Max-Product Shepard Approximation Operators

Transcription

1 Max-Product Shepard Approxiation Operators Barnabás Bede 1, Hajie Nobuhara 2, János Fodor 3, Kaoru Hirota 2 1 Departent of Mechanical and Syste Engineering, Bánki Donát Faculty of Mechanical Engineering, Budapest Tech Népszinház u 8, H-1081 Budapest, Hungary, e-ail: bedebarna@bgkbfhu 2 Departent of Coputational Intelligence and Systes Science, Tokyo Institute of Technology, 4259 Nagatsuta, Midoriku, Yokohaa , Japan, e-ail: nobuhara@hrtdistitechacjp, 3 Institute of Intelligent Engineering Systes, Budapest Tech John von Neuann Faculty of Inforatics, Budapest Tech Bécsi út 96/b, H-1034 Budapest, Hungary, e-ail: fodor@bfhu Abstract In crisp approxiation theory the operations that are used are only the usual su and product of reals We propose the following proble: are su and product the only operations that can be used in approxiation theory? As an answer to this proble we propose ax-product Shepard Approxiation operators and we prove that these operators have very siilar properties to those provided by the crisp approxiation theory In this sense we obtain unifor approxiation theore of Weierstrass type, and Jackson-type error estiate in approxiation by these operators 1 Introduction The ain proble solved in crisp approxiation theory is to approxiate a function f : [a, b] R, where [a, b] is a real interval, by soe sipler function, eg trigonoetric) polynoial, rational function or wavelet Crisp approxiation theory provides any different approxiation operators: Bernstein polynoials, Shepard-type rational approxiation operators, trigonoetric polynoials of Fejér type, wavelets, see eg [3]) to ention only a few These operators are using exclusively su and product of reals as operations, and so, the linear algebra as underlying algebraic structure Usually, the for of such an operator is n Lf, x) = l n,i x) fx i ),

2 where x i [a, b] are the knots, i = 0,, n, and l n,i x) are functions having relatively siple expression polynoials, trigonoetric polynoials, rational functions, wavelets) The ain theores in crisp approxiation theory are the Weierstrass-type unifor approxiation theores, which state that any continuous function can be approxiated uniforly by operators of a given type, and error estiates which usually are given in ters of the odulus of continuity Let us reark also that the approxiation operators provided by crisp approxiation theory are all linear Max t-nor copositions play a very iportant role in fuzzy logic and they were extensively studied, ainly in view of fuzzy relational equations see eg [4]) Also, their applications to fuzzy control are well known see [8]) The approxiation capabilities of fuzzy systes ie the capability of a fuzzy syste to approxiate soe target function) are also well known [6], [11], [5], [1], [7]) These ideas lead us to propose the following question: Are su and product the only operations that can be used in approxiation theory? The answer is surely negative and in this sense we present ax-product Shepard approxiation operators, which use ax instead of su For these operators we obtain Weierstrass-type unifor approxiation theore and for the approxiation error we obtain Jackson-type error estiates in ters of the odulus of continuity Since the operations used in the construction of these operators are ax and product, the underlying algebraic structure is a ax-product algebra and the atheatical analysis on these structures ie the etric space structure) is usually called pseudo-analysis [9]) Let us reark that these aproxiation operators are nonlinear in contrast with the operators provided by crisp approxiation theory After a preliinary section we introduce in Section 3 the ax-product approxiation operators and we study approxiation properties of these operators Soe conclusions and further research topics conclude the paper 2 Preliinaries The purpose of this paper is to approxiate a target function f : X [0, ], where X, d) is an arbitrary copact etric space, [0, ] is endowed with ax and product as algebraic operations and the usual topology induced by the Euclidean distance over the reals So, the algebraic structure over [0, ] is the ax-product algebra If we endow the ax-product algebra with the topological structure induced by the Euclidean distance, we can use the tools of atheatical analysis The atheatical analysis over this algebraic-topological structure is called pseudoanalysis see [9]) The target function f : X [0, ] is assued to be continuous Usually, the error estiates in crisp approxiation theory are provided in ters of the odulus of continuity So, let us recall it s definition and ain properties adapted to our case for the general definition see [2]) Definition 1 Let X, d) be a etric spaces and [0, ], ) the etric space of positive reals endowed with the usual Euclidean distance Let f : X [0, ] be

3 a function Then the function ω f, ) : [0, ) [0, ), defined by ω f, δ) = { f x) f y) ; x, y X, dx, y) δ} is called the odulus of continuity of f Theore 2 The following properties hold true i) f x) f y) ω f, dx, y)) for any x, y X; ii) ω f, δ) is nondecreasing in δ; iii) ω f, 0) = 0; iv) ω f, δ 1 + δ 2 ) ω f, δ 1 ) + ω f, δ 2 ) for any δ 1, δ 2, [0, ); v) ω f, nδ) nω f, δ) for any δ [0, ) and n N; vi) ω f, λδ) λ + 1) ω f, δ) for any δ, λ [0, ); vii) If f is continuous then li δ 0 ω f, δ) = 0 In order to study approxiation properties of the operators defined later in this paper we need the following lea Lea 3 For any functions A, B : {0,, n} R + we have n Ai) Bi) Ai) Bi), for any n N Proof We observe that Ai) = Bi) + Ai) Bi) Bi) + Ai) Bi), inequality which, together wit the syetric case, iplies the stateent of the lea 3 Max-product Shepard approxiation operators Let f : X [0, ] be a continuous function The Shepard-type ax-product operator associated to f is defined by Shf, n) x) = Sh x) = 1 1 f x i ) = f x i ) 1, 1)

4 where λ N It is easy to see that these are nonlinear, continuous operators For siplicity of the notation we oit the arguents f, n In what follows we obtain the ain results on these approxiation operators The following Lea is useful in obtaining the unifor approxiation theore of Weierstrass type Lea 4 For the approxiation by Shepard-type ax-product operator we have the following error bound ) Sh x) f x) dx, x i ) + 1 ω f, 1 ), 2) for any N Proof By Lea 3 we have f x i ) n dx, x Sh x) f x) = i ) λ f x) dx, x i ) λ 1 f x) f x i ) 1 By the properties of the odulus of oscillation of a function, we get for any N ω f, dx, x i )) dx, x i ) + 1) ω f, 1 ) dx, x i ) λ dx, x i ) λ Sh x) f x) 1 1 By direct coputation we get Sh x) f x) dx, x i ) + 1) dx, x i ) λ 1 dx, x i ) λ 1 dx, x i ) λ 1 1 ω f, 1 ) + 1 ω f, 1 )

5 It is easy to check that dx, x i ) and we obtain ) Sh x) f x) dx, x i ) + 1 ω f, 1 ) The following theore is a unifor approxiation theore of Weierstrass type Theore 5 Any continuous function f : X [0, ], can be uniforly approxiated by Shepard-type ax-product approxiation operators, ie for any ε > 0, there exists n N and a sequence of points x i, i = 0,, n, such that Shf, n)x) fx) < ε Proof Since X is a copact etric space, it is also totally bounded, ie for every ε > 0 there exists n N and a finite covering of X by open balls B i n having radius ε and center x i, i = 0,, If ε = 1 then dx, x i ) < 1 and by the previous Lea 4 we get Sh x) f x) 2 ω f, 1 ) By Theore 2 ω f, 1 ) 0 for and the proof is coplete In what follows, we consider the case of equally spaced data in [0, 1] interval In this case Jackson-type error estiate is obtained, that is the approxiation error is proportional to ω f, 1 n) see [3]) This result is iportant since it shows that by changing the operations we do not loose approxiation properties, since in [10] the sae order of the estiate is obtained for the classical Shepard approxiation operators Theore 6 If f : [0, 1] [0, ] is continuous and x i = i n, i = 0,, n, then we have Sh x) f x) 3 2 ω f, 1 ) n Proof Since x i n i 2n,

6 by taking = n in Lea 4, we have Sh x) f x) 3 2 ω f, 1 ) n 4 Concluding rearks and further research The above obtained results show that su and product are not the only operations that can be used in approxiation theory Indeed, by using ax and product as operations, we defined a Shepard-type approxiation operator Moreover the Weierstrass-type approxiation theore and the Jackson-type error estiates obtained in this paper show us that we do not lose approxiation properties Also, since the operator is nonlinear it is possible that it provides better approxiation for soe function it is well-known that using eg polynoial approxiation for the solution of a nonlinear differential equation leads to loss of any properties) Iage processing uses as one of its usual tools approxiation theory So if we provide an approxiation ethod then it is iediately interpreted as an iage copression ethod So we propose as a further research topic the efficient ipleentation of ax-product approxiation operators in iage copression As further reserch topic we propose also the following question: Which are the best operations for approxiation purposes for soe given class of functions As good candidates in this research we ention Frank t-nors These t-nors have a Lipschitz-type property that can be helpful for approxiation purposes References [1] G A Anastassiou, Rate of convergence of fuzzy neural network operators, univariate case, J Fuzzy Math 10, No 32002), [2] G A Anastassiou, SG Gal, Approxiation Theory: Moduli of Continuity and Global Soothness Preservation, Birkhäuser, Boston-Basel-Berlin, 2000 [3] RA Devore, GG Lorentz, Constructive Approxiation, Polynoials and Splines Approxiation, Springer-Verlag, Berlin, Heidelberg, 1993 [4] A DiNola, S Sessa, W Pedrycz, and E Sanchez, Fuzzy Relation Equation and Their Applications to Knowledge Engineering, Kluwer Acadeic Publishers, 1989 [5] G Ferrari-Trecate, R Rovatti, Fuzzy systes with overlapping Gaussian concepts: Approxiation properties in Sobolev nors, Fuzzy Sets and Systes ),

7 [6] LT Kóczy, K Hirota, Approxiate reasoning by linear rule interpolation and general approxiation, International Journal of Approxiate Reasoning 91993), [7] Puyin Liu, Universal approxiations of continuous fuzzy-valued functions by ulti-layer regular fuzzy neural networks, Fuzzy Sets and Systes, ), [8] EH Madani, S Assilian, An experient in linguistic synthesis with a fuzzy logic controller, J Man Machine Stud, 71975), 1-13 [9] E Pap, K Jegdić, Pseudo-analysis and its application in railway rooting, Fuzzy Sets and Systes ), [10] J Szabados, On a proble of R DeVore, Acta Math Hungar, )1976) [11] D Tikk, Notes on the approxiation rate of fuzzy KH interpolators, Fuzzy Sets and Systes, ),