Higher Order Multivariate Fuzzy Approximation by basic ... - SciELO

CUBO A Mathematical Journal Vol.16, N¯o 03, (21–35). October 2014

Higher Order Multivariate Fuzzy Approximation by basic Neural Network Operators George A. Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, U.S.A. [email protected] ABSTRACT Here are studied in terms of multivariate fuzzy high approximation to the multivariate unit basic sequences of multivariate fuzzy neural network operators. These operators are multivariate fuzzy analogs of earlier studied multivariate real ones. The produced results generalize earlier real ones into the fuzzy setting. Here the high order multivariate fuzzy pointwise convergence with rates to the multivariate fuzzy unit operator is established through multivariate fuzzy inequalities involving the multivariate fuzzy moduli of continuity of the Nth order (N ≥ 1) H-fuzzy partial derivatives, of the engaged multivariate fuzzy number valued function. RESUMEN Utilizando aproximaciones multivariadas difusas superiores, estudiamos la aplicaci´on a secuencias b´ asicas unitarias multivariadas de operadores de redes neuronales disfusas multivariadas. Estos operadores son an´alogos difusos multivariados de los reales multivariados estudiados anteriormente. Los resultados obtenidos generalizan los resultados reales anteriores en el marco difuso. La convergencia puntual difusa multivariada de orden superior con velocidades para los operadores unitarios difusos multivariados se establece a trav´es de desigualdades difusas multivariadas que involucran los m´odulos de continuidad difusos multivariados de las derivadas parciales H-difusas de N-´esimo orden (N ≥ 1) de las funciones con valores num´ericos difusos multivariados. Keywords and Phrases: multivariate fuzzy real analysis, multivariate fuzzy neural network operators, high order multivariate fuzzy approximation, multivariate fuzzy modulus of continuity and multivariate Jackson type inequalities. 2010 AMS Mathematics Subject Classification: 26A15, 26E50, 41A17, 41A25, 41A99, 47S40.

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George A. Anastassiou

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Fuzzy real Analysis Background

We need the following background Definition 1. (see [14]) Let µ : R → [0, 1] with the following properties (i) is normal, i.e., ∃ x0 ∈ R; µ (x0 ) = 1.

(ii) µ (λx + (1 − λ) y) ≥ min{µ (x) , µ (y)}, ∀ x, y ∈ R, ∀ λ ∈ [0, 1] (µ is called a convex fuzzy subset). (iii) µ is upper semicontinuous on R, i.e. ∀ x0 ∈ R and ∀ ε > 0, ∃ neighborhood V (x0 ) : µ (x) ≤ µ (x0 ) + ε, ∀ x ∈ V (x0 ) . (iv) The set sup p (µ) is compact in R, (where sup p (µ) := {x ∈ R : µ (x) > 0}). We call µ a fuzzy real number. Denote the set of all µ with RF . E.g. χ{x0 } ∈ RF , for any x0 ∈ R, where χ{x0 } is the characteristic function at x0 . For 0 < r ≤ 1 and µ ∈ RF define [µ]r := {x ∈ R : µ (x) ≥ r}

(1)

and [µ]0 := {x ∈ R : µ (x) ≥ 0}. Then it is well known that for each r ∈ [0, 1], [µ]r is a closed and bounded interval on R ([11]). For u, v ∈ RF and λ ∈ R, we define uniquely the sum u ⊕ v and the product λ ⊙ u by r

r

r

r

r

[u ⊕ v] = [u] + [v] , [λ ⊙ u] = λ [u] , ∀ r ∈ [0, 1] , where [u]r + [v]r means the usual addition of two intervals (as subsets of R) and λ [u]r means the usual product between a scalar and a subset of R (see, e.g. [14]). Notice 1 ⊙ u = u and it holds u ⊕ v = v ⊕ u, λ ⊙ u = u ⊙ λ. If 0 ≤ r1 ≤ r2 ≤ 1 then r

r

[u] 2 ⊆ [u] 1 . i (r) (r) (r) (r) (r) (r) r Actually [u] = u− , u+ , where u− ≤ u+ , u− , u+ ∈ R, ∀ r ∈ [0, 1]. h

(r)

(r)

For λ > 0 one has λu± = (λ ⊙ u)± , respectively. Define D : RF × RF → RF by



(r) (r) (r) (r) D (u, v) := sup max u− − v− , u+ − v+ , r∈[0,1]

(2)

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Higher Order Multivariate Fuzzy Approximation by basic Neural . . .

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where

23

i h (r) (r) r [v] = v− , v+ ; u, v ∈ RF .

We have that D is a metric on RF .

Then (RF , D) is a complete metric space, see [14], [15]. Let f, g : Rm → RF . We define the distance

D∗ (f, g) = sup D (f (x) , g (x)) . x∈Rm

Here Σ∗ stands for fuzzy summation and e 0 := χ{0} ∈ RF is the neutral element with respect to ⊕, i.e., u⊕e 0=e 0 ⊕ u = u, ∀ u ∈ RF . We need

(r)

(r)

Remark 2. ([5]). Here r ∈ [0, 1], xi , yi

∈ R, i = 1, ..., m ∈ N. Suppose that

  (r) (r) sup max xi , yi ∈ R, for i = 1, ..., m.

r∈[0,1]

Then one sees easily that sup max r∈[0,1]

m X i=1

(r) xi ,

m X

(r) yi

i=1

!

≤

m X

  (r) (r) sup max xi , yi .

(3)

i=1 r∈[0,1]

Definition 3. Let f ∈ C (Rm ), m ∈ N, which is bounded or uniformly continuous, we define (h > 0) |f (x1 , ..., xm ) − f (x′1 , ..., x′m )| .

sup

ω1 (f, h) := all

xi ,x′i ∈R,|xi −x′i |≤h,

for

Definition 4. Let f : Rm → RF , we define the fuzzy modulus of continuity of f by (F )

ω1

sup

(f, δ) =

x,y∈R,|xi −yi |≤δ,

(4)

i=1,...,m

D (f (x) , f (y)) , δ > 0,

for

(5)

i=1,...,m

where x = (x1 , ..., xm ), y = (y1 , ..., ym ) . For f : Rm → RF , we use (r)

where f± : Rm → R, ∀ r ∈ [0, 1] . We need

h i (r) (r) r [f] = f− , f+ ,

(6)

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  (r) Proposition 5. Let f : Rm → RF . Assume that ωF 1 (f, δ), ω1 f− , δ ,   (r) ω1 f+ , δ are finite for any δ > 0, r ∈ [0, 1] . Then

(F )

ω1

   

(r) (r) (f, δ) = sup max ω1 f− , δ , ω1 f+ , δ .

(7)

r∈[0,1]

Proof. By Proposition 1 of [8]. m m We define by CU → RF , F (R ) the space of fuzzy uniformly continuous functions from R m m m also CF (R ) is the space of fuzzy continuous functions on R , and Cb (R , RF ) is the fuzzy continuous and bounded functions.

We mention (F )

m Proposition 6. ([7]) Let f ∈ CU F (R ). Then ω1

Proposition 7. ([7]) It holds (F )

lim ω1

δ→0

(f, δ) < ∞, for any δ > 0.

(F )

(f, δ) = ω1

(8)

(f, 0) = 0,

m iff f ∈ CU F (R ) . (r)

Proposition 8. ([7]) Let f ∈ CF (Rm ). Then f± are equicontinuous with respect to r ∈ [0, 1] over Rm , respectively in ±. (r)

m m Note: It is clear by Propositions 5, 7, that if f ∈ CU F (R ), then f± ∈ CU (R ) (uniformly m continuous on R ).

We need Definition 9. Let x, y ∈ RF . If there exists z ∈ RF : x = y ⊕ z, then we call z the H-difference on x and y, denoted x − y. Definition 10. ([14]) Let T := [x0 , x0 + β] ⊂ R, with β > 0. A function f : T → RF is H-difference at x ∈ T if there exists an f′ (x) ∈ RF such that the limits (with respect to D) lim

h→0+

f (x + h) − f (x) , h

lim

h→0+

f (x) − f (x − h) h

(9)

exist and are equal to f′ (x) . We call f′ the H-derivative or fuzzy derivative of f at x. Above is assumed that the H-differences f (x + h) − f (x), f (x) − f (x − h) exists in RF in a neighborhood of x. m Definition 11. We denote by CN F (R ), N ∈ N, the space of all N-times fuzzy continuously m differentiable functions from R into RF .

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Here fuzzy partial derivatives are defined via Definition 10 in the obvious way as in the ordinary real case. We mention Theorem 12. ([12]) Let f : [a, b] ⊆ R → RF be H-fuzzy differentiable. Let t ∈ [a, b], 0 ≤ r ≤ 1. Clearly h i r (r) (r) [f (t)] = f (t)− , f (t)+ ⊆ R.

Then (f (t))(r) ± are differentiable and

r

′

[f (t)] = I.e.



(r) f (t)−

′  ′  (r) , f (t)+ .

′  (r) (r) , (f′ )± = f±

∀ r ∈ [0, 1] .

(10) (r)

Remark 13. (se also [6]) Let f ∈ CN (R, RF ), N ≥ 1. Then by Theorem 12 we obtain f± ∈ CN (R) and h ir  (i)  (i)  (r) (r) f(i) (t) = f (t)− , f (t)+ ,

for i = 0, 1, 2, ..., N, and in particular we have 

f(i)

(i)  (r) , = f±

(r) ±

(11)

for any r ∈ [0, 1] . m Let f ∈ CN e := F (R ), denote fα

e ∂αf e , ∂xα

e := (f g fi ∈ Z+ , i = 1, ..., m and where α α1 , ..., α m ), α

0 < |e α| :=

m X i=1

Then by Theorem 12 we get that 

(r)

f±



fi ≤ N, N > 1. α (r)

e α

= (fα e )± , ∀ r ∈ [0, 1] ,

(12)

(r) e : |e and any α α| ≤ N. Here f± ∈ CN (Rm ) .

For the definition of general fuzzy integral we follow [13] next.

Definition 14. Let (Ω, Σ, µ) be a complete σ-finite measure space. We call F : Ω → RF measurable iff ∀ closed B ⊆ R the function F−1 (B) : Ω → [0, 1] defined by F−1 (B) (w) := supF (w) (x) , all w ∈ Ω x∈B

is measurable, see [13].

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Theorem 15. ([13]) For F : Ω → RF ,

16, 3 (2014)

(r)

(r)

F (w) = {(F− (w) , F+ (w))|0 ≤ r ≤ 1},

the following are equivalent (1) F is measurable, (r)

(r)

(2) ∀ r ∈ [0, 1], F− , F+ are measurable. (r)

(r)

Following [13], given that for each r ∈ [0, 1], F− , F+ are integrable we have that the parametrized representation Z 

Z (r) (r) F− dµ, F+ dµ |0 ≤ r ≤ 1 A

A

is a fuzzy real number for each A ∈ Σ. The last fact leads to Definition 16. ([13]) A measurable function F : Ω → RF , (r)

(r)

F (w) = {(F− (w) , F+ (w))|0 ≤ r ≤ 1} (r)

(0)

is integrable if for each r ∈ [0, 1], F± are integrable, or equivalently, if F± are integrable. In this case, the fuzzy integral of F over A ∈ Σ is defined by Z 

Z Z (r) (r) Fdµ := F− dµ, F+ dµ |0 ≤ r ≤ 1 . A

(13)

A

A

By [13] F is integrable iff w → kF (w)kF is real-valeud integrable. Here   kukF := D u, e 0 , ∀ u ∈ RF .

We need also

Theorem 17. ([13]) Let F, G : Ω → RF be integrable. Then

(1) Let a, b ∈ R, then aF + bG is integrable and for each A ∈ Σ, Z Z Z (aF + bG) dµ = a Fdµ + b Gdµ; A

A

A

(2) D (F, G) is a real-valued integrable function and for each A ∈ Σ, Z  Z Z D D (F, G) dµ. Fdµ, Gdµ ≤ In particular,

A

A

A

Z

Z

Fdµ ≤ kFkF dµ.

A

F

A

(14)

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Above µ could be the Lebesgue measure, with all the basic properties valid here too. Basically here we have Z

A

i.e.

r Z  Z (r) (r) Fdµ := F− dµ, F+ dµ , Z

A

(r) Z (r) Fdµ F± dµ, = ±

(15)

A

A

(16)

A

∀ r ∈ [0, 1], respectively. We use Notation 18. We denote

!2   ∂ e → ,0 f − x := D ∂xi i=1   2   2   2 ∂ f (x1 , x2 ) e ∂ f (x1 , x2 ) e ∂ f (x1 , x2 ) e ,0 . D ,0 + D , 0 + 2D ∂x1 ∂x2 ∂x21 ∂x22 

2 X

(17)

In general we denote (j = 1, ..., N)

m X

D

i=1

X

(j1 ,...,jm )∈Zm +:

2

Pm

i=1 ji =j



!j   ∂ e → ,0 f − x := ∂xi

j! D j1 !j2 !...jm !

(18)

! ∂j f (x1 , ..., xm ) e ,0 . ∂xj11 ∂xj22 ...∂xjmm

Convergence with rates of real multivariate neural network operators

Here we follow [9]. We need the following (see [10]) definitions. Definition 19. A function b : R → R is said to be bell-shaped if b belongs to L1 and its integral is nonzero, if it is nondecreasing on (−∞, a) and nonincreasing on [a, +∞), where a belongs to R. In particular b (x) is a nonnegative number and at a, b takes a global maximum; it is the center of the bell-shaped function. A bell-shaped function is said to be centered if its center is zero. Definition 20. (see [10]) A function b : Rd → R (d ≥ 1) is said to be a d-dimensional bell-shaped function if it is integrable and its integral is not zero, and for all i = 1, ..., d, t → b (x1 , ..., t, ..., xd )

→ is a centered bell-shaped function, where − x := (x1 , ..., xd ) ∈ Rd arbitrary.

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Example 21. (from [10]) Let b be a centered bell-shaped function over R, then (x1 , ..., xd ) → b (x1 ) ...b (xd ) is a d-dimensional bell-shaped function.   Q → Assumption 22. Here b − x is of compact support B := d i=1 [−Ti , Ti ], Ti > 0 and it may have

jump discontinuities there. Let f : Rd → R be a continuous and bounded function or a uniformly continuous function.

Here we mention the study ([9]) of poitwise convergence with rates over Rd , to the unit operator I, of the ”normalized bell” real multivariate neural network operators   → Mn (f) − x := (19)


Pn k1 kd kd =−n2 f n , ... n b k1 =−n2 ... Pn2 Pn2 1−α k1 =−n2 ... kd =−n2 b n

Pn2

2


n1−α x1 − kn1 , ..., n1−α xd −


x1 − kn1 , ..., n1−α xd − knd

kd n





,

→ where 0 < α < 1 and − x := (x1 , ..., xd ) ∈ Rd , n ∈ N. Clearly, Mn is a positive linear operator.

The terms in the ratio of multiple sums (19) can be nonzero iff simultaneously   1−α ki n ≤ Ti , all i = 1, ..., d, xi − n Ti i.e., xi − kni ≤ n1−α , all i = 1, ..., d, iff nxi − Ti nα ≤ ki ≤ nxi + Ti nα , all i = 1, .., d.

(20)

− n2 ≤ nxi − Ti nα ≤ ki ≤ nxi + Ti nα ≤ n2 ,

(21)

To have the order

we need n ≥ Ti + |xi |, all i = 1, .., d. So (21) is true when we take n≥

max

i∈{1,...,d}

(Ti + |xi |) .

(22)

→ When − x ∈ B in order to have (21) it is enough to assume that n ≥ 2T ∗ , where T ∗ := max{T1 , ..., Td } > 0. Consider Iei := [nxi − Ti nα , nxi + Ti nα ] , i = 1, ..., d, n ∈ N.

The length of Iei is 2Ti nα . By Proposition 1 of [1], we get that the cardinality of ki ∈ Z that belong to Iei := card (ki ) ≥ max (2Ti nα − 1, 0), any i ∈ {1, ..., d}. In order to have card (ki ) ≥ 1, we need 1 −α

2Ti nα − 1 ≥ 1 iff n ≥ Ti

, any i ∈ {1, ..., d}.

Therefore, a sufficient condition in order to obtain the order (21) along with the interval Iei to contain at least one integer for all i = 1, ..., d is that

 −1 Ti + |xi | , Ti α . n ≥ max (23) i∈{1,...,d}

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Higher Order Multivariate Fuzzy Approximation by basic Neural . . .

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29

Clearly as n → +∞ we get that card (ki ) → +∞, all i = 1, ..., d. Also notice that card (ki ) equals to the cardinality of integers in [⌈nxi − Ti nα ⌉ , [nxi + Ti nα ]] for all i = 1, ..., d. Here, [·] denotes the integral part of the number, while ⌈·⌉ denotes its ceiling. From now on, in this article we will assume (23). Furthermore it holds 

 → (Mn (f)) − x :=

P[nx1 +T1 nα ]

k1 =⌈nx1 −T1 nα ⌉

...

P[nxd +Td nα ]

kd =⌈nxd −Td nα ⌉

  → V − x

f

k1 kd n , ... n




(24)

     k1 kd 1−α 1−α ·b n x1 − , ..., n xd − n n

→ all − x := (x1 , ..., xd ) ∈ Rd , where [nxd +Td nα ]

[nx1 +T1 nα ]

X

k1 =⌈nx1 −T1 nα ⌉

...

X

kd =⌈nxd −Td nα ⌉

  → V − x :=

     k1 kd 1−α 1−α x1 − b n xd − , ..., n . n n

(25)

From [9], we need and mention → Theorem 23. Let − x ∈ Rd ; then

      T∗ − → − → (Mn (f)) x − f x ≤ ω1 f, 1−α . n

(26)

Inequality (26) is attained by constant functions.     → → x , pointwise with rates, as n → +∞, where x → f − Inequalities (26) gives Mn (f) − − → x ∈ Rd , d ≥ 1, provided that f is uniformly continuous on Rd . In the last case it is clear that Mn → I, uniformly. From [9], we also need and mention


→ Theorem 24. Let − x ∈ Rd , f ∈ CN Rd , N ∈ N, such that all of its partial derivatives fα e of e : |e order N, α α| = N, are uniformly continuous or continuous are bounded. Then     → → (27) x ≤ x −f − (Mn (f)) −  N X 

j=1

  !j      d ∗ N N T∗ (T ∗ )j  X ∂ − →  + (T ) d · max ω1 fα , x . f e ∂xi  N!nN(1−α) α e :|e α|=N n1−α j!nj(1−α) i=1

Inequality (27) is attained by constant functions. Also, (27) gives us with rates the pointwise convergences of Mn (f) → f over Rd , as n → +∞.

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Main Results - Convergence with rates of fuzzy multivariate neural networks

Here b is as in Definition 20.   Q → Assumption 25. We suppose that b − x is of compact support B := d i=1 [−Ti , Ti ], Ti > 0, and

it may have jump discontinuities there. We consider f : Rd → RF to be fuzzy continuous and fuzzy bounded function or fuzzy uniformly continuous function.

In this section we study the D-metric pointwise convergence with rates over Rd , to the fuzzy → unit operator IF , of the fuzzy multivariate neural network operators (0 < α < 1, − x := (x1 , ..., xd ) ∈ d R , n ∈ N)   → MF (f) − x := (28) n

Pn2 ∗


⊙ b n1−α x1 − kn1 , ..., n1−α xd −


Pn2 Pn2 1−α x − k1 , ..., n1−α x − kd d 1 k1 =−n2 ... kd =−n2 b n n n

k1 =−n2

...

Pn2 ∗

kd =−n2

kd k1 n , ... n

f




[nxd +Td nα ]∗

[nx1 +T1 nα ]∗

X

=

k1 =⌈nx1 −T1

⊙

X

... nα ⌉

f

kd =⌈nxd −Td

b n1−α x1 −

nα ⌉



k1 kd , ... n n


, ..., n1−α xd −   → V − x

k1 n





kd n

kd n







(29)

,

  → x as in (25) and under the assumption (23). where V − We notice for r ∈ [0, 1] that

h  ir − → MF = n (f) x ·

=

X

...

k1 =⌈nx1 −T1 nα ⌉

· 

=

kd =⌈nxd −Td nα ⌉

k1 =⌈nx1 −T1 nα ⌉

X


, ..., n1−α xd −   → V − x

k1 n

[nxd +Td nα ]

X

kd =⌈nxd −Td nα ⌉

b n1−α x1 −

k1 =⌈nx1 −T1 nα ⌉



(r)

f−



k1 kd , ... n n


, ..., n1−α xd −   → V − x

k1 n

[nxd +Td nα ]

[nx1 +T1 nα ]

X

X

...

b n1−α x1 −

[nx1 +T1 nα ]

[nxd +Td nα ]

[nx1 +T1 nα ]

...

X

kd =⌈nxd −Td nα ⌉

(r)

f−

  r k1 kd f , ... n n

kd n



(r)

, f+

kd n







(30) 

k1 kd , ... n n





k1 kd , ... n n





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Higher Order Multivariate Fuzzy Approximation by basic Neural . . .

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·

b n1−α x1 −

[nxd +Td nα ]

[nx1 +T1 nα ]

X


, ..., n1−α xd −   → V − x

k1 n

X

...

k1 =⌈nx1 −T1 nα ⌉

kd =⌈nxd −Td nα ⌉

(r)

f+



kd n





,

k1 kd , ... n n






 , ..., n1−α xd − knd    · → V − x   i       h (r) (r) − → − → x . x , M n f+ = M n f− b n1−α x1 −

We have proved that

MF n (f)

respectively.


(r) ±

31

k1 n

  (r) = Mn f± , ∀ r ∈ [0, 1] ,

(31)

(32)

We present → Theorem 26. Let − x ∈ Rd ; then D



MF n

    
− T∗ (F ) → − → (f) x ,f x ≤ ω1 f, 1−α . n

(33)


D U d . Notice that (33) gives MF n → IF pointwise and uniformly, as n → ∞, when f ∈ CF R Proof. We observe that

D



  
− → → MF (f) x ,f − x = n

  (32)    
(r) −
(r) − (r) − (r) − → → → → F x } = x x − f x − f , sup max{ MF M (f) (f) + − n n + −

r∈[0,1]

      (26)         (r) (r) → (r) (r) → − → − → x − f+ − x } ≤ x − f− − x , M n f+ sup max{ Mn f−

r∈[0,1]

      T∗ T∗ T∗ (7) (r) (F ) (r) = ω1 f, 1−α , sup max ω1 f− , 1−α , ω1 f+ , 1−α n n n r∈[0,1]

proving the claim. We continue with
→ d Theorem 27. Let − x ∈ Rd , f ∈ CN e of F R , N ∈ N,such that all of its fuzzy partial derivatives fα e : |e order N, α α| = N, are fuzzy uniformly continuous or fuzzy continuous and fuzzy bounded. Then    
− → → D MF x ,f − x ≤ (34) n (f)

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 N X 

j=1



∗ j

(T )  j!nj(1−α)

d X

D

i=1



16, 3 (2014)

 !j    ∂ e → ,0 f − x   ∂xi

  N T∗ (T ∗ ) dN (F ) . fα max ω1 + e , 1−α e :|e n α|=N N!nN(1−α) α    
− → → As n → ∞, we get D MF (f) x ,f − x → 0 pointwise with rates. n

Proof. As before we have

D



   (32)
− → → x ,f − MF x = n (f)

      (27)         (r) (r) → (r) (r) → − → − → x − f+ − x } ≤ x − f− − x , M n f+ sup max{ Mn f−

r∈[0,1]

sup max r∈[0,1]

 N X 

j=1

∗ j

(T ) j!nj(1−α)



 !j d    X ∂ (r) − →  x  ∂xi f−  i=1

   T∗ (T ∗ )N dN (r) , f , ω max 1 − e :|e e n1−α α|=N α N!nN(1−α) α    !j d N j X    X ∂ (T ∗ ) (r) − →  x  ∂xi f+   j!nj(1−α) +

i=1

j=1

 (3) X  N j N  (T ∗ ) T∗ (T ∗ ) dN (r) ≤ f , max · ω + 1 + e :|e e n1−α α|=N α N!nN(1−α) α j!nj(1−α)

(36)

j=1

    !j !j d d       X X ∂ ∂ (r) − (r) − → → x , x  + sup max  ∂xi f− ∂xi f+   r∈[0,1] i=1

i=1

    N   (T ∗ ) dN T∗ T∗ (r) (r) f , , ω f , max sup max ω 1 1 + − e :|e e n1−α e n1−α α|=Nr∈[0,1] α α N!nN(1−α) α    !j     d N (by (3), (7), (12), (18)) X ∂ e (T ∗ )j  X → x  + f − ,0 D ≤   ∂xi j!nj(1−α) j=1

proving the claim.

i=1

  (T ∗ )N dN T∗ (F ) , ω f , max e α 1 e :|e n1−α α|=N N!nN(1−α) α

(37)

CUBO 16, 3 (2014)

4

Higher Order Multivariate Fuzzy Approximation by basic Neural . . .

33

Main Results - The fuzzy multivariate ”normalized squashing type operators” and their fuzzy convergence to the fuzzy unit with rates

We give the following definition Definition 28. Let the nonnegative function S : Rd → R, d ≥ 1, S has compact support B := d Y [−Ti , Ti ], Ti > 0 and is nondecreasing there for each coordinate. S can be continuous only on i=1

either

d Y (−∞, Ti ] or B and can have jump discontinuities. We call S the multivariate ”squashing i=1

function” (see also [10]).

Let f : Rd → RF be either fuzzy uniformly continuous or fuzzy continuous and fuzzy bounded function. → For − x ∈ Rd , we define the fuzzy multivariate ” normalized squashing type operator”,   − → LF (f) x := (38) n Pn2 ∗

k1 =−n2

...

Pn2 ∗

kd =−n2

f

kd k1 n , ... n




where 0 < α < 1 and n ∈ N: n≥ and



 → W − x :=

It is clear that

2

2

n ∗ X

k1 =−n2

...

i∈{1,...,d}

n ∗ X



1 −α

Ti + |xi | , Ti

k1 n






, ..., n1−α xd −

kd n





,

(39)

,

     kd k1 1−α 1−α xd − , ..., n . x1 − S n n n 2

(40)

kd =−n


− → LF (f) x := n where 

max

⊙ S n1−α x1 −   → W − x

 → Φ − x :=

h i − → → n− x + T nα ∗

X

f

m − → l − − → k = n→ x − T nα h i − → → n− x + T nα

X

− → l − − → k = n→ x − T nα

− → k n

  → ⊙ S n1−α − x −   → Φ − x

− →  k n

 − → k − →  . S n1−α  x − n m 



Here, we study the D-metric pointwise convergence with rates of LF n (f) − → d n → +∞, x ∈ R . This is given first by the next result.

,

(41)

(42)   
− → → x , as x → f −

34

CUBO

George A. Anastassiou

16, 3 (2014)

Theorem 29. Under the above terms and asumptions, we find that      
− T∗ (F ) − → → F ≤ ω1 x ,f x D Ln (f) f, 1−α . n

(43)


D U d Notice that (43) gives LF . n → IF pointwise and uniformly, as n → ∞, when f ∈ CF R Proof. Similar to (33). We also give
→ d Theorem 30. Let − x ∈ Rd , f ∈ CN e F R , N ∈ N, such that all of its fuzzy partial derivatives fα e : |e of order N, α α| = N, are fuzzy uniformly continuous or fuzzy continuous and fuzzy bounded. Then    
− → → D LF (f) x ,f − x ≤ (44) n    !j     N d j X X (T ∗ ) ∂ →  ,e 0 f − x  D   ∂xi j!nj(1−α) j=1

i=1

  N T∗ (T ∗ ) dN (F ) max ω1 fα . + e , 1−α e :|e α|=N n N!nN(1−α) α

  
− → → Inequality (44) gives us with rates the poitwise convergence of D( LF x ,f − x ) → 0 over n (f) Rd , as n → ∞.

Proof. Similar to (34).

Received:

November 2012.

Accepted:

May 2014.

References [1] G.A. Anastassiou, Rate of convergence of some neural network operators to the unit-univariate case, Journal of Mathematical Analysis and Application, Vol. 212 (1997), 237-262. [2] G.A. Anastassiou, Rate of Convergence of some Multivariate Neural Network Operators to the Unit, Computers and Mathematics, 40(2000), 1-19. [3] G.A. Anastassiou, Quantitative Approximation, Chapmann and Hall/CRC, Boca Raton, New York, 2001. [4] G.A. Anastassiou, Higher order Fuzzy Approximation by Fuzzy Wavelet type and Neural Network Operators, Computers and Mathematics, 48(2004), 1387-1401.

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Higher Order Multivariate Fuzzy Approximation by basic Neural . . .

35

[5] G.A. Anastassiou, Fuzzy Approximation by Fuzzy Convolution type Operators, Computers and Mathematics, 48(2004), 1369-1386. [6] G.A. Anastassiou, Higher order Fuzzy Korovkin Theory via inequalities, Communications in Applied Analysis, 10(2006), No. 2, 359-392. [7] G.A. Anastassiou, Fuzzy Korovkin Theorems and Inequalities, Journal of Fuzzy Mathematics, 15(2007), No. 1, 169-205. [8] G.A. Anastassiou, Higher order multivariate fuzzy approximation by multivariate fuzzy wavelet type and neural network operators, J. of Fuzzy Mathematics, 19(2011), no. 3, 601-618. [9] G.A. Anastassiou, Rate of convergence of some multivariate neural network operators to the unit,revisited, J. of Computational Analysis and Application, to appear 2013. [10] P. Cardaliaguet, G. Euvrard, Approximation of a function and its derivative with a neural network, Neural Networks 5(1992), 207-220. [11] R. Goetschel Jr., W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18(1986), 31-43. [12] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24(1987), 301-317. [13] Y.K. Kim, B.M. Ghil, Integrals of fuzzy-number-valued functions, Fuzzy Sets and Systems, 86(1997), 213-222. [14] C. Wu, Z. Gong, On Henstock integral of fuzzy-number-valued functions (I), Fuzzy Sets and Systems, 120, No. 3, (2001), 523-532. [15] C. Wu, M. Ma, On embedding problem of fuzzy numer spaces: Part 1, Fuzzy Sets and Systems, 44 (1991), 33-38.