EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 3, 2022, 1189-1200
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Neural Network of Multivariate Square Rational
Bernstein Operators with Positive Integer Parameter

Ibtihal J. Mohammad1,∗, Ali J. Mohammad1

1 Department of Mathematics, College of Education for Pure Science, University of Basrah,
Basrah, Iraq

Abstract. This research is defined a new neural network(NN) that depends upon a positive integer
parameter using the multivariate square rational Bernstein polynomials. Some theorems for this
network are proved, such as the pointwise and the uniform approximation theorems. Firstly,
the absolute moment for a function that belongs to Lipschitz space is defined to estimate the
order of the NN. Secondly, some numerical applications for this NN are given by taking two test
functions. Finally, the numerical results for this network are compared with the classical neural
networks(NNs). The results turn out that the new network is better than the classical one.

2020 Mathematics Subject Classifications: 41A25, 41A30, 47A58

Key Words and Phrases: Multivariate neural network, Multivariate square rational Bernstein
polynomials, Activation functions, Lipschitz Space

1. Introduction

In 2013, Costarelli and Spigler [3] introduced the artificial NN operators and studied
the behavior of this neural network in univariate Bernstein polynomials as:

For a bounded function f : [a, b] −→ R, the artificial neural networks Fn(f ;x), acti-
vated by the sigmoidal function σ and its acting on f , is defined as:

Fn(f ;x) =

⌊nb⌋∑
k=⌈na⌉

f

(
k

n

)
Φσ(x− k)

⌊nb⌋∑
k=⌈na⌉

Φσ(x− k)

, x ∈ [a, b],

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4425

Email addresses: pgs2206@uobasrah.edu.iq, ibtihaljas.moh@gmail.com (I.J. Mohammad),
ali.mohammad@uobasrah.edu.iq, alijasmoh@gmail.com (A.J. Mohammad).

https://www.ejpam.com 1189 © 2022 EJPAM All rights reserved.



I.J. Mohammad, A.J. Mohammad / Eur. J. Pure Appl. Math, 15 (3) (2022), 1189-1200 1190

where the symbols ⌊.⌋, ⌈.⌉ denote taking the ”floor” and the ”ceiling” of a given number,
respectively. And the case of multivariate in [4] it is given by the formula: The bounded
function: f : R −→ R, activated by the sigmoidal function σ and acting on f , is defined
as:

F s
n(f ;x) =

⌊nb1⌋∑
k1=⌈na1⌉

...

⌊nbs⌋∑
ks=⌈nas⌉

f

(
k

n

)
Ψσ(nx− k)

⌊nb1⌋∑
k1=⌈na1⌉

...

⌊nbs⌋∑
ks=⌈nas⌉

Ψσ(nx− k)

, (1)

where x ∈ R = [a1, b1] × . . . × [as, bs], Ψσ is a density function that is built from a
sigmoidal function σ and k = (k1, . . . , ks) ∈ Z+.

In 2014, Costarelli and Spigler [5] extended formula (1) through the use of the Kan-
torovich operator type to introduce and studied approximation theorems to this multi-
variate NN operators.

In 2016, Costarelli and Vinti [6] introduced the structure of a NN operators of type
multivariate max-product then studied the approximation theoremed and estimates the
rate of convergence to this NN operators.

In 2017, Gavrea and Ivan [7] introduced definition to square Bernstein polynomials it
is given by the formula:
For x ∈ [0, 1], f ∈ C[0, 1],

Bn,2(f ;x) =

n∑
k=0

b2n,k(x)f

(
k

n

)
n∑

k=0

b2n,k(x)

, n = 1, 2, ..., (2)

where b2n,k(x) = (bn,k(x))
2.

In 2017, Mohammad and Mohammad [9] introduced a definition of the NN operators by
using the type of summation-integral Bernstein, and then studied approximation theorems
for this NN operators.

In 2018, Hassan [8] introduce and define the new modified of Bernstein operators that
can use to build NNs.

In 2019, Bajpeyi and Kumar [1] introduced definition to the neural network of expo-
nential type and studied behavior in two case one- dimensional and multi-dimensional.In
2019, Costarelli and others [2] introduced definition to the neural network of multivariate
max-product NN of Kantorovich type.

In 2021, Mohammad and Mohammad [10] give a new modification to the formula (1)
and studied approximation theorems for this NN operators,activated by the sigmoidal
function σ and acting on f , it is given by the formula:



I.J. Mohammad, A.J. Mohammad / Eur. J. Pure Appl. Math, 15 (3) (2022), 1189-1200 1191

Gn,m(f ;x) =

∑
k

Ψσ(nx− k)f
((
n−1k− x

)m − x
)

∑
k

Ψσ(nx− k)
, (3)

This paper gives extended to the NN operators in formula (3) by using formula of
square Bernstein polynomials in formula (2) and studied approximation theorems for this
neural network. In the end, we give some numerical examples for this NNs.

2. Preliminary Results

In this part recall some preliminary results.
A sigmoidal function is measurable functions satisfying limx→−∞ σ(x) = 0 and limx→+∞ σ(x) =

1, for example logistic function σl(x) = (1 + e−x)−1, hyperbolic tangent function σh(x) =
1

2
[tanh(x) + 1].

For every non-decreasing function σ satisfying assumptions:

(i) the function such that gσ(x) = σ(x)− 1/2, is odd;

(ii) function σ ∈ C2(R) is concave for x ≥ 0;

(iii) function σ satisfying σ(x) = O(|x|−1−α) as x −→ −∞, for some α>0.

Defined the function as: Φσ(x) =
1

2
[σ(x+ 1)− σ(x− 1)], x ∈ R.

Now, gives some definitions that we will use:

Definition 1. [4] A sigmoidal function is a measurable function satisfying the following
two conditions:

limx→−∞ ζ(x) = 0;

limx→+∞ ζ(x) = 1.

Definition 2. [4]
The Lipschitz classes are defined as follows:

Lip(v) = {f ∈ C0(R) such that there exist γ>0, C>0 so that, for each x ∈ R, |f(x+ t)−
f(x)| ≤ C∥t∥v2 for every ∥t∥2 ≤ γ with (x+ t) ∈ R}.

Definition 3. [5]
The multivariate for the Φσ(x) define as follows: Ψσ(x) = Φσ(x1) ·Φσ(x2) · ... ·Φσ(xs),

for every x ∈ Rs.
Now, in the following lemmas set of properties for the functions Φσ(x) will be studied.

Lemma 1. [5]
To the function Φσ(x) for x ∈ R, then:



I.J. Mohammad, A.J. Mohammad / Eur. J. Pure Appl. Math, 15 (3) (2022), 1189-1200 1192

(i) Φσ(x) ≥ 0 for every x ∈ R and limx→±∞ Φσ(x) = 0;

(ii) Φσ(x) is a symmetrical function about the y-axis;

(iii)
∑

k∈Z Φσ(x− k) = 1, For every x ∈ R ;

(iv) For x<0 the function Φσ(x) is non-decreasing and for x ≥ 0 it is non-increasing;

(v) Φσ(x) = O(|x|−1−α) as x −→ ±∞;

(vi) The sum
∑

k∈Z Φσ(x− k) converges uniformly on subsets compact of R.

The following lemmas set of properties for the functions Ψσ(x− k) will be studied.

Lemma 2. [4]
To the function Ψσ(x− k) for x ∈ Rs, then:

(i)
∑

k Ψσ(x− k) = 1,for every x ∈ Rs;

(ii) On compact subsets of Rs the series
∑

k Ψσ(x− k) converges uniformly on compact
subsets of Rs;

(iii) For every γ>0, we get

lim
x→∞

∑
∥x−k∥>γn

Ψσ(x− k) = 0,

uniformly respect to x ∈ Rs.In a special case, for every γ>0 and 0<v<α ,∑
∥x−k∥>γn

Ψσ(x− k) = O(n−v), n −→ +∞,

where the constant α>0 as in condition (iii).

Lemma 3. [3], [4]

(i) For x ∈ [a, b] ⊂ R, n ∈ N+, ⌈na⌉ ≤ ⌊nb⌋, then:

1
⌊nb⌋∑

k=⌈na⌉

Φσ(nx− k)

≤ 1

Φσ(1)
;

(ii) For x ∈ [a1, b1]× ...× [as, bs] ⊂ Rs, n ∈ N+ so that ⌈na⌉ ≤ ⌊nb⌋ for every i = 1, ..., s,
then:

1

s∏
i=1

⌊nbi⌋∑
ki=⌈nai⌉

Φσ(nxi − ki)

≤ 1

[Φσ(1)]
s .



I.J. Mohammad, A.J. Mohammad / Eur. J. Pure Appl. Math, 15 (3) (2022), 1189-1200 1193

3. Auxiliary Results

We will define and discuss multivariate NN operators Qm(f ;x) as follows:

Definition 4. For a continuous bounded function f : R −→ R, the NN operators of mul-
tivariate square rational Bernstein operators with positive integer parameter m, Qm(f ;x)
activated by the sigmoidal function σ acting on f , defined as the following:

Qm(f ;x) =

∑
k

Ψ2
σ(nx− k)f

((
n−1k− x

)m − x
)

∑
k

Ψ2
σ(nx− k)

,m ∈ N+

∑
k

=

⌊nb1⌋∑
k1=⌈na1⌉

...

⌊nbs⌋∑
ks=⌈nas⌉

.

for sufficiently large n ∈ N, x ∈ R, Qm(1;x) = 1.

Definition 5. For v > 0, the discrete absolutely moment of the function Φ2
σ(x) of order

v is defined as

mv(Φ
2
σ) = sup

x∈R

∑
k∈Z

Φ2
σ(x− k)|x− k|v.

We will need to give some properties of the functions Φ2
σ(x) and Ψ2

σ(x) in the following
lemmas:

Lemma 4. Some properties to the function Φ2
σ(x) defined on x ∈ R, then:

(i) Φ2
σ(x) ≥ 0 for every x ∈ R and limx→±∞ Φ2

σ(x) = 0;

(ii) Φ2
σ(x) is a symmetrical function about the y-axis;

(iii)
∑

k∈Z Φ
2
σ(x− k) ≈ 0.156517, For every x ∈ R;

(iv) For x<0 the function Φ2
σ(x) is non-decreasing and for x ≥ 0 it is non-increasing;

(v) Φ2
σ(x) = O(|x|2(−1−α)) as x −→ ±∞;

(vi) The sum
∑

k∈Z Φ
2
σ(x− k) converges uniformly on subsets compact of R.

Proof. By applying Lemma 1, we can prove (i), (ii), (iv),(v) and (vi) immediately, the
consequence (iii) can be claimed by using Maple software. □

The next lemma gives some properties for the function Ψ2
σ(nx− k).

Lemma 5. To the function Ψ2
σ(x− k) for x ∈ Rs, then:



I.J. Mohammad, A.J. Mohammad / Eur. J. Pure Appl. Math, 15 (3) (2022), 1189-1200 1194

(i)
∑

k Ψ
2
σ(x− k) ≈ (0.156517)s, for every x ∈ Rs;

(ii) On compact subsets of Rs the series
∑

k Ψ
2
σ(x− k) converges uniformly on compact

subsets of Rs;

(iii) For every γ>0, we get

lim
x→∞

∑
∥x−k∥>γn

Ψ2
σ(x− k) = 0,

uniformly respect to x ∈ Rs.In a special case, for every γ>0 and 0<v<α,∑
∥x−k∥>γn

Ψ2
σ(x− k) = O(n−v), as n −→ +∞

where the constant α>0 as in condition (iii).

Proof.Using the Definition 3 and Lemma 2, the consequence (i),(ii),(iii) gets immediate.
□

Lemma 6. (i) For x ∈ [a, b] ⊂ R, n ∈ N+, ⌈na⌉ ≤ ⌊nb⌋, then:

1
⌊nb⌋∑

k=⌈na⌉

Φ2
σ(nx− k)

≤ 1

Φ2
σ(1)

;

(ii) For x ∈ [a1, b1]× ...× [as, bs] ⊂ Rs, n ∈ N+ so that ⌈na⌉ ≤ ⌊nb⌋ for every i = 1, ..., s,
then:

1

s∏
i=1

⌊nbi⌋∑
ki=⌈nai⌉

Φ2
σ(nxi − ki)

≤ 1

[Φ2
σ(1)]

s .

Proof. Using the properties of Lemma 3 the proof of this Lemma follows immediately. □

The following theorem studies the pointwise and the uniform convergence for the NN
operators, Qm(f ;x).

Theorem 1. For a bounded function f : R −→ R, and continuous at each point x ∈ R,
then

lim
n→∞

Qm(f ;x) = f(x)

if f ∈ C0(R), then

lim
n→∞

sup
x∈R

|Qm(f ;x)− f(x)| = lim
n−→∞

∥Qm(f ; .)− f(.)∥∞ = 0.



I.J. Mohammad, A.J. Mohammad / Eur. J. Pure Appl. Math, 15 (3) (2022), 1189-1200 1195

Proof. Suppose x ∈ R is a point of continuity of fwe have

|Qm(f ;x)− f(x)| =

∣∣∣∣∣∣∣∣
∑
k

Ψ2
σ(nx− k)f

((
n−1k− x

)m − x
)

∑
k

Ψ2
σ(nx− k)

− f(x)

∣∣∣∣∣∣∣∣
=

∣∣∣∣∣∣∣∣
∑
k

Ψ2
σ(nx− k)

[
f
((
n−1k− x

)m − x
)
− f(x)

]
∑
k

Ψ2
σ(nx− k)

∣∣∣∣∣∣∣∣
by using Lemma 6(ii),we get:

|Qm(f ;x)− f(x)| ≤ 1

[Φ2
σ(1)]

s

∑
k

Ψ2
σ(nx− k)

∣∣f ((
n−1k− x

)m − x
)
− f(x)

∣∣
For every n −→ ∞,n ∈ N+,x ∈ Rs are arbitrary but fixed. Suppose for a fixed ε>0,

and from the continuity of f at x, ∃γ>0 : |f(y)− f(x)|<ε, ∀y ∈ R with ∥y − x∥<ε, the
symbol ∥.∥2 denote to Euclidean norm.
Now we get

|Qm(f ;x)− f(x)| ≤ 1

[Φ2
σ(1)]

s

∑
∥(n−1k−x)m∥<

γ√
s

Ψ2
σ(nx− k)

∣∣f (
(n−1k− x)m − x

)
− f(x)

∣∣+
1

[Φ2
σ(1)]

s

∑
∥(n−1k−x)m∥≥

γ√
s

Ψ2
σ(nx− k)

∣∣f (
(n−1k− x)m − x

)
− f(x)

∣∣

:=
1

[Φ2
σ(1)]

s
(I1 + I2)

Now using the continuity of f and Lemma 5 we get that∥∥(n−1k− x
)m − x

∥∥
2
≤

√
s
∥∥(n−1k− x

)m − x
∥∥ ≤ γ

So estimation I1 is,

I1<ε
∑

∥(nx−k)m∥<
γn√
s

Ψ2
σ(nx− k) ≤ ε.

I2 ≤ 2 ∥f∥∞
∑

∥(nx−k)m∥≥
γn√
s

Ψ2
σ(nx− k) ≤ ε.



I.J. Mohammad, A.J. Mohammad / Eur. J. Pure Appl. Math, 15 (3) (2022), 1189-1200 1196

uniformly ∀x ∈ Rs.The first direction of the theorem holds because ε arbitrarily. When
f ∈ C0(R), the prove of other direction is readily followed in the same way by exchange
γ>0 with the parameter of the uniform continuity of f on R. □

Now, in the following, study the order of approximation of NN operators in f ∈ Lip(v).

Theorem 2. Suppose f ∈ Lip(v) for some v, at 0<v ≤ 1, and let sigmoidal function σ
satisfies the condition (iii) for ∥Qm(f ;x)− f(x)∥∞ = O(n−vm), as n −→ ∞.

Proof. Let f ∈ Lip(v), for every x ∈ R, for some v, with v ∈ (0, 1], and Lemma 6 one
can write as in the theorem 1

|Qm(f ;x)− f(x)| ≤ 1

[Φ2
σ(1)]

s

∑
k

Ψ2
σ(nx− k)

∣∣f ((
n−1k− x

)m − x
)
− f(x)

∣∣
Now by using the definition of Lip(v), where γ>0, C>0 are constants relative to f we
obtain Let x ∈ R the point of continuity of f
|Qm(f ;x)− f(x)| ≤

1

[Φ2
σ(1)]

s

∑
∥(n−1k−x)m∥<

γ√
s

Ψ2
σ(n

−1k− x)
∣∣f (

(n−1k− x)m − x
)
− f(x)

∣∣

+
1

[Φ2
σ(1)]

s

∑
∥(n−1k−x)m∥≥

γ√
s

Ψ2
σ(nx− k)

∣∣f (
(n−1k− x)m − x

)
− f(x)

∣∣

:=
1

[Φ2
σ(1)]

s
(J1 + J2)

since f ∈ Lip(v), we get for
∥∥(n−1k− x

)m − x
∥∥
2
≤

√
s
∥∥(n−1k− x

)m − x
∥∥ ≤ γ and

hence
|f ((nx− k)m − x)− f(x)|<C

∥∥(n−1k− x
)m∥∥v

2
≤ Cs

v
2

∥∥(n−1k− x
)m∥∥v

J1<n−vmCs
v
2

∑
∥(nx−k)m∥<

γn√
s

Ψ2
σ(nx− k)

∥∥(n−1k− x
)m∥∥v

for fixed 0<vi<α , by using Lemma 5, for a compact subset K ⊂ Rs, for every x ∈ Rs, if
n −→ ∞ can write the following:

J1 ≤ n−vmCs
v
2

∑
∥(nx−k)m∥<

γn√
s

Ψ2
σ(nx− k)

∥∥(n−1k− x
)m∥∥v



I.J. Mohammad, A.J. Mohammad / Eur. J. Pure Appl. Math, 15 (3) (2022), 1189-1200 1197

≤ n−vmCs
v
2

s∑
j=1

∑
kj∈Z

Φ2
σ(nxj − kj) |nxj − kj |vm

 ∑
k[j]∈Zs−1

Ψ2[j]
σ (nx[j] − k[j])


where Ψ2[j]

σ (nx[j] − k[j]) = Φ2
σ(nx1 − k1) · ... · Φ2

σ(nxj−1 − kj−1) · Φ2
σ(nxj+1 − kj+1) ·

... · Φ2
σ(nxs − ks), notice for every j = 1, ..., s x[j] = (x1, ..., xj−1, xj+1, ..., xs) ∈ Rs−1,

k[j] = (k1, ..., kj−1, kj+1, ..., ks) ∈ Zs−1. Now k[j] ⊂ R the set of the j-th projection of a
compact set of all elements. By using Lemma 5 and for all sufficiently large N ∈ N+ one
can obtain

≤ (0.156517)s−1n−vmCs
v
2

s∑
j=1

∑
kj∈Z

Φ2
σ(nxj − kj) |nxj − kj |vm



≤ (0.156517)s−1n−vmCs1+
v
2mvm(Φ2

σ)

note that mvm(Φ2
σ) ≤ ∞ , where mvm(Φ2

σ) give in Definition 5, since v < α one can
obtain:

J1 = O(n−vm), as n −→ ∞,

Now, we estimate J2 by using the other direction of lemma 5,

I2 ≤ 2 ∥f∥∞
∑

∥(nx−k)m∥≥
γn√
s

Ψ2
σ(nx− k) = O(n−vm), as n −→ ∞. □

Theorem 3. Let the function σ for some α ∈ (0, 1] satisfy the condition (iii), and let
f ∈ Lip(v) for some v ∈ (0, 1], then we have,

(i) ∥Qm(f ; .)− f(.)∥∞ = O(n−vm), as n −→ ∞ If v<α;

(ii) ∥Qm(f ; .)− f(.)∥∞ = O(n−(α−ε)m), as n −→ ∞, for every 0<ε<α, if α ≤ v<1.

Proof.

(i) Using the same step of the ”Theorm 2” we can obtain proving

∥Qm(f ; .)− f(.)∥∞ = O(n−vm), as n −→ ∞

(ii) As a special case for all f ∈ Lip(v) with α ≤ v ≤ 1, with ε is fixed but arbitrary
choose β := α− ε, and get 0<β<α, by based on part (i) we obtain,

∥Qm(f ; .)− f(.)∥∞ = O(n−βm) = O(n−(α−ε)m), as n −→ ∞

for function f ∈ Lip(β), 0<ε<α. □



I.J. Mohammad, A.J. Mohammad / Eur. J. Pure Appl. Math, 15 (3) (2022), 1189-1200 1198

4. Numerical Examples

This section gives numerical examples for the real value of n = 10, 30, m = 1, 2, 3 and
the functions of testing f(x, y) = cos (9xy) + 2 sin (x+ y) and g(x, y) = (2x− 1)2 − (2y −
1)2, (x, y) ∈ [0, 1]2. For the NN operators Qm(.;x, y) with the NN operators Fn(.;x, y).
We analyze the results in the figures as examples of the convergence of NN operators
Qm(.;x, y), Fn(.;x, y) with test the functions f(x, y), g(x, y). Also, we give the table to
maximum error function for Qm(.;x, y),Fn(.;x, y) as follows:

Example 1. For n = 10, 30,m = 1, 2, 3, the convergence of NN operators Qm(f ;x, y),
Fn(f ;x, y) to test function f(x, y) can be descripted in the Figure 1.

  

   𝑛 = 10, 𝑚 = 1    𝑛 = 10, 𝑚 = 2    𝑛 = 10, 𝑚 = 3 

   𝑛 = 30, 𝑚 = 1    𝑛 = 30, 𝑚 = 2    𝑛 = 30, 𝑚 = 3 

Figure 1: The numerical convergence of NN operators Fn(f ;x, y) (red) and Qm(f ;x, y) (Lime) to f(x, y)
(blue).

Example 2. Forn = 10, 30,m = 1, 2, 3, the convergence of NN operators Qm(g;x, y),
Fn(g;x, y)to test function g(x, y) can be descripted in the Figure 2.



I.J. Mohammad, A.J. Mohammad / Eur. J. Pure Appl. Math, 15 (3) (2022), 1189-1200 1199

 

 

   𝑛 = 10, 𝑚 = 1 
 

   𝑛 = 10, 𝑚 = 2 
 

   𝑛 = 10, 𝑚 = 3 

 

   𝑛 = 30, 𝑚 = 1 
 

   𝑛 = 30, 𝑚 = 2 
 

   𝑛 = 30, 𝑚 = 3 

Figure 2: The numerical convergence of NN operators Fn(g;x, y) (red) and Qm(g;x, y) (Lime) to g(x, y)
(blue).

Now, the maximum error values calculated by the flowing table between the test func-
tion f(x, y), g(x, y) and NN operators in R2:

Table 1: Maximum error.

NN n m = 1 m = 2 m = 3

Fn(f ;x, y) 10 0.757395453 0.757395451 0.757395451
Qm(f ;x, y) 0.490870074 0.187277077 0.019435682
fn(g;x, y) 0.4733567674 0.4733567674 0.4733567674
Qm(g;x, y) 0.2810197385 0.0785342121 0.0091303333
Fn(f ;x, y) 30 0.185638151 0.185638151 0.185638151
Qm(f ;x, y) 0.091746328 0.024034970 0.000333905
Fn(g;x, y) 0.1551244367 0.1551244367 0.1551244367
Qm(g;x, y) 0.0888466026 0.01026517101 0.0003333514

5. Conclusions

From Table 1 above and the two numerical examples, The NN operators Qm(.;x, y)
better the classical NN operators Fn(.;x, y) in terms of numerical results for the two test
functions f and g.



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