Multivariate Fuzzy-Random and Stochastic Arctangent, Algebraic, Gudermannian and Generalized Symmetric Activation Functions Induced Neural Network Approximations

Abstract

In this article we study the degree of approximation of multivariate pointwise and uniform convergences in the q-mean to the Fuzzy-Random unit operator of multivariate Fuzzy-Random Quasi-Interpolation arctangent, algebraic, Gudermannian and generalized symmetric activation functions based neural network operators. These multivariate Fuzzy-Random operators arise in a natural way among multivariate Fuzzy-Random neural networks. The rates are given through multivariate Probabilistic-Jackson type inequalities involving the multivariate Fuzzy-Random modulus of continuity of the engaged multivariate Fuzzy-Random function. The plain stochastic extreme analog of this theory is also met in detail for the stochastic analogs of the operators: the stochastic full quasi-interpolation operators, the stochastic Kantorovich type operators and the stochastic quadrature type operators.

1 Fuzzy-Random Functions and Stochastic Processes Background

See also [18], Ch. 22, pp. 497–501.

We start with

Definition 1

(see [35]). Let $\mu :\mathbb {R}\rightarrow \left[ 0,1\right]$ with the following properties:

• (i) is normal, i.e., $\exists$ $x_{0}\in \mathbb {R}:\mu \left( x_{0}\right) =1.$

• (ii) $\mu \left( \lambda x+\left( 1-\lambda \right) y\right) \ge \min \{ \mu \left( x\right) ,\mu \left( y\right) \}$, $\forall$ $x,y\in \mathbb {R},$ $\forall$ $\lambda \in \left[ 0,1\right]$ ($\mu$ is called a convex fuzzy subset).

• (iii) $\mu$ is upper semicontinuous on $\mathbb {R}$, i.e., $\forall$ $x_{0}\in \mathbb {R}$ and $\forall$ $\varepsilon >0$, $\exists$ neighborhood $V\left( x_{0}\right) :\mu \left( x\right) \le \mu \left( x_{0}\right) +\varepsilon$, $\forall$ $x\in V\left( x_{0}\right) .$

• (iv) the set $\overline{\text {supp}\left( \mu \right) }$ is compact in $\mathbb {R}$ (where supp$\left( \mu \right) :=\{x\in \mathbb {R};\mu \left( x\right) >0\}$).

We call $\mu$ a fuzzy real number. Denote the set of all $\mu$ with $\mathbb {R}_{\mathcal {F}}.$

E.g., $\chi _{\{x_{0}\}}\in \mathbb {R}_{\mathcal {F}}$, for any $x_{0}\in \mathbb {R},$ where $\chi _{\{x_{0}\}}$ is the characteristic function at $x_{0}$.

For $0<r\le 1$ and $\mu \in \mathbb {R}_{\mathcal {F}}$ define $\left[ \mu \right] ^{r}:=\{x\in \mathbb {R}:\mu \left( x\right) \ge r\}$ and $\left[ \mu \right] ^{0}:=\overline{\{x\in \mathbb {R}:\mu \left( x\right) >0\}}.$

Then it is well known that for each $r\in \left[ 0,1\right]$, $\left[ \mu \right] ^{r}$ is a closed and bounded interval of $\mathbb {R}$. For $u,v\in \mathbb {R}_{\mathcal {F}}$ and $\lambda \in \mathbb {R}$, we define uniquely the sum $u\oplus v$ and the product $\lambda \odot u$ by

$$\begin{aligned} \left[ u\oplus v\right] ^{r}=\left[ u\right] ^{r}+\left[ v\right] ^{r},\text { }\left[ \lambda \odot u\right] ^{r}=\lambda \left[ u\right] ^{r},\text { }\forall \text { }r\in \left[ 0,1\right] , \end{aligned}$$

where $\left[ u\right] ^{r}+\left[ v\right] ^{r}$ means the usual addition of two intervals (as subsets of $\mathbb {R}$) and $\lambda \left[ u\right] ^{r}$ means the usual product between a scalar and a subset of $\mathbb {R}$ (see, e.g., [35]). Notice $1\odot u=u$ and it holds $u\oplus v=v\oplus u$, $\lambda \odot u=u\odot \lambda$. If $0\le r_{1}\le r_{2}\le 1$ then $\left[ u\right] ^{r_{2}}\subseteq \left[ u\right] ^{r_{1}}$. Actually $\left[ u\right] ^{r}=\left[ u_{-}^{\left( r\right) },u_{+}^{\left( r\right) } \right]$, where $u_{-}^{\left( r\right) }<u_{+}^{\left( r\right) }$, $u_{-}^{\left( r\right) },u_{+}^{\left( r\right) }\in \mathbb {R}$, $\forall$ $r\in \left[ 0,1\right] .$

Define

$$\begin{aligned} D:\mathbb {R}_{\mathcal {F}}\times \mathbb {R}_{\mathcal {F}}\rightarrow \mathbb { R}_{+}\cup \{0\} \end{aligned}$$

by

$$\begin{aligned} D\left( u,v\right) :=\underset{r\in \left[ 0,1\right] }{\sup }\max \left\{ \left| u_{-}^{\left( r\right) }-v_{-}^{\left( r\right) }\right| ,\left| u_{+}^{\left( r\right) }-v_{+}^{\left( r\right) }\right| \right\} , \end{aligned}$$

where $\left[ v\right] ^{r}=\left[ v_{-}^{\left( r\right) },v_{+}^{\left( r\right) }\right] ;$ $u,v\in \mathbb {R}_{\mathcal {F}}$. We have that D is a metric on $\mathbb {R}_{\mathcal {F}}$. Then $\left( \mathbb {R}_{\mathcal {F} },D\right)$ is a complete metric space, see [35], with the properties

$$\begin{aligned} \begin{array}{c} D\left( u\oplus w,v\oplus w\right) =D\left( u,v\right) ,\text { }\forall \text { }u,v,w\in \mathbb {R}_{\mathcal {F}}, \\ D\left( k\odot u,k\odot v\right) =\left| k\right| D\left( u,v\right) \text {, }\forall \text { }u,v\in \mathbb {R}_{\mathcal {F}}\text { , }\forall \text { }k\in \mathbb {R}, \\ D\left( u\oplus v,w\oplus e\right) \le D\left( u,w\right) +D\left( v,e\right) \text {, }\forall \text { }u,v,w,e\in \mathbb {R}_{\mathcal {F}}.\end{array} \end{aligned}$$

(1)

Let $\left( M,d\right)$ metric space and $f,g:M\rightarrow \mathbb {R}_{ \mathcal {F}}$ be fuzzy real number valued functions. The distance between f, g is defined by

$$\begin{aligned} D^{*}\left( f,g\right) :=\underset{x\in M}{\sup }D\left( f\left( x\right) ,g\left( x\right) \right) . \end{aligned}$$

On $\mathbb {R}_{\mathcal {F}}$ we define a partial order by “$\le$”: $u,v\in \mathbb {R}_{\mathcal {F}}$, $u\le v$ iff $u_{-}^{\left( r\right) }\le v_{-}^{\left( r\right) }$ and $u_{+}^{\left( r\right) }\le v_{+}^{\left( r\right) }$, $\forall$ $r\in \left[ 0,1\right] .$

$\sum \limits ^{*}$ denotes the fuzzy summation, $\widetilde{o}:=\chi _{\{0\}}\in \mathbb {R}_{\mathcal {F}}$ the neutral element with respect to $\oplus$. For more see also [36, 37].

We need

Definition 2

(see also [30], Definition 13.16, p. 654). Let $\left( X, \mathcal {B},P\right)$ be a probability space. A fuzzy-random variable is a $\mathcal {B}$-measurable mapping $g:X\rightarrow \mathbb {R}_{\mathcal {F}}$ (i.e., for any open set $U\subseteq \mathbb {R}_{\mathcal {F}},$ in the topology of $\mathbb {R}_{\mathcal {F}}$ generated by the metric D, we have

$$\begin{aligned} g^{-1}\left( U\right) =\{s\in X;g\left( s\right) \in U\} \in \mathcal {B} \text {).} \end{aligned}$$

(2)

The set of all fuzzy-random variables is denoted by $\mathcal {L}_{\mathcal {F} }\left( X,\mathcal {B},P\right)$. Let $g_{n},g\in \mathcal {L}_{\mathcal {F} }\left( X,\mathcal {B},P\right)$, $n\in \mathbb {N}$ and $0<q<+\infty$. We say if

$$\begin{aligned} \underset{n\rightarrow +\infty }{\lim }\int _{X}D\left( g_{n}\left( s\right) ,g\left( s\right) \right) ^{q}P\left( ds\right) =0. \end{aligned}$$

(3)

Remark 1

(see [30], p. 654). If $f,g\in \mathcal {L}_{\mathcal {F}}\left( X,\mathcal {B},P\right)$, let us denote $F:X\rightarrow \mathbb {R}_{+}\cup \{0\}$ by $F\left( s\right) =D\left( f\left( s\right) ,g\left( s\right) \right)$, $s\in X$. Here, F is $\mathcal {B}$-measurable, because $F=G\circ H$, where $G\left( u,v\right) =D\left( u,v\right)$ is continuous on $\mathbb {R}_{\mathcal {F}}\times \mathbb {R}_{\mathcal {F}}$, and $H:X\rightarrow \mathbb {R}_{\mathcal {F}}\times \mathbb {R}_{\mathcal {F}}$, $H\left( s\right) =\left( f\left( s\right) ,g\left( s\right) \right)$, $s\in X$, is $\mathcal {B}$-measurable. This shows that the above convergence in q -mean makes sense.

Definition 3

(see [30], p. 654, Definition 13.17). Let $\left( T, \mathcal {T}\right)$ be a topological space. A mapping $f:T\rightarrow \mathcal {L}_{\mathcal {F}}\left( X,\mathcal {B},P\right)$ will be called fuzzy-random function (or fuzzy-stochastic process) on T. We denote $f\left( t\right) \left( s\right) =f\left( t,s\right)$, $t\in T$, $s\in X$.

Remark 2

(see [30], p. 655). Any usual fuzzy real function $f:T\rightarrow \mathbb {R}_{\mathcal {F}}$ can be identified with the degenerate fuzzy-random function $f\left( t,s\right) =f\left( t\right)$, $\forall$ $t\in T$, $s\in X$.

Remark 3

(see [30], p. 655). Fuzzy-random functions that coincide with probability one for each $t\in T$ will be consider equivalent.

Remark 4

(see [30], p. 655). Let $f,g:T\rightarrow \mathcal {L}_{ \mathcal {F}}\left( X,\mathcal {B},P\right)$. Then $f\oplus g$ and $k\odot f$ are defined pointwise, i.e.,

$$\begin{aligned} \left( f\oplus g\right) \left( t,s\right)= & {} f\left( t,s\right) \oplus g\left( t,s\right) , \\ \left( k\odot f\right) \left( t,s\right)= & {} k\odot f\left( t,s\right) \text { , }t\in T,s\in X,\text { }k\in \mathbb {R}. \end{aligned}$$

Definition 4

(see also Definition 13.18, pp. 655–656, [30]). For a fuzzy-random function $f:W\subseteq \mathbb {R}^{N}\rightarrow \mathcal {L}_{ \mathcal {F}}\left( X,\mathcal {B},P\right)$, $N\in \mathbb {N}$, we define the (first) fuzzy-random modulus of continuity

$$\begin{aligned} \varOmega _{1}^{\left( \mathcal {F}\right) }\left( f,\delta \right) _{L^{q}}= \end{aligned}$$

$$\begin{aligned} \sup \left\{ \left( \int _{X}D^{q}\left( f\left( x,s\right) ,f\left( y,s\right) \right) P\left( ds\right) \right) ^{\frac{1}{q}}:x,y\in W,\text { } \left\| x-y\right\| _{\infty }\le \delta \right\} , \end{aligned}$$

$0<\delta ,$ $1\le q<\infty .$

Definition 5

[16]. Here $1\le q<+\infty$. Let $f:W\subseteq \mathbb {R} ^{N}\rightarrow \mathcal {L}_{\mathcal {F}}\left( X,\mathcal {B},P\right)$, $N\in \mathbb {N}$, be a fuzzy random function. We call f a (q-mean) uniformly continuous fuzzy random function over W, iff $\forall$ $\varepsilon >0$ $\exists$ $\delta >0:$whenever $\left\| x-y\right\| _{\infty }\le \delta ,$ $x,y\in W,$ implies that

$$\begin{aligned} \int _{X}\left( D\left( f\left( x,s\right) ,f\left( y,s\right) \right) \right) ^{q}P\left( ds\right) \le \varepsilon . \end{aligned}$$

We denote it as $f\in C_{FR}^{U_{q}}\left( W\right) .$

Proposition 1

[16]. Let $f\in C_{FR}^{U_{q}}\left( W\right) ,$ where $W\subseteq \mathbb {R}^{N}$ is convex.

Then $\varOmega _{1}^{\left( \mathcal {F}\right) }\left( f,\delta \right) _{L^{q}}<\infty$, any $\delta >0.$

Proposition 2

[16]. Let $f,g:W\subseteq \mathbb {R}^{N}\rightarrow \mathcal {L}_{\mathcal {F}}\left( X,\mathcal {B},P\right)$, $N\in \mathbb {N}$, be fuzzy random functions. It holds

• (i) $\varOmega _{1}^{\left( \mathcal {F}\right) }\left( f,\delta \right) _{L^{q}}$ is nonnegative and nondecreasing in $\delta >0.$

• (ii) $\underset{\delta \downarrow 0}{\lim }\varOmega _{1}^{\left( \mathcal {F} \right) }\left( f,\delta \right) _{L^{q}}=\varOmega _{1}^{\left( \mathcal {F} \right) }\left( f,0\right) _{L^{q}}=0$, iff $f\in C_{FR}^{U_{q}}\left( W\right) .$

We mention

Definition 6

(see also [6]). Let $f\left( t,s\right)$ be a random function (stochastic process) from $W\times \left( X,\mathcal {B},P\right) ,$ $W\subseteq \mathbb {R}^{N},$ into $\mathbb {R}$, where $\left( X,\mathcal {B} ,P\right)$ is a probability space. We define the q-mean multivariate first modulus of continuity of f by

$$\begin{aligned} \varOmega _{1}\left( f,\delta \right) _{L^{q}}:= \end{aligned}$$

$$\begin{aligned} \sup \left\{ \left( \int _{X}\left| f\left( x,s\right) -f\left( y,s\right) \right| ^{q}P\left( ds\right) \right) ^{\frac{1}{q}}:x,y\in W,\text { }\left\| x-y\right\| _{\infty }\le \delta \right\} , \end{aligned}$$

(4)

$\delta >0,$ $1\le q<\infty$.

The concept of f being (q-mean) uniformly continuous random function is defined the same way as in Definition 5, just replace D by $\left| \cdot \right|$, etc. We denote it as $f\in C_{\mathbb {R} }^{U_{q}}\left( W\right) .$

Similar properties as in Propositions 1, 2 are valid for $\varOmega _{1}\left( f,\delta \right) _{L^{q}}.$

Also we have

Proposition 3

[3]. Let $Y\left( t,\omega \right)$ be a real valued stochastic process such that Y is continuous in $t\in \left[ a,b\right]$. Then Y is jointly measurable in $\left( t,\omega \right) .$

According to [28], p. 94 we have the following

Definition 7

Let $\left( Y,\mathcal {T}\right)$ be a topological space, with its $\sigma$-algebra of Borel sets $\mathcal {B}:=\mathcal {B}\left( Y,\mathcal {T}\right)$ generated by $\mathcal {T}$. If $\left( X,\mathcal {S} \right)$ is a measurable space, a function $f:X\rightarrow Y$ is called measurable iff $f^{-1}\left( B\right) \in \mathcal {S}$ for all $B\in \mathcal {B}$.

By Theorem 4.1.6 of [28], p. 89 f as above is measurable iff

$$\begin{aligned} f^{-1}\left( C\right) \in \mathcal {S}\text { for all }C\in \mathcal {T}\text {.} \end{aligned}$$

We mention

Theorem 1

(see [28], p. 95). Let $\left( X,\mathcal {S}\right)$ be a measurable space and $\left( Y,d\right)$ be a metric space. Let $f_{n}$ be measurable functions from X into Y such that for all $x\in X$, $f_{n}\left( x\right) \rightarrow f\left( x\right)$ in Y. Then f is measurable. I.e., $\underset{n\rightarrow \infty }{\lim }f_{n}=f$ is measurable.

We need also

Proposition 4

[16]. Let f, g be fuzzy random variables from $\mathcal {S}$ into $\mathbb {R}_{\mathcal {F}}$. Then

• (i) Let $c\in \mathbb {R}$, then $c\odot f$ is a fuzzy random variable.

• (ii) $f\oplus g$ is a fuzzy random variable.

Proposition 5

Let $Y\left( \overrightarrow{t},\omega \right)$ be a real valued multivariate random function (stochastic process) such that Y is continuous in $\overrightarrow{t}\in \prod \limits _{i=1}^{N}\left[ a_{i},b_{i}\right]$. Then Y is jointly measurable in $\left( \overrightarrow{t},\omega \right)$ and $\int _{\prod \limits _{i=1}^{N}\left[ a_{i},b_{i}\right] }Y\left( \overrightarrow{t},\omega \right) d \overrightarrow{t}$ is a real valued random variable.

Proof

Similar to Proposition 18.14, p. 353 of [7].

2 About Neural Networks Background

2.1 About the Arctangent Activation Function

We consider the

$$\begin{aligned} \arctan x=\int _{0}^{x}\frac{dz}{1+z^{2}},\text { }x\in \mathbb {R}. \end{aligned}$$

(5)

We will be using

$$\begin{aligned} h\left( x\right) :=\frac{2}{\pi }\arctan \left( \frac{\pi }{2}x\right) =\frac{2}{\pi }\int _{0}^{\frac{\pi x}{2}}\frac{dz}{1+z^{2}}\text {, }x\in \mathbb {R}, \end{aligned}$$

(6)

which is a sigmoid type function and it is strictly increasing. We have that

$$\begin{aligned} h\left( 0\right) =0\text {, }h\left( -x\right) =-h\left( x\right) \text {, }h\left( +\infty \right) =1\text {, }h\left( -\infty \right) =-1, \end{aligned}$$

and

$$\begin{aligned} h^{\prime }\left( x\right) =\frac{4}{4+\pi ^{2}x^{2}}>0\text {, all }x\in \mathbb {R}. \end{aligned}$$

(7)

We consider the activation function

$$\begin{aligned} \psi _{1}\left( x\right) :=\frac{1}{4}\left( h\left( x+1\right) -h\left( x-1\right) \right) \text {, }x\in \mathbb {R}, \end{aligned}$$

(8)

and we notice that

$$\begin{aligned} \psi _{1}\left( -x\right) =\psi _{1}\left( x\right) , \end{aligned}$$

(9)

it is an even function.

Since $x+1>x-1$, then $h\left( x+1\right) >h\left( x-1\right)$, and $\psi _{1}\left( x\right) >0$, all $x\in \mathbb {R}$.

We see that

$$\begin{aligned} \psi _{1}\left( 0\right) =\frac{1}{\pi }\arctan \frac{\pi }{2}\cong 0.319. \end{aligned}$$

(10)

Let $x>0$, we have that

$$\begin{aligned} \psi _{1}^{\prime }\left( x\right) =\frac{1}{4}\left( h^{\prime }\left( x+1\right) -h^{\prime }\left( x-1\right) \right) = \end{aligned}$$

$$\begin{aligned} \frac{-4\pi ^{2}x}{\left( 4+\pi ^{2}\left( x+1\right) ^{2}\right) \left( 4+\pi ^{2}\left( x-1\right) ^{2}\right) }<0. \end{aligned}$$

(11)

That is

$$\begin{aligned} \psi _{1}^{\prime }\left( x\right) <0\text {, for }x>0. \end{aligned}$$

(12)

That is $\psi _{1}$ is strictly decreasing on $[0,\infty )$ and clearly is strictly increasing on $(-\infty ,0]$, and $\psi _{1}^{\prime }\left( 0\right) =0.$

Observe that

$$\begin{aligned} \begin{array}{l} \underset{x\rightarrow +\infty }{\lim }\psi _{1}\left( x\right) =\frac{1}{4} \left( h\left( +\infty \right) -h\left( +\infty \right) \right) =0, \\ \text {and} \\ \underset{x\rightarrow -\infty }{\lim }\psi _{1}\left( x\right) =\frac{1}{4} \left( h\left( -\infty \right) -h\left( -\infty \right) \right) =0. \end{array} \end{aligned}$$

(13)

That is the x-axis is the horizontal asymptote on $\psi _{1}$.

All in all, $\psi _{1}$ is a bell symmetric function with maximum $\psi _{1}\left( 0\right) \cong 18.31.$

We need

Theorem 2

([19], p. 286). We have that

$$\begin{aligned} \sum _{i=-\infty }^{\infty }\psi _{1}\left( x-i\right) =1\text {, }\forall \text { }x\in \mathbb {R}. \end{aligned}$$

(14)

Theorem 3

([19], p. 287). It holds

$$\begin{aligned} \int _{-\infty }^{\infty }\psi _{1}\left( x\right) dx=1. \end{aligned}$$

(15)

So that $\psi _{1}\left( x\right)$ is a density function on $\mathbb {R}.$

We mention

Theorem 4

([19], p. 288). Let $0<\alpha <1$, and $n\in \mathbb {N}$ with $n^{1-\alpha }>2$. It holds

$$\begin{aligned} \sum _{\left\{ \begin{array}{l} k=-\infty \\ :\left| nx-k\right| \ge n^{1-\alpha } \end{array} \right. }^{\infty }\psi _{1}\left( nx-k\right) <\frac{2}{\pi ^{2}\left( n^{1-\alpha }-2\right) }=:c_{1}\left( \alpha ,n\right) . \end{aligned}$$

(16)

Denote by $\left\lfloor \cdot \right\rfloor$ the integral part of the number and by $\left\lceil \cdot \right\rceil$ the ceiling of the number.

We need

Theorem 5

([19], p. 289). Let $x\in \left[ a,b\right] \subset \mathbb {R}$ and $n\in \mathbb {N}$ so that $\left\lceil na\right\rceil \le \left\lfloor nb\right\rfloor$. It holds

$$\begin{aligned} \frac{1}{\sum _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }\psi _{1}\left( nx-k\right) }\mathbf {<}\frac{1}{\psi _{1}\left( 1\right) }\mathbf {\cong 4.9737=:\alpha }_{1}\textbf{,}\text { }\mathbf { \forall }\text { }\mathbf {x\in }\left[ a,b\right] \mathbf {.} \end{aligned}$$

(17)

Note 1

([19], pp. 290–291).

• i) We have that

  $$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\sum _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }\psi _{1}\left( nx-k\right) \ne 1, \end{aligned}$$

  (18)

  for at least some $x\in \left[ a,b\right] .$

• ii) For large enough $n\in \mathbb {N}$ we always obtain $\left\lceil na\right\rceil \le \left\lfloor nb\right\rfloor$. Also $a\le \frac{k}{n}\le b$, iff $\left\lceil na\right\rceil \le k\le \left\lfloor nb\right\rfloor$.

In general, by Theorem 2, it holds

$$\begin{aligned} \sum _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }\psi _{1}\left( nx-k\right) \le 1. \end{aligned}$$

(19)

We introduce (see [24])

$$\begin{aligned} Z_{1}\left( x_{1},...,x_{N}\right) :=Z_{1}\left( x\right) :=\prod _{i=1}^{N}\psi _{1}\left( x_{i}\right) \text {, }x=\left( x_{1},...,x_{N}\right) \in \mathbb {R}^{N},\text { }N\in \mathbb {N}. \end{aligned}$$

(20)

Denote by $a=\left( a_{1},...,a_{N}\right)$ and $b=\left( b_{1},...,b_{N}\right) .$

It has the properties:

• (i) $Z_{1}\left( x\right) >0$, $\forall$ $x\in \mathbb {R}^{N},$

• (ii)
  $$\begin{aligned} \sum _{k=-\infty }^{\infty }Z_{1}\left( x-k\right) :=\sum _{k_{1}=-\infty }^{\infty }\sum _{k_{2}=-\infty }^{\infty }...\sum _{k_{N}=-\infty }^{\infty }Z_{1}\left( x_{1}-k_{1},...,x_{N}-k_{N}\right) =1,\text { } \end{aligned}$$

  (21)

  where $k:=\left( k_{1},...,k_{n}\right) \in \mathbb {Z}^{N}$, $\forall$ $x\in \mathbb {R}^{N},$

  hence

• (iii)
  $$\begin{aligned} \sum _{k=-\infty }^{\infty }Z_{1}\left( nx-k\right) =1, \end{aligned}$$

  (22)

  $\forall$ $x\in \mathbb {R}^{N};$ $n\in \mathbb {N}$,

  and

• (iv)
  $$\begin{aligned} \int _{\mathbb {R}^{N}}Z_{1}\left( x\right) dx=1, \end{aligned}$$

  (23)

  that is $Z_{1}$ is a multivariate density function.

• (v) It is clear that

  $$\begin{aligned} \sum _{\left\{ \begin{array}{c} k=-\infty \\ \left\| \frac{k}{n}-x\right\| _{\infty }>\frac{1}{n^{\beta }} \end{array} \right. }^{\infty }Z_{1}\left( nx-k\right) <\frac{2}{\pi ^{2}\left( n^{1-\beta }-2\right) }=c_{1}\left( \beta ,n\right) \text {, } \end{aligned}$$

  (24)

  $0<\beta <1,$ $n\in \mathbb {N}:n^{1-\beta }>2$, $x\in \mathbb {R}^{N}.$

• (vi) By Theorem 5 we get that

  $$\begin{aligned} 0<\frac{1}{\sum _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{1}\left( nx-k\right) }<\frac{1}{\left( \psi _{1}\left( 1\right) \right) ^{N}}\cong \left( 4.9737\right) ^{N}=:\gamma _{1}\left( N\right) , \end{aligned}$$

  (25)

  $\forall$ $x\in \left( \prod _{i=1}^{N}\left[ a_{i},b_{i}\right] \right)$, $n\in \mathbb {N}$.

Furthermore it holds

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\sum _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{1}\left( nx-k\right) \ne 1, \end{aligned}$$

(26)

for at least some $x\in \left( \prod _{i=1}^{N}\left[ a_{i},b_{i}\right] \right) .$

Above it is $\left\| x\right\| _{\infty }:=\max \left\{ \left| x_{1}\right| ,...,\left| x_{N}\right| \right\}$, $x\in \mathbb {R}^{N}$, also set $\infty :=\left( \infty ,...,\infty \right)$, $-\infty =\left( -\infty ,...-\infty \right)$ upon the multivariate context.

2.2 About the Algebraic Activation Function

Here see also [20].

We consider the generator algebraic function

$$\begin{aligned} \varphi \left( x\right) =\frac{x}{\root 2m \of {1+x^{2m}}},\text { }m\in \mathbb {N}\text {, }x\in \mathbb {R}\text {,} \end{aligned}$$

(27)

which is a sigmoidal type of function and is a strictly increasing function.

We see that $\varphi \left( -x\right) =-\varphi \left( x\right)$ with $\varphi \left( 0\right) =0$. We get that

$$\begin{aligned} \varphi ^{\prime }\left( x\right) =\frac{1}{\left( 1+x^{2m}\right) ^{\frac{ 2m+1}{2m}}}>0\text {, }\forall \text { }x\in \mathbb {R}\text {,} \end{aligned}$$

(28)

proving $\varphi$ as strictly increasing over $\mathbb {R},\varphi ^{\prime }\left( x\right) =\varphi ^{\prime }\left( -x\right) .$ We easily find that $\underset{x\rightarrow +\infty }{\lim }\varphi \left( x\right) =1$, $\varphi \left( +\infty \right) =1$, and $\underset{x\rightarrow -\infty }{\lim } \varphi \left( x\right) =-1$, $\varphi \left( -\infty \right) =-1.$

We consider the activation function

$$\begin{aligned} \psi _{2}\left( x\right) =\frac{1}{4}\left[ \varphi \left( x+1\right) -\varphi \left( x-1\right) \right] . \end{aligned}$$

(29)

Clearly it is $\psi _{2}\left( x\right) =\psi _{2}\left( -x\right) ,$ $\forall$ $x\in \mathbb {R}$, so that $\psi _{2}$ is an even function and symmetric with respect to the y-axis. Clearly $\psi _{2}\left( x\right) >0$ , $\forall$ $x\in \mathbb {R}$.

Also it is

$$\begin{aligned} \psi _{2}\left( 0\right) =\frac{1}{2\root 2m \of {2}}. \end{aligned}$$

(30)

By [20], we have that $\psi _{2}^{\prime }\left( x\right) <0$ for $x>0$. That is $\psi _{2}$ is strictly decreasing over $\left( 0,+\infty \right) .$

Clearly, $\psi _{2}$ is strictly increasing over $\left( -\infty ,0\right)$ and $\psi _{2}^{\prime }\left( 0\right) =0$.

Furthermore we obtain that

$$\begin{aligned} \underset{x\rightarrow +\infty }{\lim }\psi _{2}\left( x\right) =\frac{1}{4} \left[ \varphi \left( +\infty \right) -\varphi \left( +\infty \right) \right] =0, \end{aligned}$$

(31)

and

$$\begin{aligned} \underset{x\rightarrow -\infty }{\lim }\psi _{2}\left( x\right) =\frac{1}{4} \left[ \varphi \left( -\infty \right) -\varphi \left( -\infty \right) \right] =0. \end{aligned}$$

(32)

That is the x-axis is the horizontal asymptote of $\psi _{2}$.

Conclusion, $\psi _{2}$ is a bell shape symmetric function with maximum

$$\begin{aligned} \psi _{2}\left( 0\right) =\frac{1}{2\root 2m \of {2}},\text { }m\in \mathbb {N}. \end{aligned}$$

(33)

We need

Theorem 6

[20]. We have that

$$\begin{aligned} \sum \limits _{i=-\infty }^{\infty }\psi _{2}\left( x-i\right) =1\text {, } \forall \text { }x\in \mathbb {R}. \end{aligned}$$

(34)

Theorem 7

[20]. It holds

$$\begin{aligned} \int _{-\infty }^{\infty }\psi _{2}\left( x\right) dx=1. \end{aligned}$$

(35)

Theorem 8

[20]. Let $0<\alpha <1$, and $n\in \mathbb {N}$ with $n^{1-\alpha }>2$. It holds

$$\begin{aligned} \sum \limits _{\left\{ \begin{array}{c} k=-\infty \\ :\left| nx-k\right| \ge n^{1-\alpha } \end{array} \right. }^{\infty }\psi _{2}\left( nx-k\right) <\frac{1}{4m\left( n^{1-\alpha }-2\right) ^{2m}}=:c_{2}\left( \alpha ,n\right) ,\text { }m\in \mathbb {N}. \end{aligned}$$

(36)

We need

Theorem 9

[20]. Let $\left[ a,b\right] \subset \mathbb {R}$ and $n\in \mathbb {N}$ so that $\left\lceil na\right\rceil \le \left\lfloor nb\right\rfloor$. It holds

$$\begin{aligned} \frac{1}{\sum \limits _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }\psi _{2}\left( nx-k\right) }<2\left( \root 2m \of {1+4^{m}} \right) =:\alpha _{2}, \end{aligned}$$

(37)

$\forall$ $x\in \left[ a,b\right]$, $m\in \mathbb {N}.$

Note 2

• 1) By [20] we have that

  $$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\sum \limits _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }\psi _{2}\left( nx-k\right) \ne 1, \text { } \end{aligned}$$

  (38)

  for at least some $x\in \left[ a,b\right] .$

• 2) Let $\left[ a,b\right] \subset \mathbb {R}$. For large $n\in \mathbb {N}$ we always have $\left\lceil na\right\rceil \le \left\lfloor nb\right\rfloor$. Also $a\le \frac{k}{n}\le b$, iff $\left\lceil na\right\rceil \le k\le \left\lfloor nb\right\rfloor$.

In general it holds that

$$\begin{aligned} \sum \limits _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }\psi _{2}\left( nx-k\right) \le 1. \end{aligned}$$

(39)

We introduce (see also [25])

$$\begin{aligned} Z_{2}\left( x_{1},...,x_{N}\right) :=Z_{2}\left( x\right) :=\prod _{i=1}^{N}\psi _{2}\left( x_{i}\right) \text {, }x=\left( x_{1},...,x_{N}\right) \in \mathbb {R}^{N},\text { }N\in \mathbb {N}. \end{aligned}$$

(40)

It has the properties:

• (i) $Z_{2}\left( x\right) >0$, $\forall$ $x\in \mathbb {R}^{N},$

• (ii)
  $$\begin{aligned} \sum _{k=-\infty }^{\infty }Z_{2}\left( x-k\right) :=\sum _{k_{1}=-\infty }^{\infty }\sum _{k_{2}=-\infty }^{\infty }...\sum _{k_{N}=-\infty }^{\infty }Z_{2}\left( x_{1}-k_{1},...,x_{N}-k_{N}\right) =1,\text { } \end{aligned}$$

  (41)

  where $k:=\left( k_{1},...,k_{n}\right) \in \mathbb {Z}^{N}$, $\forall$ $x\in \mathbb {R}^{N},$

  hence

• (iii)
  $$\begin{aligned} \sum _{k=-\infty }^{\infty }Z_{2}\left( nx-k\right) =1, \end{aligned}$$

  (42)

  $\forall$ $x\in \mathbb {R}^{N};$ $n\in \mathbb {N}$,

  and

• (iv)
  $$\begin{aligned} \int _{\mathbb {R}^{N}}Z_{2}\left( x\right) dx=1, \end{aligned}$$

  (43)

  that is $Z_{2}$ is a multivariate density function.

• (v) It is clear that

  $$\begin{aligned} \sum _{\left\{ \begin{array}{c} k=-\infty \\ \left\| \frac{k}{n}-x\right\| _{\infty }>\frac{1}{n^{\beta }} \end{array} \right. }^{\infty }Z_{2}\left( nx-k\right) <\frac{1}{4m\left( n^{1-\beta }-2\right) ^{2m}}=c_{2}\left( \beta ,n\right) \text {, } \end{aligned}$$

  (44)

  $0<\beta <1,$ $n\in \mathbb {N}:n^{1-\beta }>2$, $x\in \mathbb {R}^{N}$, $m\in \mathbb {N}\mathbf {.}$

• (vi) By Theorem 9 we get that

  $$\begin{aligned} 0<\frac{1}{\sum _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{2}\left( nx-k\right) }<\frac{1}{\left( \psi _{2}\left( 1\right) \right) ^{N}}\cong \left[ 2\left( \root 2m \of {1+4^{m}}\right) \right] ^{N}:=\gamma _{2}\left( N\right) , \end{aligned}$$

  (45)

  $\forall$ $x\in \left( \prod _{i=1}^{N}\left[ a_{i},b_{i}\right] \right)$, $n\in \mathbb {N}$.

Furthermore it holds

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\sum _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{2}\left( nx-k\right) \ne 1, \end{aligned}$$

(46)

for at least some $x\in \left( \prod _{i=1}^{N}\left[ a_{i},b_{i}\right] \right) .$

2.3 About the Gudermannian Activation Function

See also [21, 34].

Here we consider $gd\left( x\right)$ the Gudermannian function [34], which is a sigmoid function, as a generator function:

$$\begin{aligned} \sigma \left( x\right) =2\arctan \left( \tanh \left( \frac{x}{2}\right) \right) =\int _{0}^{x}\frac{dt}{\cosh t}=:gd\left( x\right) \text {, }x\in \mathbb {R}\text {.} \end{aligned}$$

(47)

Let the normalized generator sigmoid function

$$\begin{aligned} f\left( x\right) :=\frac{2}{\pi }\sigma \left( x\right) =\frac{2}{\pi } \int _{0}^{x}\frac{dt}{\cosh t}=\frac{4}{\pi }\int _{0}^{x}\frac{1}{ e^{t}+e^{-t}}dt,\text { }x\in \mathbb {R}. \end{aligned}$$

(48)

Here

$$\begin{aligned} f^{\prime }\left( x\right) =\frac{2}{\pi \cosh x}>0,\text { }\forall \text { }x\in \mathbb {R}, \end{aligned}$$

hence f is strictly increasing on $\mathbb {R}.$

Notice that $\tanh \left( -x\right) =-\tanh x$ and $\arctan \left( -x\right) =-\arctan x$, $x\in \mathbb {R}.$

So, here the neural network activation function will be:

$$\begin{aligned} \psi _{3}\left( x\right) =\frac{1}{4}\left[ f\left( x+1\right) -f\left( x-1\right) \right] \text {, }x\in \mathbb {R}. \end{aligned}$$

(49)

By [21], we get that

$$\begin{aligned} \psi _{3}\left( x\right) =\psi _{3}\left( -x\right) ,\text { }\forall \text { }x\in \mathbb {R}, \end{aligned}$$

(50)

i.e. it is even and symmetric with respect to the y-axis. Here we have $f\left( +\infty \right) =1$, $f\left( -\infty \right) =-1$ and $f\left( 0\right) =0$. Clearly it is

$$\begin{aligned} f\left( -x\right) =-f\left( x\right) \text {, }\forall \text { }x\in \mathbb { R}, \end{aligned}$$

(51)

an odd function, symmetric with respect to the origin. Since $x+1>x-1$, and $f\left( x+1\right) >f\left( x-1\right)$, we obtain $\psi _{3}\left( x\right) >0$, $\forall$ $x\in \mathbb {R}.$

By [21], we have that

$$\begin{aligned} \psi _{3}\left( 0\right) =\frac{1}{\pi }gd\left( 1\right) \cong 0.2757. \end{aligned}$$

(52)

By [21] $\psi _{3}$ is strictly decreasing on $\left( 0,+\infty \right)$, and strictly increasing on $\left( -\infty ,0\right)$, and $\psi _{3}^{\prime }\left( 0\right) =0$.

Also we have that

$$\begin{aligned} \underset{x\rightarrow +\infty }{\lim }\psi _{3}\left( x\right) =\underset{ x\rightarrow -\infty }{\lim }\psi _{3}\left( x\right) =0, \end{aligned}$$

(53)

that is the x-axis is the horizontal asymptote for $\psi _{3}$.

Conclusion, $\psi _{3}$ is a bell shaped symmetric function with maximum $\psi _{3}\left( 0\right) \cong 0.551$.

We need

Theorem 10

[21]. It holds that

$$\begin{aligned} \sum \limits _{i=-\infty }^{\infty }\psi _{3}\left( x-i\right) =1\text {, } \forall \text { }x\in \mathbb {R}. \end{aligned}$$

(54)

Theorem 11

[21]. We have that

$$\begin{aligned} \int _{-\infty }^{\infty }\psi _{3}\left( x\right) dx=1. \end{aligned}$$

(55)

So $\psi _{3}\left( x\right)$ is a density function.

Theorem 12

[21]. Let $0<\alpha <1$, and $n\in \mathbb {N}$ with $n^{1-\alpha }>2$. It holds

$$\begin{aligned} \sum \limits _{\left\{ \begin{array}{c} k=-\infty \\ :\left| nx-k\right| \ge n^{1-\alpha } \end{array} \right. }^{\infty }\psi _{3}\left( nx-k\right) <\frac{2}{\pi e^{\left( n^{1-\alpha }-2\right) }}=\frac{2e^{2}}{\pi e^{n^{1-\alpha }}}=:c_{3}\left( \alpha ,n\right) . \end{aligned}$$

(56)

Theorem 13

[21]. Let $\left[ a,b\right] \subset \mathbb {R}$ and $n\in \mathbb {N},$ so that $\left\lceil na\right\rceil \le \left\lfloor nb\right\rfloor$. It holds

$$\begin{aligned} \frac{1}{\sum \limits _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }\psi _{3}\left( nx-k\right) }<\frac{2\pi }{gd\left( 2\right) }\cong 4.824=:\alpha _{3}, \end{aligned}$$

(57)

$\forall$ $x\in \left[ a,b\right] .$

We make

Remark 5

[21].

• (i) We have that

  $$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\sum \limits _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }\psi _{3}\left( nx-k\right) \ne 1, \text { } \end{aligned}$$

  (58)

  for at least some $x\in \left[ a,b\right] .$

• (ii) Let $\left[ a,b\right] \subset \mathbb {R}$. For large n we always have $\left\lceil na\right\rceil \le \left\lfloor nb\right\rfloor$. Also $a\le \frac{k}{n}\le b$, iff $\left\lceil na\right\rceil \le k\le \left\lfloor nb\right\rfloor$.

In general it holds

$$\begin{aligned} \sum \limits _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }\psi _{3}\left( nx-k\right) \le 1. \end{aligned}$$

(59)

We introduce (see also [23])

$$\begin{aligned} Z_{3}\left( x_{1},...,x_{N}\right) :=Z_{3}\left( x\right) :=\prod _{i=1}^{N}\psi _{3}\left( x_{i}\right) \text {, }x=\left( x_{1},...,x_{N}\right) \in \mathbb {R}^{N},\text { }N\in \mathbb {N}. \end{aligned}$$

(60)

It has the properties:

• (i) $Z_{3}\left( x\right) >0$, $\forall$ $x\in \mathbb {R}^{N},$

• (ii)
  $$\begin{aligned} \sum _{k=-\infty }^{\infty }Z_{3}\left( x-k\right) :=\sum _{k_{1}=-\infty }^{\infty }\sum _{k_{2}=-\infty }^{\infty }...\sum _{k_{N}=-\infty }^{\infty }Z_{3}\left( x_{1}-k_{1},...,x_{N}-k_{N}\right) =1,\text { } \end{aligned}$$

  (61)

  where $k:=\left( k_{1},...,k_{n}\right) \in \mathbb {Z}^{N}$, $\forall$ $x\in \mathbb {R}^{N},$

  hence

• (iii)
  $$\begin{aligned} \sum _{k=-\infty }^{\infty }Z_{3}\left( nx-k\right) =1, \end{aligned}$$

  (62)

  $\forall$ $x\in \mathbb {R}^{N};$ $n\in \mathbb {N}$,

  and

• (iv)
  $$\begin{aligned} \int _{\mathbb {R}^{N}}Z_{3}\left( x\right) dx=1, \end{aligned}$$

  (63)

  that is $Z_{3}$ is a multivariate density function.

• (v) It is also clear that

  $$\begin{aligned} \sum _{\left\{ \begin{array}{c} k=-\infty \\ \left\| \frac{k}{n}-x\right\| _{\infty }>\frac{1}{n^{\beta }} \end{array} \right. }^{\infty }Z_{3}\left( nx-k\right) <\frac{2e^{2}}{\pi e^{n^{1-\beta }}}=c_{3}\left( \beta ,n\right) , \end{aligned}$$

  (64)

  $0<\beta <1$, $n\in \mathbb {N}:n^{1-\beta }>2$, $x\in \mathbb {R}^{N},$ $m\in \mathbb {N}.$

• (vi) By Theorem 13 we get that

  $$\begin{aligned} 0<\frac{1}{\sum _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{3}\left( nx-k\right) }<\left( \frac{2\pi }{gd\left( 2\right) }\right) ^{N}\cong \left( 4.824\right) ^{N}=:\gamma _{3}\left( N\right) , \end{aligned}$$

  (65)

  $\forall$ $x\in \left( \prod _{i=1}^{N}\left[ a_{i},b_{i}\right] \right)$, $n\in \mathbb {N}$.

Furthermore it holds

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\sum _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{3}\left( nx-k\right) \ne 1, \end{aligned}$$

(66)

for at least some $x\in \left( \prod _{i=1}^{N}\left[ a_{i},b_{i}\right] \right) .$

2.4 About the Generalized Symmetrical Activation Function

Here we consider the generalized symmetrical sigmoid function [22, 29]

$$\begin{aligned} f_{1}\left( x\right) =\frac{x}{\left( 1+\left| x\right| ^{\mu }\right) ^{\frac{1}{\mu }}},\text { }\mu >0\text {, }x\in \mathbb {R}\text {.} \end{aligned}$$

(67)

This has applications in immunology and protection from disease together with probability theory. It is also called a symmetrical protection curve.

The parameter $\mu$ is a shape parameter controling how fast the curve approaches the asymptotes for a given slope at the inflection point. When $\mu =1$ $f_{1}$ is the absolute sigmoid function, and when $\mu =2,$ $f_{1}$ is the square root sigmoid function. When $\mu =1.5$ the function approximates the arctangent function, when $\mu =2.9$ it approximates the logistic function, and when $\mu =3.4$ it approximates the error function. Parameter $\mu$ is estimated in the likelihood maximization [29]. For more see [29].

Next we study the particular generator sigmoid function

$$\begin{aligned} f_{2}\left( x\right) =\frac{x}{\left( 1+\left| x\right| ^{\lambda }\right) ^{\frac{1}{\lambda }}},\text { }\lambda \text { is an odd number, } x\in \mathbb {R}. \end{aligned}$$

(68)

We have that $f_{2}\left( 0\right) =0$, and

$$\begin{aligned} f_{2}\left( -x\right) =-f_{2}\left( x\right) , \end{aligned}$$

(69)

so $f_{2}$ is symmetric with respect to zero.

When $x\ge 0$, we get that [22]

$$\begin{aligned} f_{2}^{\prime }\left( x\right) =\frac{1}{\left( 1+x^{\lambda }\right) ^{ \frac{\lambda +1}{\lambda }}}>0, \end{aligned}$$

(70)

that is $f_{2}$ is strictly increasing on $[0,+\infty )$ and $f_{2}$ is strictly increasing on $(-\infty ,0]$. Hence $f_{2}$ is strictly increasing on $\mathbb {R}$.

We also have $f_{2}\left( +\infty \right) =f_{2}\left( -\infty \right) =1.$

Let us consider the activation function [22]:

$$\begin{aligned} \psi _{4}\left( x\right) =\frac{1}{4}\left[ f_{2}\left( x+1\right) -f_{2}\left( x-1\right) \right] = \end{aligned}$$

$$\begin{aligned} \frac{1}{4}\left[ \frac{\left( x+1\right) }{\left( 1+\left| x+1\right| ^{\lambda }\right) ^{\frac{1}{\lambda }}}-\frac{\left( x-1\right) }{ \left( 1+\left| x-1\right| ^{\lambda }\right) ^{\frac{1}{\lambda }} }\right] . \end{aligned}$$

(71)

Clearly it holds [22]

$$\begin{aligned} \psi _{4}\left( x\right) =\psi _{4}\left( -x\right) ,\text { }\forall \text { }x\in \mathbb {R}. \end{aligned}$$

(72)

and

$$\begin{aligned} \psi _{4}\left( 0\right) =\frac{1}{2\root \lambda \of {2}}, \end{aligned}$$

(73)

and $\psi _{4}\left( x\right) >0$, $\forall$ $x\in \mathbb {R}$.

Following [22], we have that $\psi _{4}$ is strictly decreasing over $[0,+\infty )$, and $\psi _{4}$ is strictly increasing on $(-\infty ,0]$, by $\psi _{4}$-symmetry with respect to y-axis, and $\psi _{4}^{\prime }\left( 0\right) =0.$

Clearly it is

$$\begin{aligned} \underset{x\rightarrow +\infty }{\lim }\psi _{4}\left( x\right) =\underset{ x\rightarrow -\infty }{\lim }\psi _{4}\left( x\right) =0, \end{aligned}$$

(74)

therefore the x-axis is the horizontal asymptote of $\psi _{4}\left( x\right) .$

The value

$$\begin{aligned} \psi _{4}\left( 0\right) =\frac{1}{2\root \lambda \of {2}},\text { }\lambda \text { is an odd number,} \end{aligned}$$

(75)

is the maximum of $\psi _{4}$, which is a bell shaped function.

We need

Theorem 14

[22]. It holds

$$\begin{aligned} \sum \limits _{i=-\infty }^{\infty }\psi _{4}\left( x-i\right) =1\text {, } \forall \text { }x\in \mathbb {R}. \end{aligned}$$

(76)

Theorem 15

[22]. We have that

$$\begin{aligned} \int _{-\infty }^{\infty }\psi _{4}\left( x\right) dx=1. \end{aligned}$$

(77)

So that $\psi _{4}\left( x\right)$ is a density function on $\mathbb {R}.$

We need

Theorem 16

[22]. Let $0<\alpha <1$, and $n\in \mathbb {N}$ with $n^{1-\alpha }>2$. It holds

$$\begin{aligned} \sum \limits _{\left\{ \begin{array}{c} j=-\infty \\ :\left| nx-j\right| \ge n^{1-\alpha } \end{array} \right. }^{\infty }\psi _{4}\left( nx-j\right) <\frac{1}{2\lambda \left( n^{1-\alpha }-2\right) ^{\lambda }}=:c_{4}\left( \alpha ,n\right) ,\text { } \end{aligned}$$

(78)

where $\lambda \in \mathbb {N}$ is an odd number.

We also need

Theorem 17

[22]. Let $\left[ a,b\right] \subset \mathbb {R}$ and $n\in \mathbb {N}$ so that $\left\lceil na\right\rceil \le \left\lfloor nb\right\rfloor$. Then

$$\begin{aligned} \frac{1}{\sum \limits _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }\psi _{4}\left( \left| nx-k\right| \right) }<2 \root \lambda \of {1+2^{\lambda }}=:\alpha _{4}, \end{aligned}$$

(79)

where $\lambda$ is an odd number, $\forall$ $x\in \left[ a,b\right] .$

We make

Remark 6

[22]. (1) We have that

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\sum \limits _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }\psi _{4}\left( nx-k\right) \ne 1, \text { for at least some }x\in \left[ a,b\right] . \end{aligned}$$

(80)

(2) Let $\left[ a,b\right] \subset \mathbb {R}$. For large enough n we always obtain $\left\lceil na\right\rceil \le \left\lfloor nb\right\rfloor$. Also $a\le \frac{k}{n}\le b$, iff $\left\lceil na\right\rceil \le k\le \left\lfloor nb\right\rfloor$.

In general it holds that

$$\begin{aligned} \sum \limits _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }\psi _{4}\left( nx-k\right) \le 1. \end{aligned}$$

(81)

We introduce (see also [26])

$$\begin{aligned} Z_{4}\left( x_{1},...,x_{N}\right) :=Z_{4}\left( x\right) :=\prod _{i=1}^{N}\psi _{4}\left( x_{i}\right) \text {, }x=\left( x_{1},...,x_{N}\right) \in \mathbb {R}^{N},\text { }N\in \mathbb {N}. \end{aligned}$$

(82)

It has the properties:

• (i) $Z_{4}\left( x\right) >0$, $\forall$ $x\in \mathbb {R}^{N},$

• (ii)
  $$\begin{aligned} \sum _{k=-\infty }^{\infty }Z_{4}\left( x-k\right) :=\sum _{k_{1}=-\infty }^{\infty }\sum _{k_{2}=-\infty }^{\infty }...\sum _{k_{N}=-\infty }^{\infty }Z_{4}\left( x_{1}-k_{1},...,x_{N}-k_{N}\right) =1,\text { } \end{aligned}$$

  (83)

  where $k:=\left( k_{1},...,k_{n}\right) \in \mathbb {Z}^{N}$, $\forall$ $x\in \mathbb {R}^{N},$

  hence

• (iii)
  $$\begin{aligned} \sum _{k=-\infty }^{\infty }Z_{4}\left( nx-k\right) =1, \end{aligned}$$

  (84)

  $\forall$ $x\in \mathbb {R}^{N};$ $n\in \mathbb {N}$,

  and

• (iv)
  $$\begin{aligned} \int _{\mathbb {R}^{N}}Z_{4}\left( x\right) dx=1, \end{aligned}$$

  (85)

  that is $Z_{4}$ is a multivariate density function.

• (v) It is clear that

  $$\begin{aligned} \sum _{\left\{ \begin{array}{c} k=-\infty \\ \left\| \frac{k}{n}-x\right\| _{\infty }>\frac{1}{n^{\beta }} \end{array} \right. }^{\infty }Z_{4}\left( nx-k\right) <\frac{1}{2\lambda \left( n^{1-\beta }-2\right) ^{\lambda }}=c_{4}\left( \beta ,n\right) , \end{aligned}$$

  (86)

  $0<\beta <1$, $n\in \mathbb {N}:n^{1-\beta }>2$, $x\in \mathbb {R}^{N},$ $\lambda$ is odd.

• (vi) By Theorem 17 we get that

  $$\begin{aligned} 0<\frac{1}{\sum _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{4}\left( nx-k\right) }<\left( 2\root \lambda \of {1+2^{\lambda }} \right) ^{N}=:\gamma _{4}\left( N\right) , \end{aligned}$$

  (87)

  $\forall$ $x\in \left( \prod _{i=1}^{N}\left[ a_{i},b_{i}\right] \right)$, $n\in \mathbb {N}$, $\lambda$ is odd.

Furthermore it holds

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\sum _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{4}\left( nx-k\right) \ne 1, \end{aligned}$$

(88)

for at least some $x\in \left( \prod _{i=1}^{N}\left[ a_{i},b_{i}\right] \right) .$

Set

$$\begin{aligned} \begin{array}{c} \left\lceil na\right\rceil :=\left( \left\lceil na_{1}\right\rceil ,...,\left\lceil na_{N}\right\rceil \right) , \\ \\ \left\lfloor nb\right\rfloor :=\left( \left\lfloor nb_{1}\right\rfloor ,...,\left\lfloor nb_{N}\right\rfloor \right) , \end{array} \end{aligned}$$

where $a:=\left( a_{1},...,a_{N}\right)$, $b:=\left( b_{1},...,b_{N}\right)$, $k:=\left( k_{1},...,k_{N}\right) .$

Let $f\in C\left( \prod _{i=1}^{N}\left[ a_{i},b_{i}\right] \right) ,$ and $n\in \mathbb {N}$ such that $\left\lceil na_{i}\right\rceil \le \left\lfloor nb_{i}\right\rfloor$, $i=1,...,N.$

We define the multivariate averaged positive linear quasi-interpolation neural network operators ($x:=\left( x_{1},...,x_{N}\right) \in \left( \prod _{i=1}^{N}\left[ a_{i},b_{i}\right] \right)$); $j=1,2,3,4$:

$$\begin{aligned} _{j}A_{n}\left( f,x_{1},...,x_{N}\right) :=\text { }_{j}A_{n}\left( f,x\right) :=\frac{\sum _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }f\left( \frac{k}{n}\right) Z_{j}\left( nx-k\right) }{ \sum _{k=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{j}\left( nx-k\right) }= \end{aligned}$$

(89)

$$\begin{aligned} \frac{\sum _{k_{1}=\left\lceil na_{1}\right\rceil }^{\left\lfloor nb_{1}\right\rfloor }\sum _{k_{2}=\left\lceil na_{2}\right\rceil }^{\left\lfloor nb_{2}\right\rfloor }...\sum _{k_{N}=\left\lceil na_{N}\right\rceil }^{\left\lfloor nb_{N}\right\rfloor }f\left( \frac{k_{1}}{n},...,\frac{k_{N}}{n}\right) \left( \prod _{i=1}^{N}\psi _{j}\left( nx_{i}-k_{i}\right) \right) }{\prod _{i=1}^{N}\left( \sum _{k_{i}=\left\lceil na_{i}\right\rceil }^{\left\lfloor nb_{i}\right\rfloor }\psi _{j}\left( nx_{i}-k_{i}\right) \right) }. \end{aligned}$$

For large enough $n\in \mathbb {N}$ we always obtain $\left\lceil na_{i}\right\rceil \le \left\lfloor nb_{i}\right\rfloor$, $i=1,...,N$. Also $a_{i}\le \frac{k_{i}}{n}\le b_{i}$, iff $\left\lceil na_{i}\right\rceil \le k_{i}\le \left\lfloor nb_{i}\right\rfloor$, $i=1,...,N$.

When $f\in C_{B}\left( \mathbb {R}^{N}\right)$ we define ($j=1,2,3,4$)

$$\begin{aligned} _{j}B_{n}\left( f,x\right) :=\text { }_{j}B_{n}\left( f,x_{1},...,x_{N}\right) :=\sum _{k=-\infty }^{\infty }f\left( \frac{k}{n} \right) Z_{j}\left( nx-k\right) := \end{aligned}$$

(90)

$$\begin{aligned} \sum _{k_{1}=-\infty }^{\infty }\sum _{k_{2}=-\infty }^{\infty }...\sum _{k_{N}=-\infty }^{\infty }f\left( \frac{k_{1}}{n},\frac{k_{2}}{n} ,...,\frac{k_{N}}{n}\right) \left( \prod _{i=1}^{N}\psi _{j}\left( nx_{i}-k_{i}\right) \right) , \end{aligned}$$

$n\in \mathbb {N}$, $\forall$ $x\in \mathbb {R}^{N},$ $N\in \mathbb {N}$, the multivariate full quasi-interpolation neural network operators.

Also for $f\in C_{B}\left( \mathbb {R}^{N}\right)$ we define the multivariate Kantorovich type neural network operators ($j=1,2,3,4$)

$$\begin{aligned} _{j}C_{n}\left( f,x\right) :=\text { }_{j}C_{n}\left( f,x_{1},...,x_{N}\right) :=\sum _{k=-\infty }^{\infty }\left( n^{N}\int _{\frac{k}{n}}^{\frac{k+1}{n}}f\left( t\right) dt\right) Z_{j}\left( nx-k\right) := \end{aligned}$$

(91)

$$\begin{aligned} \sum _{k_{1}=-\infty }^{\infty }\sum _{k_{2}=-\infty }^{\infty }...\sum _{k_{N}=-\infty }^{\infty }\left( n^{N}\int _{\frac{k_{1}}{n}}^{\frac{ k_{1}+1}{n}}\int _{\frac{k_{2}}{n}}^{\frac{k_{2}+1}{n}}...\int _{\frac{k_{N}}{n }}^{\frac{k_{N}+1}{n}}f\left( t_{1},...,t_{N}\right) dt_{1}...dt_{N}\right) \end{aligned}$$

$$\begin{aligned} \cdot \left( \prod _{i=1}^{N}\psi _{j}\left( nx_{i}-k_{i}\right) \right) , \end{aligned}$$

$n\in \mathbb {N},\ \forall$ $x\in \mathbb {R}^{N}.$

Again for $f\in C_{B}\left( \mathbb {R}^{N}\right) ,$ $N\in \mathbb {N},$ we define the multivariate neural network operators of quadrature type $_{j}D_{n}\left( f,x\right)$, $n\in \mathbb {N},$ as follows. Let $\theta =\left( \theta _{1},...,\theta _{N}\right) \in \mathbb {N}^{N},$ $\overline{r}=\left( r_{1},...,r_{N}\right) \in \mathbb {Z}_{+}^{N}$, $w_{\overline{r}}=w_{r_{1},r_{2},...r_{N}}\ge 0$, such that $\sum \limits _{\overline{r}=0}^{\theta }w_{\overline{r}}=\sum \limits _{r_{1}=0}^{\theta _{1}}\sum \limits _{r_{2}=0}^{\theta _{2}}...\sum \limits _{r_{N}=0}^{\theta _{N}}w_{r_{1},r_{2},...r_{N}}=1;$ $k\in \mathbb {Z}^{N}$ and

$$\begin{aligned} \delta _{nk}\left( f\right) :=\delta _{n,k_{1},k_{2},...,k_{N}}\left( f\right) :=\sum \limits _{\overline{r}=0}^{\theta }w_{\overline{r}}f\left( \frac{k}{n}+\frac{\overline{r}}{n\theta }\right) := \end{aligned}$$

$$\begin{aligned} \sum \limits _{r_{1}=0}^{\theta _{1}}\sum \limits _{r_{2}=0}^{\theta _{2}}...\sum \limits _{r_{N}=0}^{\theta _{N}}w_{r_{1},r_{2},...r_{N}}f\left( \frac{k_{1}}{n}+\frac{r_{1}}{n\theta _{1}},\frac{k_{2}}{n}+\frac{r_{2}}{ n\theta _{2}},...,\frac{k_{N}}{n}+\frac{r_{N}}{n\theta _{N}}\right) , \end{aligned}$$

(92)

where $\frac{\overline{r}}{\theta }:=\left( \frac{r_{1}}{\theta _{1}},\frac{ r_{2}}{\theta _{2}},...,\frac{r_{N}}{\theta _{N}}\right)$; $j=1,2,3,4.$

We put

$$\begin{aligned} _{j}D_{n}\left( f,x\right) :=\text { }_{j}D_{n}\left( f,x_{1},...,x_{N}\right) :=\sum _{k=-\infty }^{\infty }\delta _{nk}\left( f\right) Z_{j}\left( nx-k\right) := \end{aligned}$$

(93)

$$\begin{aligned} \sum _{k_{1}=-\infty }^{\infty }\sum _{k_{2}=-\infty }^{\infty }...\sum _{k_{N}=-\infty }^{\infty }\delta _{n,k_{1},k_{2},...,k_{N}}\left( f\right) \left( \prod _{i=1}^{N}\psi _{j}\left( nx_{i}-k_{i}\right) \right) , \end{aligned}$$

$\forall$ $x\in \mathbb {R}^{N}.$

For the next we need, for $f\in C\left( \prod _{i=1}^{N}\left[ a_{i},b_{i} \right] \right)$ the first multivariate modulus of continuity

$$\begin{aligned} \omega _{1}\left( f,h\right) :=\underset{ \begin{array}{c} x,y\in \prod _{i=1}^{N}\left[ a_{i},b_{i}\right] \\ \left\| x-y\right\| _{\infty }\le h \end{array} }{\sup }\left| f\left( x\right) -f\left( y\right) \right| \text {, }h>0. \end{aligned}$$

(94)

It holds that

$$\begin{aligned} \underset{h\rightarrow 0}{\lim }\omega _{1}\left( f,h\right) =0. \end{aligned}$$

(95)

Similarly it is defined for $f\in C_{B}\left( \mathbb {R}^{N}\right)$ (continuous and bounded functions on $\mathbb {R}^{N}$) the $\omega _{1}\left( f,h\right)$, and it has the property (95), given that $f\in C_{U}\left( \mathbb {R}^{N}\right)$ (uniformly continuous functions on $\mathbb {R}^{N}$).

We mention

Theorem 18

(see [23,24,25,26]). Let $f\in C\left( \prod _{i=1}^{N}\left[ a_{i},b_{i}\right] \right) ,$ $0<\beta <1$, $x\in \left( \prod _{i=1}^{N}\left[ a_{i},b_{i}\right] \right) ,$ $N,n\in \mathbb {N}$ with $n^{1-\beta }>2;$ $j=1,2,3,4$. Then

• 1)
  $$\begin{aligned} \left| _{j}A_{n}\left( f,x\right) -f\left( x\right) \right| \le \gamma _{j}\left( N\right) \left[ \omega _{1}\left( f,\frac{1}{n^{\beta }} \right) +2c_{j}\left( \beta ,n\right) \left\| f\right\| _{\infty } \right] =:\lambda _{j1}, \end{aligned}$$

  (96)

  and

• 2)
  $$\begin{aligned} \left\| _{j}A_{n}\left( f\right) -f\right\| _{\infty }\le \lambda _{j1}. \end{aligned}$$

  (97)

  We notice that $\underset{n\rightarrow \infty }{\lim }$ $_{j}A_{n}\left( f\right) =f$, pointwise and uniformly.

In this article we extend Theorem 18 to the fuzzy-random level.

We mention

Theorem 19

(see [23,24,25,26]). Let $f\in C_{B}\left( \mathbb {R}^{N}\right) ,$ $0<\beta <1$, $x\in \mathbb {R}^{N},$ $N,n\in \mathbb {N}$ with $n^{1-\beta }>2;$ $j=1,2,3,4$. Then

• 1)
  $$\begin{aligned} \left| _{j}B_{n}\left( f,x\right) -f\left( x\right) \right| \le \omega _{1}\left( f,\frac{1}{n^{\beta }}\right) +2c_{j}\left( \beta ,n\right) \left\| f\right\| _{\infty }=:\lambda _{j2}, \end{aligned}$$

  (98)
• 2)
  $$\begin{aligned} \left\| _{j}B_{n}\left( f\right) -f\right\| _{\infty }\le \lambda _{j2}. \end{aligned}$$

  (99)

  Given that $f\in \left( C_{U}\left( \mathbb {R}^{N}\right) \cap C_{B}\left( \mathbb {R}^{N}\right) \right)$, we obtain $\underset{n\rightarrow \infty }{ \lim }$ $_{j}B_{n}\left( f\right) =f$, uniformly.

We also need

Theorem 20

(see [23,24,25,26]). Let $f\in C_{B}\left( \mathbb {R}^{N}\right)$, $0<\beta <1$, $x\in \mathbb {R}^{N},$ $N,n\in \mathbb {N}$ with $n^{1-\beta }>2;$ $j=1,2,3,4$. Then

• 1)
  $$\begin{aligned} \left| _{j}C_{n}\left( f,x\right) -f\left( x\right) \right| \le \omega _{1}\left( f,\frac{1}{n}+\frac{1}{n^{\beta }}\right) +2c_{j}\left( \beta ,n\right) \left\| f\right\| _{\infty }=:\lambda _{j3}, \end{aligned}$$

  (100)
• 2)
  $$\begin{aligned} \left\| _{j}C_{n}\left( f\right) -f\right\| _{\infty }\le \lambda _{j3}. \end{aligned}$$

  (101)

  Given that $f\in \left( C_{U}\left( \mathbb {R}^{N}\right) \cap C_{B}\left( \mathbb {R}^{N}\right) \right) ,$ we obtain $\underset{n\rightarrow \infty }{ \lim }$ $_{j}C_{n}\left( f\right) =f$, uniformly.

We also need

Theorem 21

(see [23,24,25,26]). Let $f\in C_{B}\left( \mathbb {R}^{N}\right) ,$ $0<\beta <1$, $x\in \mathbb {R}^{N},$ $N,n\in \mathbb {N}$ with $n^{1-\beta }>2;$ $j=1,2,3,4$. Then

• 1)
  $$\begin{aligned} \left| _{j}D_{n}\left( f,x\right) -f\left( x\right) \right| \le \omega _{1}\left( f,\frac{1}{n}+\frac{1}{n^{\beta }}\right) +2c_{j}\left( \beta ,n\right) \left\| f\right\| _{\infty }=\lambda _{j3}, \end{aligned}$$

  (102)
• 2)
  $$\begin{aligned} \left\| _{j}D_{n}\left( f\right) -f\right\| _{\infty }\le \lambda _{j3}. \end{aligned}$$

  (103)

Given that $f\in \left( C_{U}\left( \mathbb {R}^{N}\right) \cap C_{B}\left( \mathbb {R}^{N}\right) \right) ,$ we obtain $\underset{n\rightarrow \infty }{ \lim }$ $_{j}D_{n}\left( f\right) =f$, uniformly.

In this article we extend Theorems 19, 20, 21 to the random level.

We are also motivated by [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] and continuing [17]. For general knowledge on neural networks we recommend [31,32,33].

3 Main Results

I) q -mean Approximation by Fuzzy-Random arctangent, algebraic, Gudermannian and generalized symmetric activation functions based Quasi-Interpolation Neural Network Operators

All terms and assumptions here as in Sects. 1, 2.

Let $f\in C_{\mathcal{F}\mathcal{R}}^{U_{q}}\left( \prod \limits _{i=1}^{N}\left[ a_{i},b_{i}\right] \right)$, $1\le q<+\infty$, $n,N\in \mathbb {N}$, $0<\beta <1,$ $\overrightarrow{x}\in \left( \prod \limits _{i=1}^{N}\left[ a_{i},b_{i}\right] \right)$, $\left( X,\mathcal {B},P\right)$ probability space, $s\in X$; $j=1,2,3,4.$

We define the following multivariate fuzzy random arctangent, algebraic, Gudermannian and generalized symmetric activation functions based quasi-interpolation linear neural network operators

$$\begin{aligned} \left( _{j}A_{n}^{\mathcal{F}\mathcal{R}}\left( f\right) \right) \left( \overrightarrow{x},s\right) :=\sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor *}f\left( \frac{ \overrightarrow{k}}{n},s\right) \odot \frac{Z_{j}\left( n\overrightarrow{x}- \overrightarrow{k}\right) }{\sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{j}\left( n\overrightarrow{ x}-\overrightarrow{k}\right) }, \end{aligned}$$

(104)

(see also (89).

We present

Theorem 22

Let $f\in C_{\mathcal{F}\mathcal{R}}^{U_{q}}\left( \prod \limits _{i=1}^{N} \left[ a_{i},b_{i}\right] \right) ,$ $0<\beta <1,$ $\overrightarrow{x}\in \left( \prod \limits _{i=1}^{N}\left[ a_{i},b_{i}\right] \right)$, $n,N\in \mathbb {N},$ with $n^{1-\beta }>2,$ $1\le q<+\infty .$ Assume that $\int _{X}\left( D^{*}\left( f\left( \cdot ,s\right) ,\right. \right. \left. \left. \widetilde{o}\right) \right) ^{q}P\left( ds\right) <\infty ;$ $j=1,2,3,4.$ Then

• 1)
  $$\begin{aligned} \left( \int _{X}D^{q}\left( \left( _{j}A_{n}^{\mathcal{F}\mathcal{R}}\left( f\right) \right) \left( \overrightarrow{x},s\right) ,f\left( \overrightarrow{x} ,s\right) \right) P\left( ds\right) \right) ^{\frac{1}{q}}\le \end{aligned}$$

  (105)
  $$\begin{aligned} \gamma _{j}\left( N\right) \left\{ \varOmega _{1}\left( f,\frac{1}{n^{\beta }} \right) _{L^{q}}+2c_{j}\left( \beta ,n\right) \left( \int _{X}\left( D^{*}\left( f\left( \cdot ,s\right) ,\widetilde{o}\right) \right) ^{q}P\left( ds\right) \right) ^{\frac{1}{q}}\right\} =:\lambda _{j1}^{\left( \mathcal {FR }\right) }, \end{aligned}$$
• 2)
  $$\begin{aligned} \left\| \left( \int _{X}D^{q}\left( \left( _{j}A_{n}^{\mathcal{F}\mathcal{R}}\left( f\right) \right) \left( \overrightarrow{x},s\right) ,f\left( \overrightarrow{ x},s\right) \right) P\left( ds\right) \right) ^{\frac{1}{q}}\right\| _{\infty ,\left( \prod \limits _{i=1}^{N}\left[ a_{i},b_{i}\right] \right) }\le \lambda _{j1}^{\left( \mathcal{F}\mathcal{R}\right) }, \end{aligned}$$

  (106)

  where $\gamma _{j}\left( N\right)$ as in (25), (45), (65), (87) and $c_{j}\left( \beta ,n\right)$ as in (24), (44), (64), (86).

Proof

We notice that

$$\begin{aligned} D\left( f\left( \frac{\overrightarrow{k}}{n},s\right) ,f\left( \overrightarrow{x},s\right) \right) \le D\left( f\left( \frac{ \overrightarrow{k}}{n},s\right) ,\widetilde{o}\right) +D\left( f\left( \overrightarrow{x},s\right) ,\widetilde{o}\right) \end{aligned}$$

(107)

$$\begin{aligned} \le 2D^{*}\left( f\left( \cdot ,s\right) ,\widetilde{o}\right) . \end{aligned}$$

Hence

$$\begin{aligned} D^{q}\left( f\left( \frac{\overrightarrow{k}}{n},s\right) ,f\left( \overrightarrow{x},s\right) \right) \le 2^{q}D^{*q}\left( f\left( \cdot ,s\right) ,\widetilde{o}\right) , \end{aligned}$$

(108)

and

$$\begin{aligned} \left( \int _{X}D^{q}\left( f\left( \frac{\overrightarrow{k}}{n},s\right) ,f\left( \overrightarrow{x},s\right) \right) P\left( ds\right) \right) ^{ \frac{1}{q}}\le 2\left( \int _{X}\left( D^{*}\left( f\left( \cdot ,s\right) ,\widetilde{o}\right) \right) ^{q}P\left( ds\right) \right) ^{ \frac{1}{q}}. \end{aligned}$$

(109)

We observe that

$$\begin{aligned} D\left( \left( _{j}A_{n}^{\mathcal{F}\mathcal{R}}\left( f\right) \right) \left( \overrightarrow{x},s\right) ,f\left( \overrightarrow{x},s\right) \right) = \end{aligned}$$

(110)

$$\begin{aligned} D\left( \sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor *}f\left( \frac{\overrightarrow{k}}{n} ,s\right) \odot \frac{Z_{j}\left( nx-k\right) }{\sum \limits _{ \overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{j}\left( nx-k\right) },f\left( \overrightarrow{x},s\right) \odot 1\right) = \end{aligned}$$

$$\begin{aligned} D\left( \sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor *}f\left( \frac{\overrightarrow{k}}{n} ,s\right) \odot \frac{Z_{j}\left( nx-k\right) }{\sum \limits _{ \overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{j}\left( nx-k\right) },f\left( \overrightarrow{x},s\right) \odot \frac{\sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{j}\left( nx-k\right) }{\sum \limits _{ \overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{j}\left( nx-k\right) }\right) = \end{aligned}$$

(111)

$$\begin{aligned} D\left( \sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor *}f\left( \frac{\overrightarrow{k}}{n} ,s\right) \odot \frac{Z_{j}\left( nx-k\right) }{\sum \limits _{ \overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{j}\left( nx-k\right) },\sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor *}f\left( \overrightarrow{x},s\right) \odot \frac{Z_{j}\left( nx-k\right) }{\sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{j}\left( nx-k\right) }\right) \end{aligned}$$

$$\begin{aligned} \le \sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }\left( \frac{Z_{j}\left( nx-k\right) }{\sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{j}\left( nx-k\right) }\right) D\left( f\left( \frac{ \overrightarrow{k}}{n},s\right) ,f\left( \overrightarrow{x},s\right) \right) . \end{aligned}$$

(112)

So that

$$\begin{aligned} D\left( \left( _{j}A_{n}^{\mathcal{F}\mathcal{R}}\left( f\right) \right) \left( \overrightarrow{x},s\right) ,f\left( \overrightarrow{x},s\right) \right) \le \end{aligned}$$

$$\begin{aligned} \sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }\left( \frac{Z_{j}\left( nx-k\right) }{\sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{j}\left( nx-k\right) }\right) D\left( f\left( \frac{ \overrightarrow{k}}{n},s\right) ,f\left( \overrightarrow{x},s\right) \right) = \end{aligned}$$

(113)

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }\le \frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }}\left( \frac{Z_{j}\left( nx-k\right) }{\sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{j}\left( nx-k\right) }\right) D\left( f\left( \frac{\overrightarrow{k}}{n},s\right) ,f\left( \overrightarrow{x},s\right) \right) + \end{aligned}$$

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }>\frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }}\left( \frac{Z_{j}\left( nx-k\right) }{\sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{j}\left( nx-k\right) }\right) D\left( f\left( \frac{\overrightarrow{k}}{n},s\right) ,f\left( \overrightarrow{x},s\right) \right) . \end{aligned}$$

Hence it holds

$$\begin{aligned} \left( \int _{X}D^{q}\left( \left( _{j}A_{n}^{\mathcal{F}\mathcal{R}}\left( f\right) \right) \left( \overrightarrow{x},s\right) ,f\left( \overrightarrow{x} ,s\right) \right) P\left( ds\right) \right) ^{\frac{1}{q}}\le \end{aligned}$$

(114)

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }\le \frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }}\left( \frac{Z_{j}\left( nx-k\right) }{\sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{j}\left( nx-k\right) }\right) \left( \int _{X}D^{q}\left( f\left( \frac{ \overrightarrow{k}}{n},s\right) ,f\left( \overrightarrow{x},s\right) \right) P\left( ds\right) \right) ^{\frac{1}{q}}+ \end{aligned}$$

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }>\frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }}\left( \frac{Z_{j}\left( nx-k\right) }{\sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{j}\left( nx-k\right) }\right) \left( \int _{X}D^{q}\left( f\left( \frac{ \overrightarrow{k}}{n},s\right) ,f\left( \overrightarrow{x},s\right) \right) P\left( ds\right) \right) ^{\frac{1}{q}}\le \end{aligned}$$

$$\begin{aligned} \left( \frac{1}{\sum \limits _{\overrightarrow{k}=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }Z_{j}\left( nx-k\right) }\right) \cdot \left\{ \varOmega _{1}^{\left( \mathcal {F}\right) }\left( f,\frac{1}{ n^{\beta }}\right) _{L^{q}}+\right. \end{aligned}$$

(115)

$$\begin{aligned} \left. 2\left( \int _{X}\left( D^{*}\left( f\left( \cdot ,s\right) , \widetilde{o}\right) \right) ^{q}P\left( ds\right) \right) ^{\frac{1}{q} }\left( \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x} \right\| _{\infty }>\frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k }=\left\lceil na\right\rceil }^{\left\lfloor nb\right\rfloor }} Z_{j}\left( nx-k\right) \right) \right\} \end{aligned}$$

(by (24), (25); (44), (45); (64), (65); (86), (87))

$$\begin{aligned} \le \gamma _{j}\left( N\right) \left\{ \varOmega _{1}^{\left( \mathcal {F} \right) }\left( f,\frac{1}{n^{\beta }}\right) _{L^{q}}+2c_{j}\left( \beta ,n\right) \left( \int _{X}\left( D^{*}\left( f\left( \cdot ,s\right) , \widetilde{o}\right) \right) ^{q}P\left( ds\right) \right) ^{\frac{1}{q} }\right\} . \end{aligned}$$

(116)

We have proved claim.

Conclusion 6

By Theorem 22 we obtain the pointwise and uniform convergences with rates in the q-mean and D-metric of the operator $_{j}A_{n}^{\mathcal{F}\mathcal{R}}$ to the unit operator for $f\in C_{\mathcal{F}\mathcal{R} }^{U_{q}}\left( \prod \limits _{i=1}^{N}\left[ a_{i},b_{i}\right] \right) ,$ $j=1,2,3,4.$

II) 1-mean Approximation by Stochastic arctangent, algebraic, Gudermannian and generalized symmetric activation functions based full Quasi-Interpolation Neural Network Operators

Let $g\in C_{\mathcal {R}}^{U_{1}}\left( \mathbb {R}^{N}\right)$, $0<\beta <1$ , $\overrightarrow{x}\in \mathbb {R}^{N}$, $n,N\in \mathbb {N}$, with $\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}<\infty$, $\left( X, \mathcal {B},P\right)$ probability space, $s\in X.$

We define

$$\begin{aligned} _{j}B_{n}^{\left( \mathcal {R}\right) }\left( g\right) \left( \overrightarrow{ x},s\right) :=\sum \limits _{\overrightarrow{k}=-\infty }^{\infty }g\left( \frac{\overrightarrow{k}}{n},s\right) Z_{j}\left( n\overrightarrow{x}- \overrightarrow{k}\right) ,\text { }j=1,2,3,4, \end{aligned}$$

(117)

(see also (90)).

We give

Theorem 23

Let $g\in C_{\mathcal {R}}^{U_{1}}\left( \mathbb {R}^{N}\right) ,$ $0<\beta <1$, $\overrightarrow{x}\in \mathbb {R}^{N}$, $n,N\in \mathbb {N}$, with $n^{1-\beta }>2,$ $\left\| g\right\| _{\infty ,\mathbb {R} ^{N},X}<\infty ;$ $j=1,2,3,4.$ Then

• 1)
  $$\begin{aligned} \int _{X}\left| \left( _{j}B_{n}^{\left( \mathcal {R}\right) }\left( g\right) \right) \left( \overrightarrow{x},s\right) -g\left( \overrightarrow{ x},s\right) \right| P\left( ds\right) \le \end{aligned}$$

  (118)
  $$\begin{aligned} \left\{ \varOmega _{1}\left( g,\frac{1}{n^{\beta }}\right) _{L^{1}}+2c_{j}\left( \beta ,n\right) \left\| g\right\| _{\infty ,\mathbb {R}^{N},X}\right\} =:\mu _{j1}^{\left( \mathcal {R}\right) }, \end{aligned}$$
• 2)
  $$\begin{aligned} \left\| \int _{X}\left| \left( _{j}B_{n}^{\left( \mathcal {R}\right) }\left( g\right) \right) \left( \overrightarrow{x},s\right) -g\left( \overrightarrow{x},s\right) \right| P\left( ds\right) \right\| _{\infty ,\mathbb {R}^{N}}\le \mu _{j1}^{\left( \mathcal {R}\right) }. \end{aligned}$$

  (119)

Proof

Since $\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}<\infty$, then

$$\begin{aligned} \left| g\left( \frac{\overrightarrow{k}}{n},s\right) -g\left( \overrightarrow{x},s\right) \right| \le 2\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}<\infty . \end{aligned}$$

(120)

Hence

$$\begin{aligned} \int _{X}\left| g\left( \frac{\overrightarrow{k}}{n},s\right) -g\left( \overrightarrow{x},s\right) \right| P\left( ds\right) \le 2\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}<\infty . \end{aligned}$$

(121)

We observe that

$$\begin{aligned} \left( _{j}B_{n}^{\left( \mathcal {R}\right) }\left( g\right) \right) \left( \overrightarrow{x},s\right) -g\left( \overrightarrow{x},s\right) = \end{aligned}$$

$$\begin{aligned} \sum \limits _{\overrightarrow{k}=-\infty }^{\infty }g\left( \frac{ \overrightarrow{k}}{n},s\right) Z_{j}\left( nx-k\right) -g\left( \overrightarrow{x},s\right) \sum \limits _{\overrightarrow{k}=-\infty }^{\infty }Z_{j}\left( nx-k\right) = \end{aligned}$$

(122)

$$\begin{aligned} \left( \sum \limits _{\overrightarrow{k}=-\infty }^{\infty }g\left( \frac{ \overrightarrow{k}}{n},s\right) -g\left( \overrightarrow{x},s\right) \right) Z_{j}\left( nx-k\right) . \end{aligned}$$

However it holds

$$\begin{aligned} \sum \limits _{\overrightarrow{k}=-\infty }^{\infty }\left| g\left( \frac{\overrightarrow{k}}{n},s\right) -g\left( \overrightarrow{x},s\right) \right| Z_{j}\left( nx-k\right) \le 2\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}<\infty . \end{aligned}$$

(123)

Hence

$$\begin{aligned} \left| \left( _{j}B_{n}^{\left( \mathcal {R}\right) }\left( g\right) \right) \left( \overrightarrow{x},s\right) -g\left( \overrightarrow{x} ,s\right) \right| \le \end{aligned}$$

$$\begin{aligned} \sum \limits _{\overrightarrow{k}=-\infty }^{\infty }\left| g\left( \frac{\overrightarrow{k}}{n},s\right) -g\left( \overrightarrow{x},s\right) \right| Z_{j}\left( nx-k\right) = \end{aligned}$$

(124)

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }\le \frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =-\infty }^{\infty }}\left| g\left( \frac{\overrightarrow{k}}{n} ,s\right) -g\left( \overrightarrow{x},s\right) \right| Z_{j}\left( nx-k\right) + \end{aligned}$$

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }>\frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =-\infty }^{\infty }}\left| g\left( \frac{\overrightarrow{k}}{n} ,s\right) -g\left( \overrightarrow{x},s\right) \right| Z_{j}\left( nx-k\right) . \end{aligned}$$

Furthermore it holds

$$\begin{aligned} \left( \int _{X}\left| \left( _{j}B_{n}^{\left( \mathcal {R}\right) }\left( g\right) \right) \left( \overrightarrow{x},s\right) -g\left( \overrightarrow{x},s\right) \right| P\left( ds\right) \right) \le \end{aligned}$$

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }\le \frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =-\infty }^{\infty }}\left( \int _{X}\left| g\left( \frac{ \overrightarrow{k}}{n},s\right) -g\left( \overrightarrow{x},s\right) \right| P\left( ds\right) \right) Z_{j}\left( nx-k\right) + \end{aligned}$$

(125)

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }>\frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =-\infty }^{\infty }}\left( \int _{X}\left| g\left( \frac{ \overrightarrow{k}}{n},s\right) -g\left( \overrightarrow{x},s\right) \right| P\left( ds\right) \right) Z_{j}\left( nx-k\right) \le \end{aligned}$$

$$\begin{aligned} \varOmega _{1}\left( g,\frac{1}{n^{\beta }}\right) _{L^{1}}+2\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}\underset{\left\| \frac{ \overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }>\frac{1}{ n^{\beta }}}{\sum \limits _{\overrightarrow{k}=-\infty }^{\infty }} Z_{j}\left( nx-k\right) \le \end{aligned}$$

$$\begin{aligned} \varOmega _{1}\left( g,\frac{1}{n^{\beta }}\right) _{L^{1}}+2c_{j}\left( \beta ,n\right) \left\| g\right\| _{\infty ,\mathbb {R}^{N},X}, \end{aligned}$$

proving the claim.

Conclusion 7

By Theorem 23 we obtain pointwise and uniform convergences with rates in the 1-mean of random operators $_{j}B_{n}^{\left( \mathcal {R}\right) }$ to the unit operator for $g\in C_{ \mathcal {R}}^{U_{1}}\left( \mathbb {R}^{N}\right)$, $j=1,2,3,4.$

III) 1-mean Approximation by Stochastic arctangent, algebraic, Gudermannian and generalized symmetric activation functions based multivariate Kantorovich type neural network operator

Let $g\in C_{\mathcal {R}}^{U_{1}}\left( \mathbb {R}^{N}\right)$, $0<\beta <1$ , $\overrightarrow{x}\in \mathbb {R}^{N}$, $n,N\in \mathbb {N}$, with $\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}<\infty$, $\left( X,\mathcal {B},P\right)$ probability space, $s\in X.$

We define ($j=1,2,3,4$):

$$\begin{aligned} _{j}C_{n}^{\left( \mathcal {R}\right) }\left( g\right) \left( \overrightarrow{ x},s\right) :=\sum \limits _{\overrightarrow{k}=-\infty }^{\infty }\left( n^{N}\int _{\frac{\overrightarrow{k}}{n}}^{\frac{\overrightarrow{k}+1}{n} }g\left( \overrightarrow{t},s\right) d\overrightarrow{t}\right) Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k}\right) , \end{aligned}$$

(126)

(see also (91).

We present

Theorem 24

Let $g\in C_{\mathcal {R}}^{U_{1}}\left( \mathbb {R}^{N}\right) ,$ $0<\beta <1$, $\overrightarrow{x}\in \mathbb {R}^{N}$, $n,N\in \mathbb {N}$, with $n^{1-\beta }>2;$ $j=1,2,3,4,$ $\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}<\infty .$ Then

• 1)
  $$\begin{aligned} \int _{X}\left| \left( _{j}C_{n}^{\left( \mathcal {R}\right) }\left( g\right) \right) \left( \overrightarrow{x},s\right) -g\left( \overrightarrow{ x},s\right) \right| P\left( ds\right) \le \end{aligned}$$
  $$\begin{aligned} \left[ \varOmega _{1}\left( g,\frac{1}{n}+\frac{1}{n^{\beta }}\right) _{L^{1}}+2c_{j}\left( \beta ,n\right) \left\| g\right\| _{\infty , \mathbb {R}^{N},X}\right] =:\gamma _{j1}^{\left( \mathcal {R}\right) }, \end{aligned}$$

  (127)
• 2)
  $$\begin{aligned} \left\| \int _{X}\left| \left( _{j}C_{n}^{\left( \mathcal {R}\right) }\left( g\right) \right) \left( \overrightarrow{x},s\right) -g\left( \overrightarrow{x},s\right) \right| P\left( ds\right) \right\| _{\infty ,\mathbb {R}^{N}}\le \gamma _{j1}^{\left( \mathcal {R}\right) }. \end{aligned}$$

  (128)

Proof

Since $\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}<\infty$, then

$$\begin{aligned} \left| n^{N}\int _{\frac{\overrightarrow{k}}{n}}^{\frac{\overrightarrow{k}+1}{n}}g\left( \overrightarrow{t},s\right) d\overrightarrow{t}-g\left( \overrightarrow{x},s\right) \right| =\left| n^{N}\int _{\frac{ \overrightarrow{k}}{n}}^{\frac{\overrightarrow{k}+1}{n}}\left( g\left( \overrightarrow{t},s\right) -g\left( \overrightarrow{x},s\right) \right) d\overrightarrow{t}\right| \le \end{aligned}$$

$$\begin{aligned} n^{N}\int _{\frac{\overrightarrow{k}}{n}}^{\frac{\overrightarrow{k}+1}{n} }\left| g\left( \overrightarrow{t},s\right) -g\left( \overrightarrow{x} ,s\right) \right| d\overrightarrow{t}\le 2\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}<\infty . \end{aligned}$$

(129)

Hence

$$\begin{aligned} \int _{X}\left| n^{N}\int _{\frac{\overrightarrow{k}}{n}}^{\frac{ \overrightarrow{k}+1}{n}}g\left( \overrightarrow{t},s\right) d\overrightarrow{t}-g\left( \overrightarrow{x},s\right) \right| P\left( ds\right) \le 2\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}<\infty . \end{aligned}$$

(130)

We observe that

$$\begin{aligned} \left( _{j}C_{n}^{\left( \mathcal {R}\right) }\left( g\right) \right) \left( \overrightarrow{x},s\right) -g\left( \overrightarrow{x},s\right) = \end{aligned}$$

$$\begin{aligned} \sum \limits _{\overrightarrow{k}=-\infty }^{\infty }\left( n^{N}\int _{\frac{ \overrightarrow{k}}{n}}^{\frac{\overrightarrow{k}+1}{n}}g\left( \overrightarrow{t},s\right) d\overrightarrow{t}\right) Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k}\right) -g\left( \overrightarrow{x} ,s\right) = \end{aligned}$$

$$\begin{aligned} \sum \limits _{\overrightarrow{k}=-\infty }^{\infty }\left( n^{N}\int _{\frac{ \overrightarrow{k}}{n}}^{\frac{\overrightarrow{k}+1}{n}}g\left( \overrightarrow{t},s\right) d\overrightarrow{t}\right) Z_{j}\left( n \overrightarrow{x}-\overrightarrow{k}\right) -g\left( \overrightarrow{x} ,s\right) \sum \limits _{\overrightarrow{k}=-\infty }^{\infty }Z_{j}\left( n \overrightarrow{x}-\overrightarrow{k}\right) = \end{aligned}$$

(131)

$$\begin{aligned} \sum \limits _{\overrightarrow{k}=-\infty }^{\infty }\left[ \left( n^{N}\int _{ \frac{\overrightarrow{k}}{n}}^{\frac{\overrightarrow{k}+1}{n}}g\left( \overrightarrow{t},s\right) d\overrightarrow{t}\right) -g\left( \overrightarrow{x},s\right) \right] Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k}\right) = \end{aligned}$$

$$\begin{aligned} \sum \limits _{\overrightarrow{k}=-\infty }^{\infty }\left[ n^{N}\int _{\frac{ \overrightarrow{k}}{n}}^{\frac{\overrightarrow{k}+1}{n}}\left( g\left( \overrightarrow{t},s\right) -g\left( \overrightarrow{x},s\right) \right) d\overrightarrow{t}\right] Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k} \right) . \end{aligned}$$

However it holds

$$\begin{aligned} \sum \limits _{\overrightarrow{k}=-\infty }^{\infty }\left[ n^{N}\int _{\frac{ \overrightarrow{k}}{n}}^{\frac{\overrightarrow{k}+1}{n}}\left| g\left( \overrightarrow{t},s\right) -g\left( \overrightarrow{x},s\right) \right| d\overrightarrow{t}\right] Z_{j}\left( n\overrightarrow{x}- \overrightarrow{k}\right) \le 2\left\| g\right\| _{\infty ,\mathbb {R }^{N},X}<\infty . \end{aligned}$$

(132)

Hence

$$\begin{aligned} \left| \left( _{j}C_{n}^{\left( \mathcal {R}\right) }\left( g\right) \right) \left( \overrightarrow{x},s\right) -g\left( \overrightarrow{x} ,s\right) \right| \le \end{aligned}$$

$$\begin{aligned} \sum \limits _{\overrightarrow{k}=-\infty }^{\infty }\left[ n^{N}\int _{\frac{ \overrightarrow{k}}{n}}^{\frac{\overrightarrow{k}+1}{n}}\left| g\left( \overrightarrow{t},s\right) -g\left( \overrightarrow{x},s\right) \right| d\overrightarrow{t}\right] Z_{j}\left( n\overrightarrow{x}- \overrightarrow{k}\right) = \end{aligned}$$

(133)

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }\le \frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =-\infty }^{\infty }}\left[ n^{N}\int _{\frac{\overrightarrow{k}}{n}}^{\frac{ \overrightarrow{k}+1}{n}}\left| g\left( \overrightarrow{t},s\right) -g\left( \overrightarrow{x},s\right) \right| d\overrightarrow{t}\right] Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k}\right) + \end{aligned}$$

(134)

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }>\frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =-\infty }^{\infty }}\left[ n^{N}\int _{\frac{\overrightarrow{k}}{n}}^{\frac{ \overrightarrow{k}+1}{n}}\left| g\left( \overrightarrow{t},s\right) -g\left( \overrightarrow{x},s\right) \right| d\overrightarrow{t}\right] Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k}\right) = \end{aligned}$$

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }\le \frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =-\infty }^{\infty }}\left[ n^{N}\int _{0}^{\frac{1}{n}}\left| g\left( \overrightarrow{t}+\frac{\overrightarrow{k}}{n},s\right) -g\left( \overrightarrow{x},s\right) \right| d\overrightarrow{t}\right] Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k}\right) + \end{aligned}$$

(135)

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }>\frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =-\infty }^{\infty }}\left[ n^{N}\int _{0}^{\frac{1}{n}}\left| g\left( \overrightarrow{t}+\frac{\overrightarrow{k}}{n},s\right) -g\left( \overrightarrow{x},s\right) \right| d\overrightarrow{t}\right] Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k}\right) . \end{aligned}$$

Furthermore it holds

$$\begin{aligned} \left( \int _{X}\left| \left( _{j}C_{n}^{\left( \mathcal {R}\right) }\left( g\right) \right) \left( \overrightarrow{x},s\right) -g\left( \overrightarrow{x},s\right) \right| P\left( ds\right) \right) \underset{ \text {(by Fubini's theorem)}}{\le } \end{aligned}$$

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }\le \frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =-\infty }^{\infty }}\left[ n^{N}\int _{0}^{\frac{1}{n}}\left( \int _{X}\left| g\left( \overrightarrow{t}+\frac{\overrightarrow{k}}{n},s\right) -g\left( \overrightarrow{x},s\right) \right| P\left( ds\right) \right) d \overrightarrow{t}\right] Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k} \right) + \end{aligned}$$

(136)

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }>\frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =-\infty }^{\infty }}\left[ n^{N}\int _{0}^{\frac{1}{n}}\left( \int _{X}\left| g\left( \overrightarrow{t}+\frac{\overrightarrow{k}}{n},s\right) -g\left( \overrightarrow{x},s\right) \right| P\left( ds\right) \right) d \overrightarrow{t}\right] Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k} \right) \le \end{aligned}$$

$$\begin{aligned} \varOmega _{1}\left( g,\frac{1}{n}+\frac{1}{n^{\beta }}\right) _{L^{1}}+2\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}\underset{\left\| \frac{ \overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }>\frac{1}{ n^{\beta }}}{\sum \limits _{\overrightarrow{k}=-\infty }^{\infty }} Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k}\right) \le \end{aligned}$$

$$\begin{aligned} \varOmega _{1}\left( g,\frac{1}{n}+\frac{1}{n^{\beta }}\right) _{L^{1}}+2c_{j}\left( \beta ,n\right) \left\| g\right\| _{\infty , \mathbb {R}^{N},X}, \end{aligned}$$

(137)

proving the claim.

Conclusion 8

By Theorem 24 we obtain pointwise and uniform convergences with rates in the 1-mean of random operators $_{j}C_{n}^{\left( \mathcal {R}\right) }$ to the unit operator for $g\in C_{ \mathcal {R}}^{U_{1}}\left( \mathbb {R}^{N}\right)$, $j=1,2,3,4.$

IV) 1-mean Approximation by Stochastic arctangent, algebraic, Gudermannian and generalized symmetric activation functions based multivariate quadrature type neural network operator

Let $g\in C_{\mathcal {R}}^{U_{1}}\left( \mathbb {R}^{N}\right)$, $0<\beta <1$ , $\overrightarrow{x}\in \mathbb {R}^{N}$, $n,N\in \mathbb {N}$, with $\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}<\infty$, $\left( X, \mathcal {B},P\right)$ probability space, $s\in X$, $j=1,2,3,4.$

We define

$$\begin{aligned} _{j}D_{n}^{\left( \mathcal {R}\right) }\left( g\right) \left( \overrightarrow{ x},s\right) :=\sum \limits _{\overrightarrow{k}=-\infty }^{\infty }\left( \delta _{n\overrightarrow{k}}\left( g\right) \right) \left( s\right) Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k}\right) , \end{aligned}$$

(138)

where

$$\begin{aligned} \left( \delta _{n\overrightarrow{k}}\left( g\right) \right) \left( s\right) :=\sum \limits _{\overrightarrow{\overline{r}}=0}^{\overrightarrow{\theta } }w_{\overrightarrow{\overline{r}}}g\left( \frac{\overrightarrow{k}}{n}+\frac{ \overrightarrow{\overline{r}}}{n\overrightarrow{\theta }},s\right) , \end{aligned}$$

(139)

(see also (92), (93)).

We finally give

Theorem 25

Let $g\in C_{\mathcal {R}}^{U_{1}}\left( \mathbb {R}^{N}\right) ,$ $0<\beta <1$, $\overrightarrow{x}\in \mathbb {R}^{N}$, $n,N\in \mathbb {N}$, with $n^{1-\beta }>2;$ $j=1,2,3,4,$ $\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}<\infty .$ Then

• 1)
  $$\begin{aligned} \int _{X}\left| \left( _{j}D_{n}^{\left( \mathcal {R}\right) }\left( g\right) \right) \left( \overrightarrow{x},s\right) -g\left( \overrightarrow{ x},s\right) \right| P\left( ds\right) \le \end{aligned}$$
  $$\begin{aligned} \left\{ \varOmega _{1}\left( g,\frac{1}{n}+\frac{1}{n^{\beta }}\right) _{L^{1}}+2c_{j}\left( \beta ,n\right) \left\| g\right\| _{\infty , \mathbb {R}^{N},X}\right\} =:\gamma _{j1}^{\left( \mathcal {R}\right) }, \end{aligned}$$

  (140)
• 2)
  $$\begin{aligned} \left\| \int _{X}\left| \left( _{j}D_{n}^{\left( \mathcal {R}\right) }\left( g\right) \right) \left( \overrightarrow{x},s\right) -g\left( \overrightarrow{x},s\right) \right| P\left( ds\right) \right\| _{\infty ,\mathbb {R}^{N}}\le \gamma _{j1}^{\left( \mathcal {R}\right) }. \end{aligned}$$

  (141)

Proof

Notice that

$$\begin{aligned} \left| \left( \delta _{n\overrightarrow{k}}\left( g\right) \right) \left( s\right) -g\left( \overrightarrow{x},s\right) \right| = \end{aligned}$$

$$\begin{aligned} \left| \sum \limits _{\overrightarrow{\overline{r}}=0}^{\overrightarrow{ \theta }}w_{\overrightarrow{\overline{r}}}\left( g\left( \frac{ \overrightarrow{k}}{n}+\frac{\overrightarrow{\overline{r}}}{n\overrightarrow{ \theta }},s\right) -g\left( \overrightarrow{x},s\right) \right) \right| \le \end{aligned}$$

$$\begin{aligned} \sum \limits _{\overrightarrow{\overline{r}}=0}^{\overrightarrow{\theta }}w_{ \overrightarrow{\overline{r}}}\left| g\left( \frac{\overrightarrow{k}}{n }+\frac{\overrightarrow{\overline{r}}}{n\overrightarrow{\theta }},s\right) -g\left( \overrightarrow{x},s\right) \right| \le 2\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}<\infty . \end{aligned}$$

(142)

Hence

$$\begin{aligned} \int _{X}\left| \left( \delta _{n\overrightarrow{k}}\left( g\right) \right) \left( s\right) -g\left( \overrightarrow{x},s\right) \right| P\left( ds\right) \le 2\left\| g\right\| _{\infty ,\mathbb {R} ^{N},X}<\infty . \end{aligned}$$

(143)

We observe that

$$\begin{aligned} \left( _{j}D_{n}^{\left( \mathcal {R}\right) }\left( g\right) \right) \left( \overrightarrow{x},s\right) -g\left( \overrightarrow{x},s\right) = \end{aligned}$$

$$\begin{aligned} \sum \limits _{\overrightarrow{k}=-\infty }^{\infty }\left( \delta _{n \overrightarrow{k}}\left( g\right) \right) \left( s\right) Z_{j}\left( n \overrightarrow{x}-\overrightarrow{k}\right) -g\left( \overrightarrow{x} ,s\right) = \end{aligned}$$

$$\begin{aligned} \sum \limits _{\overrightarrow{k}=-\infty }^{\infty }\left( \left( \delta _{n \overrightarrow{k}}\left( g\right) \right) \left( s\right) -g\left( \overrightarrow{x},s\right) \right) Z_{j}\left( n\overrightarrow{x}- \overrightarrow{k}\right) . \end{aligned}$$

(144)

Thus

$$\begin{aligned} \left| _{j}D_{n}^{\left( \mathcal {R}\right) }\left( g\right) \left( \overrightarrow{x},s\right) -g\left( \overrightarrow{x},s\right) \right| \le \end{aligned}$$

$$\begin{aligned} \sum \limits _{\overrightarrow{k}=-\infty }^{\infty }\left| \left( \delta _{n\overrightarrow{k}}\left( g\right) \right) \left( s\right) -g\left( \overrightarrow{x},s\right) \right| Z_{j}\left( n \overrightarrow{x}-\overrightarrow{k}\right) \le 2\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}<\infty . \end{aligned}$$

(145)

Hence it holds

$$\begin{aligned} \left| \left( _{j}D_{n}^{\left( \mathcal {R}\right) }\left( g\right) \right) \left( \overrightarrow{x},s\right) -g\left( \overrightarrow{x} ,s\right) \right| \le \end{aligned}$$

$$\begin{aligned} \sum \limits _{\overrightarrow{k}=-\infty }^{\infty }\left| \left( \delta _{n\overrightarrow{k}}\left( g\right) \right) \left( s\right) -g\left( \overrightarrow{x},s\right) \right| Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k}\right) = \end{aligned}$$

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }\le \frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =-\infty }^{\infty }}\left| \left( \delta _{n\overrightarrow{k}}\left( g\right) \right) \left( s\right) -g\left( \overrightarrow{x},s\right) \right| Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k}\right) + \end{aligned}$$

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }>\frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =-\infty }^{\infty }}\left| \left( \delta _{n\overrightarrow{k}}\left( g\right) \right) \left( s\right) -g\left( \overrightarrow{x},s\right) \right| Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k}\right) . \end{aligned}$$

(146)

Furthermore we derive

$$\begin{aligned} \left( \int _{X}\left| \left( _{j}D_{n}^{\left( \mathcal {R}\right) }\left( g\right) \right) \left( \overrightarrow{x},s\right) -g\left( \overrightarrow{x},s\right) \right| P\left( ds\right) \right) \le \end{aligned}$$

$$\begin{aligned} \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x}\right\| _{\infty }\le \frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k} =-\infty }^{\infty }}\sum \limits _{\overrightarrow{\overline{r}}=0}^{ \overrightarrow{\theta }}w_{\overrightarrow{\overline{r}}}\left( \int _{X}\left| g\left( \frac{\overrightarrow{k}}{n}+\frac{ \overrightarrow{\overline{r}}}{n\overrightarrow{\theta }},s\right) -g\left( \overrightarrow{x},s\right) \right| P\left( ds\right) \right) Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k}\right) \end{aligned}$$

(147)

$$\begin{aligned} +\left( \underset{\left\| \frac{\overrightarrow{k}}{n}-\overrightarrow{x} \right\| _{\infty }>\frac{1}{n^{\beta }}}{\sum \limits _{\overrightarrow{k }=-\infty }^{\infty }}Z_{j}\left( n\overrightarrow{x}-\overrightarrow{k} \right) \right) 2\left\| g\right\| _{\infty ,\mathbb {R}^{N},X}\le \end{aligned}$$

$$\begin{aligned} \varOmega _{1}\left( g,\frac{1}{n}+\frac{1}{n^{\beta }}\right) _{L^{1}}+2c_{j}\left( \beta ,n\right) \left\| g\right\| _{\infty , \mathbb {R}^{N},X}, \end{aligned}$$

(148)

proving the claim.

Conclusion 9

From Theorem 25 we obtain pointwise and uniform convergences with rates in the 1-mean of random operators $_{j}D_{n}^{\left( \mathcal {R}\right) }$ to the unit operator for $g\in C_{ \mathcal {R}}^{U_{1}}\left( \mathbb {R}^{N}\right)$, $j=1,2,3,4$.