

()


CUBO A Mathematical Journal

Vol.16, No¯ 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.

ganastss@memphis.edu

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 multi-

variate 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 aplicación a

secuencias básicas unitarias multivariadas de operadores de redes neuronales disfusas

multivariadas. Estos operadores son análogos difusos multivariados de los reales multi-

variados 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 través de desigualdades difusas multivariadas que involucran los módulos

de continuidad difusos multivariados de las derivadas parciales H-difusas de N-ésimo

orden (N ≥ 1) de las funciones con valores numéricos 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.


22 George A. Anastassiou CUBO
16, 3 (2014)

1 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

[u ⊕ v]
r
= [u]

r
+ [v]

r
, [λ ⊙ u]

r
= λ [u]

r
, ∀ 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

[u]
r2 ⊆ [u]

r1 .

Actually [u]
r
=
[
u
(r)
− , u

(r)
+

]
, where u

(r)
− ≤ u

(r)
+ , u

(r)
− , u

(r)
+ ∈ R, ∀ r ∈ [0, 1].

For λ > 0 one has λu
(r)
± = (λ ⊙ u)

(r)
± , respectively.

Define D : RF × RF → RF by

D (u, v) := sup
r∈[0,1]

max
{∣∣∣u(r)− − v

(r)
−

∣∣∣ ,
∣∣∣u(r)+ − v

(r)
+

∣∣∣
}
, (2)


CUBO
16, 3 (2014)

Higher Order Multivariate Fuzzy Approximation by basic Neural . . . 23

where

[v]
r
=
[
v
(r)
− , v

(r)
+

]
; 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
x∈Rm

D (f (x) , g (x)) .

Here Σ∗ stands for fuzzy summation and 0̃ := χ{0} ∈ RF is the neutral element with respect to ⊕,

i.e.,

u ⊕ 0̃ = 0̃ ⊕ u = u, ∀ u ∈ RF.

We need

Remark 2. ([5]). Here r ∈ [0, 1], x
(r)
i

, y
(r)
i

∈ R, i = 1, ..., m ∈ N. Suppose that

sup
r∈[0,1]

max
(
x
(r)
i

, y
(r)
i

)
∈ R, for i = 1, ..., m.

Then one sees easily that

sup
r∈[0,1]

max

(
m∑

i=1

x
(r)
i

,

m∑

i=1

y
(r)
i

)
≤

m∑

i=1

sup
r∈[0,1]

max
(
x
(r)
i

, y
(r)
i

)
. (3)

Definition 3. Let f ∈ C (Rm), m ∈ N, which is bounded or uniformly continuous, we define

(h > 0)

ω1 (f, h) := sup
all xi,x′i∈R,|xi−x

′

i
|≤h, for i=1,...,m

|f (x1, ..., xm) − f (x
′
1, ..., x

′
m)| . (4)

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

ω
(F)
1

(f, δ) = sup
x,y∈R,|xi−yi|≤δ, for i=1,...,m

D (f (x) , f (y)) , δ > 0, (5)

where x = (x1, ..., xm), y = (y1, ..., ym) .

For f : Rm → RF, we use

[f]
r
=
[
f
(r)
− , f

(r)
+

]
, (6)

where f
(r)
± : R

m → R, ∀ r ∈ [0, 1] .

We need


24 George A. Anastassiou CUBO
16, 3 (2014)

Proposition 5. Let f : Rm → RF. Assume that ω
F
1 (f, δ), ω1

(
f
(r)
− , δ

)
,

ω1

(
f
(r)
+ , δ

)
are finite for any δ > 0, r ∈ [0, 1] .

Then

ω
(F)
1

(f, δ) = sup
r∈[0,1]

max
{
ω1

(
f
(r)
− , δ

)
, ω1

(
f
(r)
+ , δ

)}
. (7)

Proof. By Proposition 1 of [8].

We define by CUF (R
m) the space of fuzzy uniformly continuous functions from Rm → RF,

also CF (R
m) is the space of fuzzy continuous functions on Rm, and Cb (R

m, RF) is the fuzzy

continuous and bounded functions.

We mention

Proposition 6. ([7]) Let f ∈ CUF (R
m). Then ω

(F)
1

(f, δ) < ∞, for any δ > 0.

Proposition 7. ([7]) It holds

lim
δ→0

ω
(F)
1

(f, δ) = ω
(F)
1

(f, 0) = 0, (8)

iff f ∈ CUF (R
m) .

Proposition 8. ([7]) Let f ∈ CF (R
m). Then f

(r)
± are equicontinuous with respect to r ∈ [0, 1]

over Rm, respectively in ±.

Note: It is clear by Propositions 5, 7, that if f ∈ CUF (R
m), then f

(r)
± ∈ CU (R

m) (uniformly

continuous on Rm).

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.

Definition 11. We denote by CNF (R
m), N ∈ N, the space of all N-times fuzzy continuously

differentiable functions from Rm into RF.


CUBO
16, 3 (2014)

Higher Order Multivariate Fuzzy Approximation by basic Neural . . . 25

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

[f (t)]
r
=
[
f (t)

(r)
−

, f (t)
(r)
+

]
⊆ R.

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

[f′ (t)]
r
=

[(
f (t)

(r)
−

)′
,
(
f (t)

(r)
+

)′]
.

I.e.

(f′)
(r)

± =
(
f
(r)
±

)′
, ∀ r ∈ [0, 1] . (10)

Remark 13. (se also [6]) Let f ∈ CN (R, RF), N ≥ 1. Then by Theorem 12 we obtain f
(r)
± ∈

CN (R) and
[
f(i) (t)

]r
=

[(
f (t)

(r)
−

)(i)
,
(
f (t)

(r)
+

)(i)]
,

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

(
f(i)
)(r)
±

=
(
f
(r)
±

)(i)
, (11)

for any r ∈ [0, 1] .

Let f ∈ CNF (R
m), denote fα̃ :=

∂
α̃f

∂xα̃
, where α̃ := (α̃1, ..., α̃m), α̃i ∈ Z

+, i = 1, ..., m and

0 < |α̃| :=

m∑

i=1

α̃i ≤ N, N > 1.

Then by Theorem 12 we get that

(
f
(r)
±

)
α̃
= (fα̃)

(r)
± , ∀ r ∈ [0, 1] , (12)

and any α̃ : |α̃| ≤ N. Here f
(r)
± ∈ C

N (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) := sup
x∈B

F (w) (x) , all w ∈ Ω

is measurable, see [13].


26 George A. Anastassiou CUBO
16, 3 (2014)

Theorem 15. ([13]) For F : Ω → RF,

F (w) = {(F
(r)
− (w) , F

(r)
+ (w))|0 ≤ r ≤ 1},

the following are equivalent

(1) F is measurable,

(2) ∀ r ∈ [0, 1], F
(r)
− , F

(r)
+ are measurable.

Following [13], given that for each r ∈ [0, 1], F
(r)
− , F

(r)
+ are integrable we have that the

parametrized representation
{(∫

A

F
(r)
− dµ,

∫

A

F
(r)
+ dµ

)
|0 ≤ r ≤ 1

}

is a fuzzy real number for each A ∈ Σ.

The last fact leads to

Definition 16. ([13]) A measurable function F : Ω → RF,

F (w) = {(F
(r)
− (w) , F

(r)
+ (w))|0 ≤ r ≤ 1}

is integrable if for each r ∈ [0, 1], F
(r)
± are integrable, or equivalently, if F

(0)
± are integrable.

In this case, the fuzzy integral of F over A ∈ Σ is defined by
∫

A

Fdµ :=

{(∫

A

F
(r)
− dµ,

∫

A

F
(r)
+ dµ

)
|0 ≤ r ≤ 1

}

. (13)

By [13] F is integrable iff w → ‖F (w)‖F is real-valeud integrable. Here

‖u‖F := D
(
u, 0̃

)
, ∀ 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,
∫

A

(aF + bG) dµ = a

∫

A

Fdµ + b

∫

A

Gdµ;

(2) D (F, G) is a real-valued integrable function and for each A ∈ Σ,

D

(∫

A

Fdµ,

∫

A

Gdµ

)
≤

∫

A

D (F, G) dµ. (14)

In particular, ∥∥∥∥
∫

A

Fdµ

∥∥∥∥
F

≤

∫

A

‖F‖F dµ.


CUBO
16, 3 (2014)

Higher Order Multivariate Fuzzy Approximation by basic Neural . . . 27

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

Basically here we have
[∫

A

Fdµ

]r
:=

[∫

A

F
(r)
− dµ,

∫

A

F
(r)
+ dµ

]
, (15)

i.e. (∫

A

Fdµ

)(r)

±

=

∫

A

F
(r)
± dµ, (16)

∀ r ∈ [0, 1], respectively.

We use

Notation 18. We denote (
2∑

i=1

D

(
∂

∂xi
, 0̃

))2
f
(
−→
x
)
:= (17)

D

(
∂2f (x1, x2)

∂x2
1

, 0̃

)
+ D

(
∂2f (x1, x2)

∂x2
2

, 0̃

)
+ 2D

(
∂2f (x1, x2)

∂x1∂x2
, 0̃

)
.

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

(
m∑

i=1

D

(
∂

∂xi
, 0̃

))j
f
(
−→
x
)
:= (18)

∑

(j1,...,jm)∈Z
m
+
:
∑

m

i=1
ji=j

j!

j1!j2!...jm!
D

(
∂jf (x1, ..., xm)

∂x
j1
1
∂x

j2
2
...∂x

jm
m

, 0̃

)
.

2 Convergence with rates of real multivariate neural net-

work 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) ∈ R
d arbitrary.


28 George A. Anastassiou CUBO
16, 3 (2014)

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.

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)

∑n2
k1=−n

2 ...
∑n2

kd=−n
2 f
(
k1
n
, ...kd

n

)
b
(
n1−α

(
x1 −

k1
n

)
, ..., n1−α

(
xd −

kd
n

))
∑n2

k1=−n
2 ...

∑n2
kd=−n

2 b
(
n1−α

(
x1 −

k1
n

)
, ..., n1−α

(
xd −

kd
n

)) ,

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

The terms in the ratio of multiple sums (19) can be nonzero iff simultaneously

∣∣∣∣n
1−α

(
xi −

ki

n

)∣∣∣∣ ≤ Ti, all i = 1, ..., d,

i.e.,
∣∣xi − kin

∣∣ ≤ Ti
n1−α

, all i = 1, ..., d, iff

nxi − Tin
α ≤ ki ≤ nxi + Tin

α, all i = 1, .., d. (20)

To have the order

− n2 ≤ nxi − Tin
α ≤ ki ≤ nxi + Tin

α ≤ n2, (21)

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

Ĩi := [nxi − Tin
α, nxi + Tin

α
] , i = 1, ..., d, n ∈ N.

The length of Ĩi is 2Tin
α. By Proposition 1 of [1], we get that the cardinality of ki ∈ Z that belong

to Ĩi := card (ki) ≥ max (2Tin
α − 1, 0), any i ∈ {1, ..., d}. In order to have card (ki) ≥ 1, we need

2Tin
α − 1 ≥ 1 iff n ≥ T

− 1
α

i
, any i ∈ {1, ..., d}.

Therefore, a sufficient condition in order to obtain the order (21) along with the interval Ĩi to

contain at least one integer for all i = 1, ..., d is that

n ≥ max
i∈{1,...,d}

{
Ti + |xi| , T

− 1
α

i

}
. (23)


CUBO
16, 3 (2014)

Higher Order Multivariate Fuzzy Approximation by basic Neural . . . 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 − Tin
α⌉ , [nxi + Tin

α]] 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
)
:=

∑[nx1+T1nα]
k1=⌈nx1−T1nα⌉

...
∑[nxd+Tdnα]

kd=⌈nxd−Tdnα⌉
f
(
k1
n
, ...kd

n

)

V
(
−→
x
) (24)

·b

(
n1−α

(
x1 −

k1

n

)
, ..., n1−α

(
xd −

kd

n

))

all −→x := (x1, ..., xd) ∈ R
d, where

V
(
−→
x
)
:=

[nx1+T1n
α
]∑

k1=⌈nx1−T1nα⌉

...

[nxd+Tdn
α
]∑

kd=⌈nxd−Tdnα⌉

b

(
n1−α

(
x1 −

k1

n

)
, ..., n1−α

(
xd −

kd

n

))
. (25)

From [9], we need and mention

Theorem 23. Let −→x ∈ Rd; then

∣∣∣(Mn (f))
(
−→
x
)
− f
(
−→
x
)∣∣∣ ≤ ω1

(
f,

T∗

n1−α

)
. (26)

Inequality (26) is attained by constant functions.

Inequalities (26) gives Mn (f)
(
−→
x
)

→ f
(
−→
x
)
, pointwise with rates, as n → +∞, where

−→
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
(
R

d
)
, N ∈ N, such that all of its partial derivatives fα̃ of

order N, α̃ : |α̃| = N, are uniformly continuous or continuous are bounded. Then

∣∣∣(Mn (f))
(
−→
x
)
− f
(
−→
x
)∣∣∣ ≤ (27)






N∑

j=1

(T∗)
j

j!nj(1−α)



(

d∑

i=1

∣∣∣∣
∂

∂xi

∣∣∣∣

)j
f
(
−→
x
)






+

(T∗)
N
dN

N!nN(1−α)
· max
α̃:|α̃|=N

ω1

(
fα̃,

T∗

n1−α

)
.

Inequality (27) is attained by constant functions. Also, (27) gives us with rates the pointwise

convergences of Mn (f) → f over R
d, as n → +∞.


30 George A. Anastassiou CUBO
16, 3 (2014)

3 Main Results - Convergence with rates of fuzzy multivari-

ate neural networks

Here b is as in Definition 20.

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) ∈

R
d, n ∈ N)

MFn (f)
(
−→
x
)
:= (28)

∑n2∗
k1=−n

2 ...
∑n2∗

kd=−n
2 f
(
k1
n
, ...kd

n

)
⊙ b

(
n1−α

(
x1 −

k1
n

)
, ..., n1−α

(
xd −

kd
n

))
∑n2

k1=−n
2 ...

∑n2
kd=−n

2 b
(
n1−α

(
x1 −

k1
n

)
, ..., n1−α

(
xd −

kd
n

))

=

[nx1+T1n
α
]∗∑

k1=⌈nx1−T1nα⌉

...

[nxd+Tdn
α
]∗∑

kd=⌈nxd−Tdnα⌉

f

(
k1

n
, ...

kd

n

)
(29)

⊙
b
(
n1−α

(
x1 −

k1
n

)
, ..., n1−α

(
xd −

kd
n

))

V
(
−→
x
) ,

where V
(
−→
x
)
as in (25) and under the assumption (23).

We notice for r ∈ [0, 1] that

[
MFn (f)

(
−→
x
)]r

=

[nx1+T1n
α
]∑

k1=⌈nx1−T1nα⌉

...

[nxd+Tdn
α
]∑

kd=⌈nxd−Tdnα⌉

[
f

(
k1

n
, ...

kd

n

)]r

·
b
(
n1−α

(
x1 −

k1
n

)
, ..., n1−α

(
xd −

kd
n

))

V
(
−→
x
) (30)

=

[nx1+T1n
α
]∑

k1=⌈nx1−T1nα⌉

...

[nxd+Tdn
α
]∑

kd=⌈nxd−Tdnα⌉

[
f
(r)
−

(
k1

n
, ...

kd

n

)
, f

(r)
+

(
k1

n
, ...

kd

n

)]

·
b
(
n1−α

(
x1 −

k1
n

)
, ..., n1−α

(
xd −

kd
n

))

V
(
−→
x
)

=




[nx1+T1n
α
]∑

k1=⌈nx1−T1nα⌉

...

[nxd+Tdn
α
]∑

kd=⌈nxd−Tdnα⌉

f
(r)
−

(
k1

n
, ...

kd

n

)


CUBO
16, 3 (2014)

Higher Order Multivariate Fuzzy Approximation by basic Neural . . . 31

·
b
(
n1−α

(
x1 −

k1
n

)
, ..., n1−α

(
xd −

kd
n

))

V
(
−→
x
) ,

[nx1+T1n
α
]∑

k1=⌈nx1−T1nα⌉

...

[nxd+Tdn
α
]∑

kd=⌈nxd−Tdnα⌉

f
(r)
+

(
k1

n
, ...

kd

n

)

·
b
(
n1−α

(
x1 −

k1
n

)
, ..., n1−α

(
xd −

kd
n

))

V
(
−→
x
)


 (31)

=
[(

Mn

(
f
(r)
−

))(
−→
x
)
,
(
Mn

(
f
(r)
+

))(
−→
x
)]

.

We have proved that (
MFn (f)

)(r)
±

= Mn

(
f
(r)
±

)
, ∀ r ∈ [0, 1] , (32)

respectively.

We present

Theorem 26. Let −→x ∈ Rd; then

D
((

MFn (f)
)(−→

x
)
, f
(
−→
x
))

≤ ω
(F)
1

(
f,

T∗

n1−α

)
. (33)

Notice that (33) gives MFn
D
→ IF pointwise and uniformly, as n → ∞, when f ∈ C

U
F

(
R

d
)
.

Proof. We observe that

D
((

MFn (f)
)(−→

x
)
, f
(
−→
x
))

=

sup
r∈[0,1]

max{
∣∣∣
(
MFn (f)

)(r)
−

(
−→
x
)
− f

(r)
−

(
−→
x
)∣∣∣ ,
∣∣∣
(
MFn (f)

)(r)
+

(
−→
x
)
− f

(r)
+

(
−→
x
)∣∣∣}

(32)
=

sup
r∈[0,1]

max{
∣∣∣
(
Mn

(
f
(r)
−

))(
−→
x
)
− f

(r)
−

(
−→
x
)∣∣∣ ,
∣∣∣
(
Mn

(
f
(r)
+

))(
−→
x
)
− f

(r)
+

(
−→
x
)∣∣∣}

(26)
≤

sup
r∈[0,1]

max

{

ω1

(
f
(r)
− ,

T∗

n1−α

)
, ω1

(
f
(r)
+ ,

T∗

n1−α

)}
(7)
= ω

(F)
1

(
f,

T∗

n1−α

)
,

proving the claim.

We continue with

Theorem 27. Let −→x ∈ Rd, f ∈ CNF
(
R

d
)
, N ∈ N,such that all of its fuzzy partial derivatives fα̃ of

order N, α̃ : |α̃| = N, are fuzzy uniformly continuous or fuzzy continuous and fuzzy bounded. Then

D
((

MFn (f)
)(−→

x
)
, f
(
−→
x
))

≤ (34)


32 George A. Anastassiou CUBO
16, 3 (2014)






N∑

j=1

(T∗)
j

j!nj(1−α)



(

d∑

i=1

D

(
∂

∂xi
, 0̃

))j
f
(
−→
x
)







+
(T∗)

N
dN

N!nN(1−α)
max

α̃:|α̃|=N
ω

(F)
1

(
fα̃,

T∗

n1−α

)
.

As n → ∞, we get D
((

MFn (f)
)(−→

x
)
, f
(
−→
x
))

→ 0 pointwise with rates.

Proof. As before we have

D
((

MFn (f)
)(−→

x
)
, f
(
−→
x
))

(32)
=

sup
r∈[0,1]

max{
∣∣∣
(
Mn

(
f
(r)
−

))(
−→
x
)
− f

(r)
−

(
−→
x
)∣∣∣ ,
∣∣∣
(
Mn

(
f
(r)
+

))(
−→
x
)
− f

(r)
+

(
−→
x
)∣∣∣}

(27)
≤

sup
r∈[0,1]

max











N∑

j=1

(T∗)
j

j!nj(1−α)



(

d∑

i=1

∣∣∣∣
∂

∂xi

∣∣∣∣

)j
f
(r)
−

(
−→
x
)







+
(T∗)

N
dN

N!nN(1−α)
max

α̃:|α̃|=N
ω1

((
f
(r)
−

)
α̃
,

T∗

n1−α

)
,






N∑

j=1

(T∗)
j

j!nj(1−α)



(

d∑

i=1

∣∣∣∣
∂

∂xi

∣∣∣∣

)j
f
(r)
+

(
−→
x
)







+
(T∗)

N
dN

N!nN(1−α)
max

α̃:|α̃|=N
ω1

((
f
(r)
+

)
α̃
,

T∗

n1−α

)}
(3)
≤

N∑

j=1

(T∗)
j

j!nj(1−α)
· (36)

sup
r∈[0,1]

max








(

d∑

i=1

∣∣∣∣
∂

∂xi

∣∣∣∣

)j
f
(r)
−

(
−→
x
)

 ,



(

d∑

i=1

∣∣∣∣
∂

∂xi

∣∣∣∣

)j
f
(r)
+

(
−→
x
)






+

(T∗)
N
dN

N!nN(1−α)
max

α̃:|α̃|=N
sup

r∈[0,1]

max

{

ω1

((
f
(r)
−

)
α̃
,

T∗

n1−α

)
, ω1

((
f
(r)
+

)
α̃
,

T∗

n1−α

)}

(by (3), (7), (12), (18))
≤






N∑

j=1

(T∗)
j

j!nj(1−α)



(

d∑

i=1

D

(
∂

∂xi
, 0̃

))j
f
(
−→
x
)






+ (37)

(T∗)
N
dN

N!nN(1−α)
max

α̃:|α̃|=N
ω

(F)
1

(
fα̃,

T∗

n1−α

)
,

proving the claim.


CUBO
16, 3 (2014)

Higher Order Multivariate Fuzzy Approximation by basic Neural . . . 33

4 Main Results - The fuzzy multivariate ”normalized squash-

ing 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∏

i=1

[−Ti, Ti], Ti > 0 and is nondecreasing there for each coordinate. S can be continuous only on

either
d∏

i=1

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

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”,

LFn (f)
(
−→
x
)
:= (38)

∑n2∗
k1=−n

2 ...
∑n2∗

kd=−n
2 f
(
k1
n
, ...kd

n

)
⊙ S

(
n1−α

(
x1 −

k1
n

)
, ..., n1−α

(
xd −

kd
n

))

W
(
−→
x
) ,

where 0 < α < 1 and n ∈ N:

n ≥ max
i∈{1,...,d}

{
Ti + |xi| , T

− 1
α

i

}
, (39)

and

W
(
−→
x
)
:=

n
2∗∑

k1=−n
2

...

n
2∗∑

kd=−n
2

S

(
n1−α

(
x1 −

k1

n

)
, ..., n1−α

(
xd −

kd

n

))
. (40)

It is clear that

(
LFn (f)

)(−→
x
)
:=

[
n−→x +

−→
T n

α

]
∗

∑

−→
k =

⌈
n−→x −

−→
T nα

⌉

f
(−→

k
n

)
⊙ S

(
n1−α

(
−→
x −

−→
k
n

))

Φ
(
−→
x
) , (41)

where

Φ
(
−→
x
)
:=

[
n−→x +

−→
T n

α

]

∑

−→
k =

⌈
n−→x −

−→
T nα

⌉
S


n1−α


−→x −

−→
k

n




 . (42)

Here, we study the D-metric pointwise convergence with rates of
(
LFn (f)

)(−→
x
)

→ f
(
−→
x
)
, as

n → +∞, −→x ∈ Rd.

This is given first by the next result.


34 George A. Anastassiou CUBO
16, 3 (2014)

Theorem 29. Under the above terms and asumptions, we find that

D
((

LFn (f)
)(−→

x
)
, f
(
−→
x
))

≤ ω
(F)
1

(
f,

T∗

n1−α

)
. (43)

Notice that (43) gives LFn
D
→ IF pointwise and uniformly, as n → ∞, when f ∈ C

U
F

(
R

d
)
.

Proof. Similar to (33).

We also give

Theorem 30. Let −→x ∈ Rd, f ∈ CNF
(
R

d
)
, N ∈ N, such that all of its fuzzy partial derivatives fα̃

of order N, α̃ : |α̃| = N, are fuzzy uniformly continuous or fuzzy continuous and fuzzy bounded.

Then

D
((

LFn (f)
)(−→

x
)
, f
(
−→
x
))

≤ (44)






N∑

j=1

(T∗)
j

j!nj(1−α)



(

d∑

i=1

D

(
∂

∂xi
, 0̃

))j
f
(
−→
x
)







+
(T∗)

N
dN

N!nN(1−α)
max

α̃:|α̃|=N
ω

(F)
1

(
fα̃,

T∗

n1−α

)
.

Inequality (44) gives us with rates the poitwise convergence of D(
(
LFn (f)

)(−→
x
)
, f
(
−→
x
)
) → 0 over

R
d, 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 Net-

work Operators, Computers and Mathematics, 48(2004), 1387-1401.


CUBO
16, 3 (2014)

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.


	Fuzzy real Analysis Background
	Convergence with rates of real multivariate neural network operators
	Main Results - Convergence with rates of fuzzy multivariate neural networks
	Main Results - The fuzzy multivariate ''normalized squashing type operators'' and their fuzzy convergence to the fuzzy unit with rates