m, h F(th) to be S m, h F(th) = F m 1 j. (10) Frst we prove some basc results, whch are used n the proof of the approxmaton theorem. Lemma 4.1. Gven an ntal sequence of compact sets {F 0, Z} K(Rn ), we defne the sets at level k by repeated applcaton of (9). Let d k = sup haus(f k 1, Fk ). (11) Z Then d k 1 d 0, k = 1,..., m 2. m 1 5

6 Proof. It follows from (9) and (2) that haus(f k, F k 1 ) = haus(f 1 k 1 λ k F k 1, F k 1 ) = λ k k 1 haus(f 1, F k 1 ) + t d k 1. Thus haus(f k, F k 1 ) + t d k 1. (12) In the same way we obtan Therefore haus(f k 1, F+1 k k 1 ) = haus(f, F k 1 λ k F+1 k 1 ) +1 = (1 λ k k 1 +1 )haus(f, F k 1 +1 ) t 1 dk 1. haus(f k 1, F+1 k ) t 1 dk 1. (13) By the trangle nequalty and usng the estmates (12) and (13) we get: haus(f k, F +1 k k 1 ) haus(f Ths leads to, F k k 1 ) + haus(f, F+1 k ) 1 d k 1. d k 1 d k 1. (14) Now, usng (14) repeatedly, we obtan the clam of the lemma d k 1 = 1 m 1 d k 1 1 d 0. m (k 1) 1 m (k 1) m 2 1 m 2 m 1 1 m 1 d 0 Lemma 4.2. Let S m, h F, be defne by (9) and (10). Then for any pont t [j, j + 1) haus( S m, h F(th), F 0 j ) d0m 2. (15) 6

7 Proof. By (10), the trangle nequalty, (12) and Lemma 4.1 haus(s m, h F(th), F 0 j m 1 k=1 = d 0 ( + j t ) = haus(f m 1 j m + j t 1 m 1 m 1 Fnally we obtan m 1, Fj 0 ) k=1 m (k 1) 1 d 0 = m 1 ) k k=1 haus(f k 1 j d0 m 1 m 1 k=1 = d 0 ( m + j t m 2 m 1, Fj k ) k=1 ( + j t) ). haus(s m, h F(th), F 0 j ) d0 ( m 2 + j t ) d 0m 2. + j t d k 1 As a consequence of the last lemma, we get the approxmaton result. Theorem 4.1. Let the set-valued functon F : [0, 1] K(R n ) be Hölder contnuous wth exponent ν (0, 1], haus(f(x), F(z)) C ν x z ν, x, z [0, 1]. Let F 0 = F(h), = 0, 1,..., N wth hn = 1, and F 0 = {0} otherwse. Then for any x [h(m 1), 1] ( m ) haus(s m, h F(x), F(x)) C ν h ν. (16) Proof. For x [(m 1)h, 1], let l x Z be such that x [l x h, (l x + 1)h). Note that for such x, the value S m, h F(x) depends on values F 0 for {l x m + 1, l x m + 2,..., l x } {0, 1,..., N}. By the trangle nequalty we have haus(s m, h F(x), F(x)) haus(s m, h F(x), F 0 l x ) + haus(f 0 l x, F(x)) Hence by Lemma 4.2 we obtan haus(s m, h F(x), F(x)) d 0m 2 + haus(f 0 l x, F(x)) (17) Now, by the Hölder contnuty of F, d 0 C ν h ν and haus(fl 0 x, F(x)) = haus(f(l x h), F(x)) C ν h ν Ths together wth (17) leads to the clam of the theorem. 7

8 Example 4.1. We construct Schoenberg approxmatons to the multfuncton F(x) defned by F(x) = { y : max{0, (r/2) 2 (x 0.5) 2 } y 2 r 2 (x 0.5) 2}, r = 0.5, x [0, 1]. (18) (a) Approxmaton wth S 3, h F. The orgnal set-valued functon s presented n gray on the left-hand sde of Fgure 4.1, 40 cross-sectons of the reconstructed shape, S 3, 0.01 F, s depcted n black. The graph of e h (x) = haus(s 3,h F(x), F(x)) at x = as functon of h, s shown on the rght-hand sde of Fgure 4.1. (a) (b) Fgure 4.1. (a) F - n gray. Forty cross-sectons of S 3, 0.01 F - n black. (b) Error between the orgnal and the reconstructed cross-sectons at x = as functon of h. We note that e h (0.425) changes almost lnearly wth h. Ths s n accordance wth Theorem 4.1, snce at x = F s Lpschtz contnuous (ν = 1). The graph of the maxmal error between cross-sectons of the reconstructed shape, S 3, h F and the correspondng cross-sectons of (18) as a 8

functon of h s presented n Fgure 4.2 (a). The maxmal error s obtaned at the ponts of change of topology of the cross-sectons of (18), whch are depcted n Fgure 4.2 (b).

9 functon of h s presented n Fgure 4.2 (a). The maxmal error s obtaned at the ponts of change of topology of the cross-sectons of (18), whch are depcted n Fgure 4.2 (b). To verfy that the decay of the error n ths fgure s n accordance wth Theorem 4.1, we show that F n (18) s Hölder contnuous wth exponent 1/2, at ponts of change of topology. (a) Fgure 4.2. (a) Maxmal error between the orgnal and the reconstructed cross-sectons as functon of h. (b) Ponts of change of topology, where the maxmal error s attaned. (b) Consder the boundary of the rng n 2D determned by (18). Locally near the ponts of change of topology of cross-sectons the boundary can be descrbed by a scalar functon y = f(x), or by x = g(y). One can see easly that the dervatve of f tends to nfnty at ponts of change of topology (see Fgure 4.2 (b)). Let x = g(y) be the nverse functon of f and let (x 0, y 0 ) be a pont of topology change. Snce g (y 0 ) = 1/f (x 0 ) = 0, we get by the Taylor expanson of degree 2 of g(y) about y 0, g(y) g(y 0 ) = ( y) 2 g (y 0 ) 2 + R 3, y = y y 0. Thus for x x 0 =h and snce R 3 / y 2 = o( y ), we obtan y 2h/g (y 0 ), from whch t can be concluded that F s Hölder contnuous wth exponent 1/2 at the ponts of change of topology. 9

(b) Approxmaton wth S 4, h F. (a) Fgure 4.3. (b) (a) F - n gray. Forty cross-sectons of S4, 0.01 F - n black. (b) Error between the orgnal and the reconstructed cross-sectons at x = 0.

10 (b) Approxmaton wth S 4, h F. (a) Fgure 4.3. (b) (a) F - n gray. Forty cross-sectons of S4, 0.01 F - n black. (b) Error between the orgnal and the reconstructed cross-sectons at x = as functon of h. Fgure 4.3 s smlar to Fgure 4.1 but wth S 4, h F replacng S 3, h F. It s easy to observe that the behavor of the error functon s almost quadratc n h. We conjecture that F s smooth enough at x = n a sense yet to be defned, and that ẽ h (x) = haus(f(x), S 4, h F(x)) = O(h 2 ), (19) n ponts of smoothness of F. Moreover, we conjecture that (19) holds for S 2 em, h F for all m 2. Ths s an mprovement over the approxmaton rate n Theorem 4.1, as n the case of real-valued functons. 10

11 5 Bernsten polynomals of real-valued functons and ther evaluaton by repeated bnary averages. For f C[0, 1], the Bernsten polynomal of degree m s B m (f, u) = m =0 ( ) ( ) m u (1 u) m f. (20) m The value B m (f, u) can be calculated recursvely by usng the de Casteljau algorthm [9] n terms of repeated bnary averages. The algorthm s based on the followng recurrence relaton, B,m (u) = (1 u)b,m 1 (u) + u B 1,m 1 (u), (21) ( ) m where B,m (u) = u (1 u) m. B m (f, u) n (20) for u [0, 1] can be presented by a repeated applcaton of (21) as: B m (f, u) = m =0 ( ) m u (1 u) m f 0 = m k =0 ( ) u (1 u) m k f k, (22) wth the values f k gven recursvely by f k = (1 u)f k 1 and wth f 0 = f(/m), = 0, 1,..., m. + u f+1 k 1, = 0, 1,...,, k = 1,..., m, (23) Comparng formulas (23) wth formulas (5) one can easly see that the de Boor algorthm s a generalzaton of the de Casteljau algorthm. Takng k = m n (22) we obtan B m (f, u) = f0 m. Thus the Bernsten polynomal of a real-valued functon can be defned by repeated bnary averages. 6 Bernsten operators for set-valued functons Let F : [0, 1] K(R n ) be a set-valued functon wth compact mages. Let F 0 = F(/m) be the ntal cross-sectons, F 0 K(R n ), = 0, 1,..., m. Consder the Bernsten polynomal of a set-valued functon, havng the form of 11

12 the Bernsten polynomal of a real-valued functon wth sums of numbers replaced by Mnkowsk sums of sets, m ( ) ( ) m Bm M (F, u) = u (1 u) m F (24) m =0 It s shown n [5] that the lmt of B M m (F, u), for a fxed u (0, 1), when m, s the convex hull of F(u). Therefore, the set-valued polynomal (24) s a good approxmaton for functons wth convex compact mages. To obtan an operator, whch does not convexfy the ntal data, we defne constructvely the Bernsten approxmaton of F n terms of the de Casteljau algorthm wth the metrc average as the basc bnary operaton. Thus to calculate the value of the Bernsten polynomal of degree m at the pont u [0, 1], B m (F, u), we use the followng extenson of (23): and defne Frst we show, F k = F k 1 1 u F k 1 +1, = 0, 1,...,, k = 1,..., m (25) B m (F, u) = F m 0. (26) Lemma 6.1. Let F k = {F k, = 0,..., } be defne as above, and let Then d k = Proof. From (25) and (2) In the same way we obtan sup haus(f k Z T 1, Fk ), k = 0, 1,..., m 1. (27) [1,m k] d k d 0, k = 1,..., m 1. haus(f k, F k 1 ) = haus(f k 1, F k 1 1 u F k 1 = u haus(f k 1 +1 ), F k 1 +1 ) u dk 1. haus(f k 1, F 1) k = haus(f k u F k 1, F k 1 ) = (1 u) haus(f k 1 1, F k 1 ) (1 u) d k 1. Now, by the trangle nequalty, (28) and (29) we get, Thus haus(f 1 k, F k k 1 ) haus(f, F 1 k k 1 ) + haus(f, F k (1 u)d k 1 + u d k 1 = d k 1. d k d k 1, whch mples the clam of the lemma. 12 ) (28) (29)

13 We do not have a proof of the convergence of B m (F, u) to F(u) as m. Yet we have a proof n the case of set-valued functons wth crosssectons n R all of the same topology. Our proof s based on the followng result from [10]: Result 6.1. For F : [0, 1] Co(R n ) Lpschtz contnuous haus(b M m (F, u), F(u)) C/ m, u [0, 1], where B M m (F, u) s defned by (24) and the constant C depends only on the Lpschtz constant of F. Any set A n R conssts of a number of dsjont ntervals, some possbly wth empty nteror. Thus A can be wrtten n the form A = J j=1 A j wth A j, j = 1,..., J ordered and dsjont ntervals, namely a j < a j+1 for any a j A j and a j+1 A j+1, j = 1,..., J 1. We denote ths by A 1 <... < A J. We ntroduce a measure of separaton of such a set wth J > 1: s(a) = nf {dst(a, A j) : a A l } (30) l,j {1,...,J},l j In the followng we assume that J s fnte. We dscuss only the case J > 1, snce J = 1 s a specal case of Result (6.1). Defnton 6.1. Two sets A, B K(R) are called topologcally equvalent f each s a unon of the same number of dsjont ntervals, namely A = J A j, B = j=1 J B j, (31) wth A j, j = 1,..., J and B j, j = 1,..., J dsjont ordered ntervals. Defnton 6.2. Let A, B K(R) be topologcally equvalent. The sets A, B are called metrcally equvalent f j=1 Π B (A j ) B j and Π A (B j ) A j, j = 1,..., J. (32) Ths relaton between the two sets s denoted by A B. Lemma 6.2. Let A, B K(R) be topologcally equvalent. If haus(a, B) < then A and B are metrcally equvalent. mn(s(a), s(b)) 2 (33) 13

14 Proof. Assume the opposte,.e. that (33) holds, but A, B are not metrcally equvalent, namely there exsts a subset B l B such that two ponts from B l have ther closest ponts n A n two subsets of A, say A j and A j+1. By the contnuty of the projecton mappng there exsts a pont b B l such that {a 1, a 2 } Π A ( b), a 1 A j, a 2 A j+1. By the trangle nequalty, dst(a 1, a 2 ) dst( b, a 1 ) + dst( b, a 2 ) = 2 dst( b, A). (34) Now, by the defnton of the Hausdorff dstance, (34) and (30) we obtan: haus(a, B) dst( b, A) 1 2 dst(a 1, a 2 ) 1 2 s(a) n contradcton to assumpton (33). Thus Π A (B l ) A j, and by symmetry Π B (A j ) B k. It remans to prove that k = l. Let a A j and b l B l be such that a Π A (b l ). Let b k B k be such that b k Π B (a). By the trangle nequalty and by the defnton of the Hausdorff dstance dst(b l, b k ) dst(b l, a) + dst(a, b k ) dst(b l, A) + dst(a, B) 2 haus(a, B). (35) Now by (30) we have f k l that s(b) dst(b l, b k ). Ths together wth (35) contradcts (33). Hence Π A (B l ) A j and Π B (A j ) B l. Snce A and B are both of the form (31), we conclude that l = j. Thus A B. Corollary 6.1. The metrc average of two topologcally equvalent sets A and B, satsfyng (33), s gven by A t B = J A j t B j. (36) j=1 Lemma 6.3. Let {F 0 R, = 0, 1,..., m} be topologcally equvalent, of the form F 0 = J j=1 F 0,j, 14

15 wth F 0,j, j = 1,..., J dsjont ordered ntervals. Defne {F k } and d k by (25) and (27) respectvely, and defne If d 0 < s 0 /2, then s k = mn{s(f k ) : = 0, 1,..., }, k = 0, 1,..., m 1. (37) d k < s k /2, k = 1,..., m 1, (38) and the sets {F k, = 0,...,, k = 0,..., m} are topologcally equvalent. Proof. We prove the lemma by nducton. We assume that the sets {F l : = 0,..., m l, l = 0,..., k 1} are topologcally equvalent and that d k 1 s k 1 /2. Note that the nducton hypothess s satsfed for k = 1. Snce two consecutve sets of {F k 1 } are metrcally equvalent, by the nducton hypothess and by Lemma 6.2, we get by Corollary 6.1 that F k = J j=1 F k 1,j 1 u F k 1 +1,j, = 0,...,. (39) Now, by property 5 of the metrc average (see Secton 2) F k 1,j where I,j k s an nterval. Frst we show that 1 u F k 1 k 1 +1,j = (1 u)f,j + u F k 1 +1,j = Ik,j, (40) σ k = mn{ c 1 c 2 : c 1 I k,j, c 2 I k,j+1, j {1,..., J 1}} s k 1 (41) Let σ k = c 1 c 2. By (39) and (40), we have c l = (1 u)a l + u b l, l = 1, 2 wth a 1 F k 1, j, b 1 F k 1 +1, j j {1,..., J 1}. Thus, and a 2 F k 1, j+1, b 2 F k 1 +1, j+1 for some c 1 c 2 = (1 u)(a 1 a 2 ) + u(b 1 b 2 ). Snce the dfferences (a 1 a 2 ) and (b 1 b 2 ) have the same sgn, we can wrte c 1 c 2 = (1 u) a 1 a 2 + u b 1 b 2 ). Fnally, usng (30) we obtan: σ k = c 1 c 2 = (1 u) a 1 a 2 + u b 1 b 2 (1 u) s(f k 1 ) + u s(f k 1 k 1 ) mn(s(f ), s(f k 1 )) sk 1 +1 It follows from (41) that I, k j I k, j+1 = for j {1,..., J 1}, and n vew of (39) and (40) we conclude that F k s topologcally equvalent to F k 1, F k

16 Moreover by (41) s k = mn σ k, and s k s k 1. Ths together wth Lemma 6.1 and the nducton hypothess leads to d k d k 1 < s k 1 /2 s k /2. Thus the nducton hypothess holds for k whch concludes the proof of the lemma. Lemmas 6.3 and 6.2 lead to Corollary 6.2. Let the sets {F k : = 0,...,, k = 0,..., m 1} be as n Lemma 6.3. Then F k F 1 k, = 1,...,, k = 0,..., m 1. Now we can prove, Theorem 6.1. Let the set-valued functon F : [0, 1] K(R) be Lpschtz contnuous, such that for each t [0, 1], F(t) = J j=1 F j(t), wth J > 1, where {F j (t)} are dsjont ordered ntervals. Then for m large enough haus(b m (F, u), F(u)) C/ m, u [0, 1]. (42) Proof. Let m be such that for F 0 = F(/m), = 0,..., m, wth d 0 defned by (27) and d 0 < s /2, s = nf s(f(t)) > 0. 0 t 1 Such m exsts snce F s Lpschtz contnuous. In fact m has to be large enough. Obvously s s 0, where s 0 s defned n (37). Thus d 0 s 0 /2. Now, by Corollary 6.1 and Property 5 of the metrc average we get Therefore B m (F, u) = J Bm M (F j, u). j=1 haus(f(u), B m (F, u)) = max 1 j J haus(f j(u), B M m (F j, u)), and (42) follows from Result

Example 6.1. To llustrate Theorem 6.1, we consder the functon F(x) defned by F(x) = { {y : 1 y 0.06x 2 + 2} {y : 0.1x 2 + 2.5 y 13.5} }, x [0, 10].

17 Example 6.1. To llustrate Theorem 6.1, we consder the functon F(x) defned by F(x) = { {y : 1 y 0.06x 2 + 2} {y : 0.1x y 13.5} }, x [0, 10]. (43) Ths functon s depcted n gray n (a), (b), (c) of Fgure 6.1. Ffty crosssectons of the reconstructed shapes, B 12 (F, u), B 13 (F, u) and B 30 (F, u), are colored by black and presented n (a), (b) and (c) of Fgure 6.1 respectvely. Note that (33) does not hold for m = 12, whle for m = 13 and m = 30 (33) holds. Fgure 6.1 shows that for m = 12 there s no approxmaton, whle B 13 (F, u) s already approxmatng the shape. The approxmaton by B 30 (F, u) s better than that by B 13 (F, u). (a) (b) (c) Fgure 6.1. (a) F(x) - n gray. Ffty cross-sectons of B 12 (F, u) - n black. (b) F(x) - n gray. Ffty cross-sectons of B 13 (F, u) - n black. (a) F(x) - n gray. Ffty cross-sectons of B 30 (F, u) - n black. 7 Concluson We expect that the approxmaton methods studed n ths paper wll become useful for practcal applcatons. For ths, an effectve algorthm for the evaluaton of the metrc average s needed. An algorthm for computng the metrc 17

18 average of two compact sets n R, whch has lnear complexty n the total number of ntervals, s presented n [2]. Ths algorthm can be appled to the reconstructon of 2D shapes from ther 1D cross-sectons. The computaton of the metrc average of compact sets n R 2, requred for the reconstructon of 3D objects from ther 2D cross-sectons, s much more complcated. As a frst attempt, [7] presents an algorthm for the computaton of the metrc average of two ntersectng convex polygons havng lnear complexty n the number of vertces of the two polygons. Ths algorthm s generalzed for the case of two ntersectng regular polygons, but wth quadratc computaton tme [8]. The authors stpulate that the lack of a general approxmaton result n the case of the Bernsten operators n contrast to the cases of the Schoenberg operators and splne subdvson operators [4] s due to the global nature of the Bernsten operators. In the Bernsten operators the approxmaton at a pont depends on values of the approxmated functon over all the nterval of approxmaton, whle n the two other operators t depends on a fnte number of samples of approxmated functon near the pont. Ths falure of the adaptaton method, based on the metrc average, lead the authors to extend the metrc average to a new set-operaton actng on a fnte sequence of compact sets. Wth ths operaton, most known approxmaton methods for real-valued functons, are adapted to set-valued functons successfully [6]. Yet at ths stage the results are manly theoretcal. References [1] Z.Artsten, Pecewse lnear approxmatons of set-valued maps, Journal of Approxmaton Theory 56, (1989). [2] R. Baer, N. Dyn, E. Farkh, Metrc averages of 1D compact sets, n Approxmaton theory X, C. Chu, L. L. Schumaker and J. Stoeckler (eds.), Vanderblt Unv. Press. Nashvlle, TN, 9-22 (2002). [3] C. de Boor, A practcal gude to splne. Sprnger-Verlag (2001) [4] N. Dyn, E. Farkh, Splne subdvson schemes for compact sets wth metrc averages, n Trends n Approxmaton Theory, K.Kopotun, T.Lyche and M.Neamtu (eds.), Vanderblt Unv. Press, (2001). [5] N. Dyn, E. Farkh, Set-valued approxmatons wth Mnkowsk averages - convergence and convexfcaton rates, Numercal Functonal Analyss and Optmzaton 25, (2004). 18

19 [6] N. Dyn, E. Farkh, A. Mokhov, Approxmatons of Set-Valued Functons by Metrc Lnear Operators, submtted. [7] N. Dyn, E. Lpovetsk, An effcent algorthm for the computaton of the metrc average of two ntersectng convex polygons, wth applcaton to morphng, to appear n Advances n Computatonal Math. [8] N. Dyn, E. Lpovetsk, An algorthm for the computaton of the metrc average of two ntersectng regular polygons, prvate communcaton. [9] H.Prautzsch, W.Boehm and M.Paluszny, Bezer and B-Splne Technques, Sprnger (2002). [10] R.A. Vtale, Approxmaton of convex set-valued functons, Journal of Approxmaton Theory 26, (1979). 19

Approximation of Univariate Set-Valued Functionsan Overview

Approximation of Univariate Set-Valued Functionsan Overview 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