Approximation of fuzzy numbers by nonlinear Bernstein operators of max-product kind

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1 EUSFLAT-LFA 20 July 20 Aix-les-Bais, Frace Approximatio of fuzzy umbers by oliear Berstei operators of max-product kid Lucia Coroiau Sori G. Gal 2 Barabás Bede 3,2 Departmet of Mathematics ad Computer Sciece, The Uiversity of Oradea, Uiversitatii, 40087, Oradea, Romaia, s : lcoroiau@uoradea.ro, 2 galso@uoradea.ro 3 Departmet of Mathematics, The Uiversity of Texas-Pa America, 20 West Uiversity, Ediburg, Tx, 78539, USA, bedeb@utpa.edu Abstract I this paper firstly we exted from [0, ] to a arbitrary compact iterval [a, b], the defiitio of the oliear Berstei operators of max-product kid, (f, N, by provig that their order of uiform approximatio to f is ω (f,/ ad that they preserve the quasi-cocavity of f. Sice (f geerates i a simple way a fuzzy umber of the same support [a, b] withf, it turs out that these results are very suitable i the approximatio of the fuzzy umbers. Thus, besides the approximatio properties, for sufficietly large, we prove that these oliear operators preserve the o-degeerate segmet core of the fuzzy umber f ad, i additio, the segmet cores of (f, N, approximate the segmet core of f with the order /. Keywords: fuzzy umbers, oliear Berstei operator of max-product kid. Itroductio Recetly, may papers made ivestigatios o the approximatio of fuzzy umbers by trapezoidal or triagular fuzzy members (see []-[3], [5]-[6], [2], [6]-[9], [24]-[27], [29] ad by o-liear side fuctios (see [4], [7], [20], [23], [28]. The mai aim of this ote is to use the so-called Berstei operator of max-product kid, firstly itroduced (ad formally studied i the book [4], p ad completely studied i the papers [9], [8], [3], for approximatig fuzzy umbers with cotiuous membership fuctios. The study of max-product type operators started with the papers [0]-[], where the authors itroduced ad studied the Shepard oliear operator of max-product kid. For a positive cotiuous fuctio f :[0, ] R, the max-product Berstei operator was defied i [4], by (f(x = p,k(xf(k/ p,x [0, ] (,k(x where p,k (x = ( k x k ( x k. Here meas maximum. 20. The authors - Published by Atlatis Press 734 Notice that the max-product Berstei operator is obtaied from the liear Berstei polyomial writte i the form (f(x = p,k(xf(k/ p,,k(x replacig the sum operator by the maximum operator. As it was proved i [8], [9], (f isacotiuous oliear (more exactly subliear o the space of positive fuctios operator, well-defied for all x R, ad a piecewise ratioal fuctio o R. I additio we have (f(0 = f(0 ad (f( = f(. Also, i [8] it was proved that (f preserves the mootoicity ad the quasicovexityof f o [0, ] ad that the order of uiform approximatio by (f is 2ω (f,/ +, where for g :[0, ] R ad δ>0, we take ω (f,δ =sup{ g(x g(y : x, y [0, ], x y δ}. The, i the paper [3] it was proved that the order of uiform approximatio i the whole class C + ([0, ] of positive cotiuous fuctios o [0, ], caot be improved, i the sese that there exists a fuctio f i C + ([0, ], for which the approximatio order by the max-product Berstei operator is exactly ω (f,/. However, for the class of strictly positive Lipschitz fuctios, a Jackso type estimate, ω (f,/, was obtaied. Fially, i the same paper it was proved that preserves the quasi-cocavity too. Now, sice the restrictio of a cotiuous fuzzy umber to its compact support is a quasi-cocave fuctio, aturally it is suggested that could be used to approximate a fuzzy umber (more correctly the restrictio of a fuzzy umber to its support. O the other had, sice preserves the mootoicity, we ca use the operator to approximate fuzzy umbers give i the parametric LU-form too. The pla of the paper goes as follows. I Sectio 2 we defie the Berstei max-product operator o a compact iterval [a, b] ad prove that the approximatio ad shape preservig properties oe trasfer from the case of the [0, ] iterval.

2 I Sectio 3, we discuss all the aspects coected to the approximatio of fuzzy umbers by these oliear operators. Fially, we preset o cocrete examples, two graphs which illustrate how the max-product Berstei operators approximate a fuzzy umber ad what advatages preset with respect to the approximatio of fuzzy umbers by the associated liear Berstei polyomials. 2. Berstei max-product operators defied o compact itervals From ow oe, through out this paper, we deote by C(I adc + (I respectively, the space of cotiuous fuctios defied o a iterval I ad the space of positive cotiuous fuctios defied o I respectively. For a fuctio f C + ([a, b], we defie the correspodig max-product Berstei operator by (f(x = p,k(xf(a + k b a p,x [a, b],k(x where p,k (x = ( ( k ( k x a k b a b x b a. Sice p,k(x = for all x [a, b], it is immediate that that B p,k(x > 0 for all x [a, b] whichmeas (f is well defied. Also, by simple calcu- (f(a =f(aad (f(b = latios we get f(b. The, sice the maximum of a fiite umber of cotiuous fuctios is a cotiuous fuctio, we get that for ay f C + ([a, b], (f C + ([a, b] too. Actually, if f :[a, b] R + is bouded (ot ecessarily cotiuous the oe ca easily prove that (f C + ([a, b]. I this sectio we will prove that : C + ([a, b] C + ([a, b] has the same order of uiform approximatio as the liear Berstei operator ad that it preserves the quasi-cocavity too. First we eed the followig results ad defiitios. Theorem ([8], Theorem 4.. If f :[0, ] R + is cotiuous the we have the estimate B (f(x f(x 2ω (f; for all N,x [0, ]., + Theorem 2 ([3], Theorem 5.. Let us cosider the fuctio f :[0, ] R + ad let us fix N,. Suppose, i additio, that there exists c [0, ] such that f is odecreasig o [0, c] ad oicreasig o [c, ]. The, there exists c [0, ] such that (f is odecreasig o [0,c ] ad oicreasig o [c, ]. I additio we have c c + ad (f(c f(c ω (f, Defiitio 3 Let f :[a, b] R be cotiuous o [a, b]. The fuctio f is called: (i quasi-covex if f(λx +( λy max{f(x,f(y}, for all x, y [a, b],λ [0, ], (see, e.g. [4], page 4, (iv; (ii quasi-cocave, if f is quasi-covex. Remark 4 By [22], the cotiuous fuctio f is quasi-covex o [a, b] equivaletly meas that there exists a poit c [a, b] such that f is oicreasig o [a, c] ad odecreasig o [c, b]. From the above defiitio, we easily get that the cotiuous fuctio f is quasi-cocave o [a, b], equivaletly meas that there exists a poit c [a, b] such that f is odecreasig o [a, c] ad oicreasig o [c, b]. We ca ow preset the mai results of this sectio. Theorem 5 If a, b R, a<bad f :[a, b] R + is cotiuous, the we have the estimate (f(x f(x 2([b a]+ω (f;, + for all N,x [a, b]. Here ω deotes the modulus of cotiuity of f o [a, b]. Proof. Let us cosider the fuctio g :[0, ] R, g(y =f(a +(b ay. It is easy to check that g(k/ =f(a+k b a for all k {0,,..., }. Now, let us choose arbitrary x [a, b] adlety [0, ] be such that x = a +(b ay. This implies y = (x a/(b aad y =(b x/(b a. From these equalities ad otig the expressios for g(k/ we obtai (f(x = (g(y. By Theorem we get (f(x f(x = (g(y g(y 2ω (g;. + Sice ω (g; b a + ω (f; + ad from the property ω (f; λδ ([λ] +ω (f; δ weobtai ω (g; ([b a] +ω (f; theorem is proved. + + ad the Theorem 6 Let us cosider the fuctio f : [a, b] R + ad let us fix N,. Suppose i additio that there exists c [a, b] such that f is odecreasig o [a, c] ad oicreasig o [c, b]. The, there exists c [a, b] such that B (f is odecreasig o [a, c ] ad oicreasig o [c,b]. I additio we have c c b a + ad B (f(c f(c ([b a]+ω (f, +.

3 Proof. We costruct the fuctio g as i the previous theorem. Let c [0, ] be such that g(c =c. Sice g is the compositio betwee f ad the liear odecreasig fuctio t a +(b at, we get that g is odecreasig o [0,c ] ad oicreasig o [c, ]. By Theorem 2 it results that there exists c [0, ] such that B (g is odecreasig o [0,c ], oicreasig o [c, ] ad i additio we have (g(c g(c ω (g, + ad c c +. Let c = a +(b ac. If x,x 2 [a, c ]withx x 2 the let y,y 2 [0,c ] be such that x = a+(b ay ad x 2 = a+(b ay 2. The, it follows that (f(x = (g(y ad (f(x 2 = (g(y 2. The mootoicity of (g implies (g(y (g(y 2, that is (f(x (f(x 2. We thus obtai that (f is odecreasig o [a, c ]. Usig the same type of reasoig we obtai that (f is oicreasig o [c,b]. For the rest of the proof, otig that c c + we get c c = (b a(c c b a +. Fially, otig that (g(c g(c ω (g, ad takig ito + accout that ω (g, + ([b a]+ω (f, we obtai (f(c f(c = (g(c g(c ad the proof is complete. ω (g, + ([b a]+ω (f, +, + Remark 7 From the above theorem ad by Remark 4, it results that if f :[a, b] R + is cotiuous ad quasi-cocave the (f is quasi-cocave too. Remark 8 As we have metioed i the Itroductio, for fuctios i the space C + ([0, ], preserves the mootoicity ad the quasi-covexity. Reasoig similar as i the proof of Theorem 6, it ca be proved that these preservatio properties hold i the geeral case of the space C + ([a, b]. 3. Applicatios i the approximatio of fuzzy umbers Defiitio 9 A fuzzy umber u is characterized by a upper semicotiuous fuctio µ u : R [0, ] with the followig properties: (i There exists a, b R, a b such that µ u (x = 0 outside [a, b]. (ii There exists c, d R, c d such that: (ii µ u is odecreasig o [a, c]; (ii 2 µ u (x =for all x [c, d]; (ii 2 µ u is oicreasig o [d, b]. The set {x R : µ u (x =} is called the core of u ad it is deoted by core(u. The closure of the set {x R : µ u (x > 0} is called the support of u ad it is deoted by supp(u. From the above defiitio it is immediate that supp(u is a bouded iterval. If µ u is cotiuous ad supp(u =[a, b], core(u = [c, d], the we ecessarily have a < c d < b. For simplicity, from ow oe we will use the same otatio for a fuzzy umber ad for its membership fuctio. We eed the followig auxiliary results. Lemma 0 Let a, b R, a < b. For N, k, j {0,,..., } ad x (a+j b a b a +,a+(j + +, let m k,,j (x = p,k(x p,j (x. The m k,,j (x < for all j {0,,..., } ad k {0,,..., }\{j}. Proof. Without ay loss of geerality we may suppose that a =0adb =, because usig the same reasoig as i the proof of Theorems 5-6 we easily obtai the coclusio of the lemma i the geeral case. So, let us fix x (j/( +, (j + /( +. Accordig to Lemma 3.2. i [8] we have m 0,,j (x m,,j (x... m j,,j (x m j,,j (x m j+,,j (x... m,,j (x Sice m j,,j (x =, it suffices to prove that m j+,,j (x < adm j,,j (x <. By direct calculatios we get m j,,j (x m j+,,j (x = j + j x x. Sice the fuctio g(y =( y/y is strictly decreasig o the iterval [j/( +, (j +/( +], it results that x x > (j +/( + (j +/( + = j j +. Clearly, this implies m j,,j (x/m j+,,j (x >, that is m j+,,j (x <. By similar reasoigs we get that m j,,j (x < ad the proof is complete. Lemma If a, b R, a<bad f :[a, b] R + is bouded the, for all j {0,,..., }, we have (f(a + j(b a/ f(a + j(b a/. B Proof. From Lemma 0, sice a + j(b a/ (a + j(b a/( +,a+(j +(b a/( + ad sice m k,,j (a + j(b a/ = p,k(a+j(b a/ p,j(a+j(b a/ for all k {0,,..., }, it follows that p,k(a + j(b a/ =p,j (a + j(b a/. The, we have 736

4 (f(a + j(b a/ = p,k(a + j(b a/f(a + k(b a/ p,j (a + j(b a/ p,j(a + j(b a/f(a + j(b a/ p,j (a + j(b a/ = f(a + j(b a/ ad the lemma is proved. Now, suppose that u is a fuzzy umber such that supp(u = [a, b] ad core(u = [c, d]. For N we itroduce the fuctio (u : R [0, ], (u(x = 0 for all x outside [a, b] ad (u(x = (u(x for all x [a, b]. From Theorem 5, it results that the order of uiform approximatio of the fuzzy umber u by (u is ω (u, /. The, sice the restrictio of u o the iterval [a, b] is a fuctio like those cosidered i Theorem 6, it results that (u isaquasicocave fuctio o [a,b]. Moreover, we have the followig. Theorem 2 Let u be a fuzzy umber with supp(u = [a, b] ad core(u = [c, d] such that a c < d b. The for sufficietly large, it results that (u is ( a fuzzy umber such that : (i supp(u =supp (u ; (ii If core( (u = [c,d ],the c c b a ad d d b a ; (iii If, i additio, u is cotiuous o [a, b], the (u(x u(x 2([b a]+ω (u;, + for all x R. Proof. Let N, such that b a <d c. By Theorem 6 it follows that there exists c [a, b] such that (u is odecreasig o [a, c ] ad oicreasig o [c,b]. O the other had, from the defiitio of (u, it results that (u u ad sice u =, it follows that (u. (Here deotes the uiform orm o B([a, b]- the space of bouded fuctios o [a, b]. Therefore, to prove that (u is a fuzzy umber, it suffices to prove the existece of α [a, b] such that (u =. Let α = a + j(b a/ where j is chose such that c<α<d.such j exists sice b a <d c. Sice α core(u, it results u(α =. O the other had, by Lemma it follows that (u(α u(α ad clearly this implies that (u is a fuzzy umber. I what follows we prove puctually the rest of the theorem. (i As we have metioed at the begiig of Sectio 2, (u(a =u(a ad Notig the defiitios of u ad (u(b =u(b. (u, it fol- (u(x = 0 outside [a, b]. Now, sice lows that 737 u(x > 0 ad (u(x = (u(x for all x (a, b, we easily get that (u(x > 0for all x (a, b, which proves (i. (ii Sice u is odecreasig o the iterval [a, c] ad oicreasig o the iterval [c, b], by Theorem 6 it follows that there exists c ( [a, b] such that (u is odecreasig o the iterval [a, c (] ad odecreasig o the iterval [c (,b] ad, i additio, we have c c ( (b a/( +. O the other had, u is odecreasig o the iterval [a,d] ad oicreasig o the iterval [d, b]. It follows that there exists d ( [a, b] such that (u is oicreasig o the iterval [a, d (] ad odecreasig o the iterval [d (,b] ad, i additio, we have d d ( (b a/( +. For N satisfyig (b a/ < (d c/2, it results that c ( < d ( ad this implies that (u is costat o the iterval [c (,d (], that is (u(x = for all x [c (,d (]. This implies that [c (,d (] core( (u, that is c( c ( ad d ( d(. But we ecessarily have c( > a + j (b a/ ad d( <a+ j 2 (b a/ where j ad j 2 are chose such that a + j (b a/ < c a +(j +(b a/ ad a +(j 2 (b a/ d<a+ j 2 (b a/. To prove the first statemet, let us observe that (see the proof of Lemma p,k(a + j (b a/ =p,j (a + j (b a/, which implies that (u(a + j (b a/ = p,k(a + j (b a/u(a + k(b a/. p,j (a + j (b a/ Let k {0,,..., } be such that p,k(a + j (b a/u(a + k(b a/ =p,k (a + j (b a/u(a + k (b a/. If k = j the we get (u(a + j (b a/ =u(a + j (b a/ ad sice a + j (b a/ / core(u, we obtai that (u(a + j (b a/ <. If k j the By Lemma 0, it results m k,,j (a + j (b a/ < ad sice (u(a + j (b a/ =m k,,j (a + j (b a/ u(a+k (b a/ we reach to the same coclusio, that is (u(a + j (b a/ <. Now, sice a + j (b a/, c( [a, d (] (this is immediate sice for (b a <(d c/2 wehave c<d ( ad sice (u is odecreasig o [a, d (] ad otig that (u(a+j (b a/ < (u(c, it follows that we ecessarily have c( > a + j (b a/. The proof of the statemet d( <a+ j 2 (b a/ is similar ad therefore we omit the details. Havig i mid the above iequalities, we obtai c (b a/ < c( c ( c +(b a/( +. This clearly implies that c( c (b a/. By similar reasoigs we obtai that d( d (b a/ ad the proof of statemet (ii is complete. (iii The proof is immediate by Theorem 5, takig

5 (u ito accout the cotiuity of u. Remarks (i If the fuzzy umber u is uimodal, that is c = d, the (u is ot ecessarily a fuzzy umber. But if we ormalize (u, the we obtai the fuzzy umber (u. (Recall that deotes the uiform orm. (u u uiformly, we easily get that (u (u u, Sice uiformly. Or, for N we itroduce the fuzzy umber u as follows. First, we choose k(c, such that a +(b a k(c, ( + (k(c, + c a +(b a. ( + For x outside the iterval (a +(b a (k(c, /( +,a+(b a (k(c, +2/( +, we take u (x =u(x. For x [a +(b a k(c, /( +,a+(b a (k(c, +/(+] we take u (x =. Fially, i the missig itervals we take liear fuctios so that the cotiuity of u is esured. I additio, it follows that there exists a costat C idepedet of, such that ω (u ; Cω (u;. + + Ideed, it is clear that it suffices to compare the two moduli oly o oe of the two subitervals (each of them of legth (b a/( + where u (x isa liear fuctio. If ω (u ;/ + is attaied o the left-had side iterval, it easily follows that it is less tha u(c u[c 2(b a/( +] [2(b a+]ω (u;/( + [2(b a+]ω (u;/ +. If ω (u ;/ + is attaied i a iterval where u (x is ot etirely liear, by decomposig that iterval ito two cosecutive subitervals, such that o oe u (x is liear ad o the other oe coicides with u(x (by costructio, by the triagle iequality it easily follows that ω (u ;/ + ω (u;/ + +[2(b a+]ω (u;/ +. Now, sice a +(b a k(c, / core(u, it follows that u (a +(b a k(c, / =,whichby Lemma implies (u (a +(b a k(c, / =. Cosequetly, we get that (u is a proper fuzzy umber. Moreover, sice lim core(u = c, by Theorem 2, (ii, it results that lim (core (u = c. We prove ow that (u u, uiformly o [a, b]. We have (u (x u(x (u (x u (x + u (x u(x + 2([b a]+ω (u ; + u (x u(x 2C([b a]+ω (u; + + u (x u(x Sice u (x u(x 2([b a] +ω (u; +, we obtai 738 (u (x u(x 2C([b a]+ω (u; + + 2([b a]+ω (u; + ad this proves that (u u, uiformly o [a, b]. (ii From Theorem 2 it follows that the maxproduct Berstei operator,, is more coveiet for approximatig fuzzy umbers tha the classical liear Berstei operator,. While the order of uiform approximatio is the same, the max-product Berstei operator preserves better the shape of the approximated fuzzy umber. I fact, it is easy to prove that if the fuzzy umber u has a cotiuous membership fuctio, the as icreases to we have (u <. Of course, if we ormalize (u the we obtai a fuzzy umber (it is kow that the liear Berstei operator preserves the quasi-cocavity, see e.g. [2], but the core of the ormalized liear Berstei operator oe reduces to a poit which is icoveiet i the case whe the core of u is a proper iterval. (iii For practical cosideratios it is useful to study the problem of approximatig fuzzy umbers that are of Lipschitz-type. For example let us suppose that the fuzzy umber u is α-lipschitz o [a, b], of order α (0, ], i.e., u(x u(y M x y α, x, y [a, b] with some absolute costat M. By Theorem 2, (iii, we have (u(x u(x 2([b a]+m( + α/2, Now let ε>0 be arbitrary. The we have (2(b a+m( + α/2 <ε,

6 Figure : Approximatio of a fuzzy umber (solid lie by classical (dotted lie ad oliear Berstei operators (dashed lie of degree =30. Figure 2: Approximatio of a fuzzy umber (solid lie by classical (dotted lie ad oliear Berstei operators (dashed lie of degree =80. for ay 0 =[( C ε (2/α ]+, with C = (2(b a+m, where [ ] stads for the iteger part of x. Example 3 We approximate the fuzzy umber 4x 2 if 0 x</2 /2 x 3/4 u(x =. 4 4x 3/4 <x 0 otherwise usig both the classical ad the oliear maxproduct Berstei operators. I Figures ad 2 we ca compare the classical ad oliear maxproduct operators i approximatig the above fuzzy umber. Wecaeasilyseethattheclassicalliear operator marked with dotted lie is outperformed by the max-product operator marked with dashed lie, this beig almost coicidet with the target fuzzy umber at its core. The theoretical coclusios of the paper are well illustrated by this particular example. So far, i this sectio we studied fuzzy umbers give explicitly. It is kow that a fuzzy umber ca be represeted i parametric form too. I this case the fuzzy umber u is give by a pair of fuctios (u,u + whereu,u + :[0, ] R satisfy the followig requiremets: (i u is odecreasig; (ii u + is oicreasig; (iii u ( u + (. It is kow that for u = (u,u +, we have core(u = [u (,u + (] ad supp(u = [u (0,u + (0]. If u =(u,u + adv =(v,v + the (see [5] d 2 (u, v = ( u (α v (α 2 dα ( u + (α v + (α 2 dα 739 deotes the Euclidea distace betwee u ad v. For a fuzzy umber u = (u,u + we attache the Berstei max-product operators (u ad (u +. Sice preserves the mootoicity, it is immediate that (u is odecreasig ad (u + is oicreasig. I additio we have (u (0 = u (0, (u ( = u (, (v (0 = v (0 ad (v ( = v (. I coclusio we obtai that B (u = ( (u, (u + is a proper fuzzy umber which i additio preserves the core ad the support of u. We have the followig. Theorem 4 If u = (u,u + is a fuzzy umber such that u ad u + are cotiuous the d(u, B (u 2 ( 2max {ω u ; +,ω (u + ; }. + Proof. The proof follows from Theorem by applyig the mea value theorem for both itegrals i the expressio of d(u, B (u. Remark 5 Sice the liear Berstei operator preserves the mootoicity, coicides at the ed poits with the approximated fuctio ad has the same order of uiform approximatio, it follows that we obtai a similar estimatio as i the above theorem if istead of the Berstei max-product operator we use the liear Berstei operator. 4. Coclusios I this paper we proved that the Berstei maxproduct operators might be very useful tools i ap-

7 proximatig fuzzy umbers. They preserve the support ad almost etirely the core of the approximated fuzzy umber i the case whe the core of the fuzzy umber is a proper iterval. If the fuzzy umber u has a cotiuous membership fuctio, the the sequece (u covergestou with respect to the uiform orm ad moreover for sufficietly large, (u is a proper fuzzy umber. For uimodal fuzzy umbers, we preseted two methods of approximatio by the Berstei max-product operator. The approximatio properties together with the shape preservig properties, idicate that whe we approximate a fuzzy umber by the Berstei max-product operator, most of the iformatio associated to a fuzzy umber is maitaied, which is very importat i practice. The complexity of the calculus is at the same level as, for example, i the case of the liear Berstei operator. For fuzzy umbers give i parametric form with cotiuous sides, agai we obtai the covergece property with respect to the Euclidea distace. Ackowledgemet The cotributio of the first author was possible with the fiacial support of the Sectorial Operatioal Programme for Huma Resources Developmet , co-fiaced by the Europea Social Fud, uder the project umber POS- DRU/07/.5/S/7684 with the title Moder Doctoral Studies: Iteratioalizatio ad Iterdiscipliarity. Refereces [] S. Abbasbady, M.Amirfakhria, The earest approximatio of a fuzzy quatity i parametric form, Applied Mathematics ad Computatio, 72 (2006, [2] S. Abbasbady, M.Amirfakhria, The earest trapezoidal form of a geeralized left right fuzzy umber, Iteratioal Joural of Approximate Reasoig, 43 (2006, [3] A. I. Ba, Approximatio of fuzzy umbers by trapezoidal fuzzy umbers preservig the expected iterval, Fuzzy Sets ad Systems, 59 (2008, [4] A. I. Ba, O the earest parametric approximatio of a fuzzy umber-revisited, Fuzzy Sets ad Systems, 60 (2009, [5] A. I. Ba, Trapezoidal ad triagular approximatios of fuzzy umbers-iadverteces ad correctios, Fuzzy Sets ad Systems 60 (2009, [6] A. I. Ba, L. Coroiau, Cotiuity ad Liearity of the trapezoidal approximatio preservig the expected iterval operator, Iteratioal Fuzzy Systems Associatio World Cogres, July 2009, [7] A. I. Ba, L. Coroiau, Metric properties of the eraest exteded parametric fuzzy ymbers ad applicatios, Iteratioal Joural of Approximate Reasoig, 52 ( [8] Bede, B., Coroiau, L., Gal, S.G., Approximatio ad shape preservig properties of the Berstei operator of max-product kid, Iter. J. Math. ad Math. Sci., volume 2009, Article ID , 26 pages, doi:0.55/2009/ [9] B. Bede, S.G. Gal, Approximatio by oliear Berstei ad Favard-Szász-Mirakja operators of max-product kid, Joural of Cocrete ad Applicable Mathematics, 8 (200, No. 2, [0] B. Bede, H. Nobuhara, M. Daňková, A. Di Nola, Approximatio by pseudo-liear operators, Fuzzy Sets ad Systems, 59 (2008, [] B. Bede, H. Nobuhara, J. Fodor, K. Hirota, Max-product Shepard approximatio operators, Joural of Advaced Computatioal Itelligece ad Itelliget Iformatics, 0 (2006, [2] L. Coroiau, Best Lipschitz costat of the trapezoidal approximatio operator preservig the expected iterval, Fuzzy Sets ad Systems, 65 (20, [3] L. Coroiau, S. G. Gal, Improved Estimates i Approximatio by Berstei Operators of Max-product kid, Aalysis ad Applicatios (to appear. [4] S.G. Gal, Shape-Preservig Approximatio by Real ad Complex Polyomials, Birkhäuser, Bosto-Basel-Berli, [5] P. Grzegorzewski, Metrics ad orders i space of fuzzy umbers, Fuzzy Sets ad Systems, 97 (998, [6] P. Grzegorzewski, Nearest iterval approximatio of a fuzzy umber, Fuzzy Sets ad Systems, 30 (2002, [7] P. Grzegorzewski, Trapezoidal approximatios of fuzzy umbers preservig the expected iterval-algorithms ad properties, Fuzzy Sets ad Systems, 47 (2008, [8] P. Grzegorzewski, E. Mrówka, Trapezoidal approximatios of fuzzy umbers, Fuzzy Sets ad Systems, 53 (2005, [9] P. Grzegorzewski, E. Mrówka, Trapezoidal approximatios of fuzzy umbers-revisited, Fuzzy Sets ad Systems, 58 (2007, [20] E. N. Nasibov, S. Peker, O the earest parametric approximatio of a fuzzy umber, Fuzzy Sets ad Systems, 59 (2008, [2] R. Păltăea, The preservatio of the property of quasicovexity of higher order by Berstei polyomials, Revue d Aalyse Numér. Théor. Approx., 25 (996, o. -2, [22] T. Popoviciu, Deux remarques sur les fuctio covexes, Bull. Soc. Sci. Acad. Roumaie, 220

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