[EN] Let C(K, E1) be the space of continuous functions defined between a compact Hausdorff space K and the space of fuzzy numbers E1 endowed
with the supremum metric. We provide a sufficient set of conditions on a subspace of C(K, E1) in order that it be dense. We also obtain a similar result for interpolating families of C(K, E1).This research is supported by Spanish Government (MTM2016-77143-P), Universitat Jaume I (Projecte P1-1B2014-35) and Generalitat Valenciana (Projecte AICO/2016/030).Font, JJ.; Sanchis, D.; Sanchis, M. (2017). A version of Stone-Weierstrass theorem in Fuzzy Analysis. En Proceedings of the Workshop on Applied Topological Structures. Editorial Universitat Politècnica de València. 41-46. http://hdl.handle.net/10251/128026OCS414

Font, Juan J.

Sanchis, Delia

Sanchis, Manuel

[EN] Let C(K, E1) be the space of continuous functions defined between a compact Hausdorff space K and the space of fuzzy numbers E1 endowed
with the supremum metric. We provide a sufficient set of conditions on a subspace of C(K, E1) in order that it be dense. We also obtain a similar result for interpolating families of C(K, E1).This research is supported by Spanish Government (MTM2016-77143-P), Universitat Jaume I (Projecte P1-1B2014-35) and Generalitat Valenciana (Projecte AICO/2016/030).Font, JJ.; Sanchis, D.; Sanchis, M. (2017). A version of Stone-Weierstrass theorem in Fuzzy Analysis. En Proceedings of the Workshop on Applied Topological Structures. Editorial Universitat Politècnica de València. 41-46. http://hdl.handle.net/10251/128026414

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WATS'17Workshop on Applied Topological StructuresValencia, 11-12 July 2017pp. 41 – 46A version of Stone-Weierstrass theorem in FuzzyAnalysisJuan J. Font, Delia Sanchis and Manuel Sanchis 1Departament de Matema`tiques, and Institut Universitari de Matema`tiques i Aplicacions de Castello´ (IMAC),Universitat Jaume I, Campus del Riu Sec. s/n, 12071 Castello´, Spain (font@mat.uji.es, dsanchis@uji.es,manuel.sanchis@mat.uji.es)AbstractLet C(K,E1) be the space of continuous functions defined between acompact Hausdorff spaceK and the space of fuzzy numbers E1 endowedwith the supremum metric. We provide a sufficient set of conditionson a subspace of C(K,E1) in order that it be dense. We also obtain asimilar result for interpolating families of C(K,E1).Keywords: fuzzy Analysis; fuzzy numbers; fyzzy functions.MSC: 54E35; 54E40.1. IntroductionFuzzy numbers provide formalized tools to deal with non-precise quantities. Theyare indeed fuzzy sets in the real line and were introduced in 1978 by Dubois andPrade ([3]), who also defined their basic operations. Since then, Fuzzy Analysis1This research is supported by Spanish Government (MTM2016-77143-P), Universitat JaumeI (Projecte P1-1B2014-35) and Generalitat Valenciana (Projecte AICO/2016/030).This work is licensed under a Creative Commons License CC BY-NC-ND 4.0Editorial Universitat Polite`cnica de Vale`ncia41J. J. Font, D. Sanchis and M. Sanchishas developed based on the notion of fuzzy number just as much as classical RealAnalysis did based on the concept of real number. Such development was eased bya characterization of fuzzy numbers provided in 1986 by Goetschel and Voxman([5]) leaning on their level sets.As real-valued functions do in the classical setting, fuzzy-number-valued functions,that is, functions defined on a topological space taking values in the space offuzzy numbers, play a central role in Fuzzy Analysis. Namely, fuzzy-number-valued functions have become the main tool in several fuzzy contexts, such asfuzzy differential equations ([1]), fuzzy integrals ([12]) or fuzzy optimization ([6]).However the main difficulty of dealing with these functions is the fact that thespace they form is not a linear space; indeed it is not a group with respect toaddition.In this paper we focus on the conditions under which continuous (with respect tothe supremum metric) fuzzy-number-valued functions defined on a compact Haus-dorff space can be (uniformly) approximated to any degree of accuracy. Moreprecisely and based on ideas of R. I. Jewett ([8]) and J. B. Prolla ([11]), we pro-vide a sufficient set of conditions on a subspace of the space of fuzzy-number-valuedfunctions in order that it be dense, which is to say a Stone-Weierstrass type result.The celebrated Stone-Weierstrass theorem is one of the most important results inclassical Analysis, plays a key role in the development of General ApproximationTheory and, particularly, is in the essence of the approximation capabilities of neu-ral networks. We also obtain a similar result for interpolating families of continuousfuzzy-number-valued functions in the sense that the uniform approximation canalso demand exact agreement at any finite number of points.2. PreliminariesLet F (R) denote the family of all fuzzy subsets on the real numbers R. Foru ∈ F (R) and λ ∈ [0, 1], the λ-level set of u is defined by[u]λ := { x ∈ R : u(x) ≥ λ } , λ ∈]0, 1],[u]0 := cl R {x ∈ R : u(x) > 0 } .42A version of Stone-Weierstrass theorem in Fuzzy AnalysisThe fuzzy number space E1 is the set of elements u of F (R) satisfying the followingproperties:(1) u is normal, i.e., there exists an x0 ∈ R with u(x0) = 1;(2) u is convex, i.e., u(λx+ (1 − λ)y) ≥ min {u(x), u(y)} for all x, y ∈ R, λ ∈[0, 1];(3) u is upper-semicontinuous;(4) [u]0 is a compact set in R.Notice that if u ∈ E1, then the λ-level set [u]λ of u is a compact interval for eachλ ∈ [0, 1]. We denote [u]λ = [u−(λ), u+(λ)]. Every real number r can be considereda fuzzy number since r can be identified with the fuzzy number r˜ defined asr˜(t) :={1 if t = r,mean0 if t 6= r.We can now state the characterization of fuzzy numbers provided by Goetscheland Voxman ([5]):Theorem 1. Let u ∈ E1 and [u]λ = [u−(λ), u+(λ)], λ ∈ [0, 1]. Then the pair offunctions u−(λ) and u+(λ) has the following properties:• u−(λ) is a bounded left continuous nondecreasing function on (0, 1];• u+(λ) is a bounded left continuous nonincreasing function on (0, 1];• u−(λ) and u+(λ) are right continuous at λ = 0;• u−(1) ≤ u+(1).Conversely, if a pair of functions α(λ) and β(λ) satisfy the above conditions (i)-(iv), then there exists a unique u ∈ E1 such that [u]λ = [α(λ), β(λ)] for eachλ ∈ [0, 1].Given u, v ∈ E1 and k ∈ R, we can define u+ v := [u−(λ), u+(λ)] + [v−(λ), v+(λ)]and ku := k[u−(λ), u+(λ)]. It is well-known that E1 endowed with this two naturaloperations is not a vector space. Indeed (E1,+) is not a group.On the other hand, we can endow E1 with the following metric:43J. J. Font, D. Sanchis and M. SanchisDefinition 2 ([5, 2]). For u, v ∈ E1, we can defined∞(u, v) := supλ∈[0,1]max{|u−(λ)− v−(λ)|, |u+(λ) − v+(λ)|}.It is called the supremum metric on E1, and (E1, d∞) is well-known to be acomplete metric space. Notice that, by the definition of d∞, R endowed withthe euclidean topology can be topologically identified with the closed subspaceR˜ = { x˜ : x ∈ R } of (E1, d∞) where x˜+(λ) = x˜−(λ) = x for all λ ∈ [0, 1]. As ametric space, we shall always consider E1 equipped with the metric d∞.Proposition 3. The metric d∞ satisfies the following properties:(1) d∞(∑mi=1 ui,∑mi=1 vi) ≤∑mi=1 d∞(ui, vi) where ui, vi ∈ E1 for i = 1, ...,m.(2) d∞(ku, kv) = kd∞(u, v) where u, v ∈ E1 and k > 0.(3) d∞(ku, µu) =| k − µ | d∞(u, 0), where u ∈ E1, k ≥ 0 and µ ≥ 0.(4) d∞(ku, µv) ≤| k − µ | d∞(u, 0) + µd∞(u, v), where u, v ∈ E1, k ≥ 0 andµ ≥ 0.We shall denote by C(K,E1) the space of continuous functions defined betweenthe compact Hausdorff space K and the metric space (E1, d∞). In C(K,E1) weshall consider the following metric:D(f, g) = supt∈Kd∞(f(t), g(t)),which induces the uniform convergence topology on C(K,E1).Proposition 4. Let φ ∈ C(K,R+) and f ∈ C(K,E1). Then the function k 7→φ(k)f(k), k ∈ K, belongs to C(K,E1).3. A version of the Stone-Weierstrass theorem in Fuzzy Analysis.Let us first introduce a basic tool to obtain our main theorem (Theorem 11).Definition 5. Let W be a nonempty subset of C(K,E1). We defineConv(W ) = {ϕ ∈ C(K, [0, 1]) : ϕf + (1− ϕ)g ∈Wfor all f, g ∈W}.44A version of Stone-Weierstrass theorem in Fuzzy AnalysisProposition 6. Let W be a nonempty subset of C(K,E1). Then we have:(1) φ ∈ Conv(W ) implies that 1− φ ∈ Conv(W ).(2) If φ, ϕ ∈ Conv(W ), then φ · ϕ ∈ Conv(W ).(3) If φ belongs to the uniform closure of Conv(W ), then so does 1− φ.(4) If φ, ϕ belong to the uniform closure of Conv(W ), then so does φ · ϕ.(5) Uniform closureDefinition 7. It is said that M ⊂ C(K, [0, 1]) separates the points of K if givens, t ∈ K, there exists φ ∈M such that φ(s) 6= φ(t).Next we state two technical lemmas which will used in the sequel:Lemma 8 ([8, Lemma 2]). Let 0 < a < b < 1 and 0 < δ < 12 . There exists apolynomial p(x) = (1− xm)n such that(1) p(x) > 1− δ for all 0 ≤ x ≤ a,(2) p(x) < δ for all b ≤ x ≤ 1.Lemma 9 ([8, Theorem 1]). Let W ⊂ C(K,E1). The maximum of two elementsof Conv(W ) belongs to the uniform closure of Conv(W ).Lemma 10. Let W ⊆ C(K,E1). If Conv(W ) separates the points of K, then,given x0 ∈ K and a open neighborhood N of x0, there exists a neighborhood U ofx0 such that for all 0 < δ <12 , there is ϕ ∈ Conv(W ) such that(1) ϕ(t) > 1− δ, for all t ∈ U ;(2) ϕ(t) < δ, for all t /∈ N .Gathering the information obtained so far, we can now state and prove a versionof the Stone-Weierstrass theorem for fuzzy-number-valued continuous functions:Theorem 11. LetW be a nonempty subset of C(K,E1) and assume that Conv(W )separates points. If given f ∈ C(K,E1) and ε > 0, there exists, for each x ∈ K,gx ∈W such that d∞(f(x), gx(x)) < ε, then W is dense in (C(K,E1), D).45J. J. Font, D. Sanchis and M. Sanchis4. ConclusionWe have proved that, under certain natural assumptions, continuous (with re-spect to the supremum metric) fuzzy-number-valued functions defined on a com-pact Hausdorff space can be (uniformly) approximated to any degree of accuracy,which yields a Stone-Weierstrass type result in this setting. A similar result forinterpolating families of continuous fuzzy-number-valued functions in the sensethat the uniform approximation can also demand exact agreement at any finitenumber of points.References[1] B. Bede and S. Gal, Generalizations of the differentiability of fuzzy-number-valued functionswith applications to fuzzy differential equations, Fuzzy Sets and Systems 151 (2005) 581–599.[2] P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets: Theory and Applications, WorldScientific, Singapore, 1994.[3] D. Dubois and H. Prade, Operations on fuzzy numbers, Internat. J. of Systems Sci. 9 (1978)613-626.[4] J-X. Fang and Q-Y. Xue, Some properties of the space of fuzzy-valued continuous functionson a compact set, Fuzzy Sets and Systems 160 (2009) 1620–1631.[5] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems 18 (1986)31–42.[6] S. Hai, Z. Gong and H. Li, Generalized differentiability for n-dimensional fuzzy-number-valued functions and fuzzy optimization, Inf. Sci. 374 (2016) 151–163.[7] H. Huang and C. Wu, Approximation of level fuzzy-valued functions by multilayer regularfuzzy neural networks, Math. Comp. Model. 49 (2009) 1311–1318.[8] R. I. Jewett, A variation on the Stone-Weierstrass theorem, Proc. Amer. Math. Soc. 14(1963) 690–693.[9] M. Leshno, V. Y. Lin, A. Pinkus and S. Schocken, Multilayer feedforward networks with anonpolynomial activation function can approximate any function, Neural Networks 6 (1993)861–867.[10] P. Y. Liu, Universal approximations of continuous fuzzy-valued function by multi-layer reg-ular fuzzy neural networks, Fuzzy Sets and Systems 119 (2001) 313-320.[11] J. B. Prolla, On the Weierstrass-Stone Theorem, J. Approx. Theory, 78 (1994), 299–313.[12] C. Wu and Z. Gong, On Henstock integral of fuzzy-number-valued functions (I), Fuzzy Setsand Systems 120 (2001) 523–532.46

A version of Stone-Weierstrass theorem in Fuzzy Analysis

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A version of Stone-Weierstrass theorem in Fuzzy Analysis

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