CUBO A Mathematical Journal
Vol.16, No¯ 02, (33–47). June 2014

Voronovskaya type asymptotic expansions for multivariate
quasi-interpolation neural network operators

George A. Anastassiou

Department of Mathematical Sciences,

University of Memphis,

Memphis, TN 38152, U.S.A.

ganastss@memphis.edu

ABSTRACT

Here we study further the multivariate quasi-interpolation of sigmoidal and hyperbolic

tangent types neural network operators of one hidden layer. We derive multivariate

Voronovskaya type asymptotic expansions for the error of approximation of these op-

erators to the unit operator.

RESUMEN

Aqúı estudiamos extensiones de la cuasi-interpolación multivariada de operadores de

redes neuronales de tipo sigmoidal y tangente hiperbólica de una capa oculta. Obten-

emos expansiones asintóticas del tipo Voronovskaya para el error de aproximación de

estos operadores para el operador unidad.

Keywords and Phrases: Multivariate Neural Network Approximation, multivariate Voronovskaya

type asymptotic expansion.

2010 AMS Mathematics Subject Classification: 41A25, 41A36, 41A60.



34 George A. Anastassiou CUBO
16, 2 (2014)

1 Background

Here we follow [5], [6].

We consider here the sigmoidal function of logarithmic type

si (xi) =
1

1 + e−xi
, xi ∈ R, i = 1, ..., N; x := (x1, ..., xN) ∈ R

N,

each has the properties lim
xi→+∞

si (xi) = 1 and lim
xi→−∞

si (xi) = 0, i = 1, ..., N.

These functions play the role of activation functions in the hidden layer of neural networks.

As in [7], we consider

Φi (xi) :=
1

2
(si (xi + 1) − si (xi − 1)) , xi ∈ R, i = 1, ..., N.

We notice the following properties:

i) Φi (xi) > 0, ∀ xi ∈ R,

ii)
∑∞

ki=−∞
Φi (xi − ki) = 1, ∀ xi ∈ R,

iii)
∑∞

ki=−∞
Φi (nxi − ki) = 1, ∀ xi ∈ R; n ∈ N,

iv)
∫∞
−∞

Φi (xi) dxi = 1,

v) Φi is a density function,

vi) Φi is even: Φi (−xi) = Φi (xi), xi ≥ 0, for i = 1, ..., N.

We see that ([7])

Φi (xi) =

(

e2 − 1

2e2

)

1

(1 + exi−1) (1 + e−xi−1)
, i = 1, ..., N.

vii) Φi is decreasing on R+, and increasing on R−, i = 1, ..., N.

Let 0 < β < 1, n ∈ N. Then as in [6] we get

viii)
∞∑

⎧
⎪⎨

⎪⎩

ki = −∞

: |nxi − ki| > n
1−β

Φi (nxi − ki) ≤ 3.1992e
−n(1−β), i = 1, ..., N.



CUBO
16, 2 (2014)

Voronovskaya type asymptotic expansions for multivariate . . . 35

Denote by ⌈·⌉ the ceiling of a number, and by ⌊·⌋ the integral part of a number. Consider

here x ∈
(∏N

i=1 [ai, bi]
)

⊂ RN, N ∈ N such that ⌈nai⌉ ≤ ⌊nbi⌋, i = 1, ..., N; a := (a1, ..., aN),

b := (b1, ..., bN) .

As in [6] we obtain

ix)

0 <
1

∑⌊nbi⌋
ki=⌈nai⌉

Φi (nxi − ki)
<

1

Φi (1)
= 5.250312578,

∀ xi ∈ [ai, bi] , i = 1, ..., N.

x) As in [6], we see that

lim
n→∞

⌊nbi⌋∑

ki=⌈nai⌉

Φi (nxi − ki) ̸= 1,

for at least some xi ∈ [ai, bi], i = 1, ..., N.

We will use here

Φ (x1, ..., xN) := Φ (x) :=
N∏

i=1

Φi (xi) , x ∈ R
N. (1)

It has the properties:

(i)’ Φ (x) > 0, ∀ x ∈ RN,

We see that
∞∑

k1=−∞

∞∑

k2=−∞

...
∞∑

kN=−∞

Φ (x1 − k1, x2 − k2, ..., xN − kN) =

∞∑

k1=−∞

∞∑

k2=−∞

...
∞∑

kN=−∞

N∏

i=1

Φi (xi − ki) =
N∏

i=1

(

∞∑

ki=−∞

Φi (xi − ki)

)

= 1.

That is

(ii)’
∞∑

k=−∞

Φ (x − k) :=
∞∑

k1=−∞

∞∑

k2=−∞

...
∞∑

kN=−∞

Φ (x1 − k1, ..., xN − kN) = 1,

k := (k1, ..., kN), ∀ x ∈ R
N.

(iii)’
∞∑

k=−∞

Φ (nx − k) :=

∞∑

k1=−∞

∞∑

k2=−∞

...
∞∑

kN=−∞

Φ (nx1 − k1, ..., nxN − kN) = 1,

∀ x ∈ RN; n ∈ N.



36 George A. Anastassiou CUBO
16, 2 (2014)

(iv)’
∫

RN

Φ (x) dx = 1,

that is Φ is a multivariate density function.

Here ∥x∥
∞

:= max {|x1| , ..., |xN|}, x ∈ R
N, also set ∞ := (∞, ..., ∞), −∞ := (−∞, ..., −∞)

upon the multivariate context, and

⌈na⌉ : = (⌈na1⌉ , ..., ⌈naN⌉) ,

⌊nb⌋ : = (⌊nb1⌋ , ..., ⌊nbN⌋) .

For 0 < β < 1 and n ∈ N, fixed x ∈ RN, have that

⌊nb⌋∑

k=⌈na⌉

Φ (nx − k) =

⌊nb⌋∑

⎧
⎪⎨

⎪⎩

k = ⌈na⌉
∥

∥

k
n
− x
∥

∥

∞
≤ 1

nβ

Φ (nx − k) +

⌊nb⌋∑

⎧
⎪⎨

⎪⎩

k = ⌈na⌉
∥

∥

k
n
− x
∥

∥

∞
> 1

nβ

Φ (nx − k) .

In the last two sums the counting is over disjoint vector of k’s, because the condition
∥

∥

k
n
− x
∥

∥

∞
>

1
nβ

implies that there exists at least one
∣

∣

kr
n

− xr
∣

∣ > 1
nβ

, r ∈ {1, ..., N}.

It holds

(v)’
⌊nb⌋∑

⎧
⎪⎨

⎪⎩

k = ⌈na⌉
∥

∥

k
n
− x
∥

∥

∞
> 1

nβ

Φ (nx − k) ≤ 3.1992e−n
(1−β)

,

0 < β < 1, n ∈ N, x ∈
(∏N

i=1 [ai, bi]
)

.

Furthermore it holds

(vi)’

0 <
1

∑⌊nb⌋
k=⌈na⌉

Φ (nx − k)
< (5.250312578)

N
,

∀ x ∈
(∏N

i=1 [ai, bi]
)

, n ∈ N.

It is clear also that



CUBO
16, 2 (2014)

Voronovskaya type asymptotic expansions for multivariate . . . 37

(vii)’
∞∑

⎧
⎪⎨

⎪⎩

k = −∞
∥

∥

k
n
− x
∥

∥

∞
> 1

nβ

Φ (nx − k) ≤ 3.1992e−n
(1−β)

,

0 < β < 1, n ∈ N, x ∈ RN.

By (x) we obviously see that

(viii)’

lim
n→∞

⌊nb⌋∑

k=⌈na⌉

Φ (nx − k) ̸= 1

for at least some x ∈
(∏N

i=1 [ai, bi]
)

.

Let f ∈ C
(∏N

i=1 [ai, bi]
)

and n ∈ N such that ⌈nai⌉ ≤ ⌊nbi⌋, i = 1, ..., N.

We define the multivariate positive linear neural network operator

(x := (x1, ..., xN) ∈
(∏N

i=1 [ai, bi]
)

)

Gn (f, x1, ..., xN) := Gn (f, x) :=

∑⌊nb⌋
k=⌈na⌉

f
(

k
n

)

Φ (nx − k)
∑⌊nb⌋

k=⌈na⌉
Φ (nx − k)

(2)

:=

∑⌊nb1⌋
k1=⌈na1⌉

∑⌊nb2⌋
k2=⌈na2⌉

...
∑⌊nbN⌋

kN=⌈naN⌉
f
(

k1
n
, ..., kN

n

)

(∏N
i=1 Φi (nxi − ki)

)

∏N
i=1

(∑⌊nbi⌋
ki=⌈nai⌉

Φi (nxi − ki)
) .

For large enough n we always obtain ⌈nai⌉ ≤ ⌊nbi⌋, i = 1, ..., N. Also ai ≤
ki
n

≤ bi, iff

⌈nai⌉ ≤ ki ≤ ⌊nbi⌋, i = 1, ..., N.

Notice here that for large enough n ∈ N we get that

e−n
(1−β)

< n−βj, j = 1, ..., m ∈ N, 0 < β < 1.

Thus be given fixed A, B > 0, for the linear combination
(

An−βj + Be−n
(1−β)

)

the (dominant)

rate of convergence to zero is n−βj. The closer β is to 1 we get faster and better rate of convergence

to zero.

By ACm
(∏N

i=1 [ai, bi]
)

, m, N ∈ N, we denote the space of functions such that all par-

tial derivatives of order (m − 1) are coordinatewise alsolutely continuous functions, also f ∈

Cm−1
(∏N

i=1 [ai, bi]
)

.



38 George A. Anastassiou CUBO
16, 2 (2014)

Let f ∈ ACm
(∏N

i=1 [ai, bi]
)

, m, N ∈ N. Here fα denotes a partial derivative of f, α :=

(α1, ..., αN), αi ∈ Z
+, i = 1, ..., N, and |α| :=

∑N
i=1 αi = l, where l = 0, 1, ..., m. We write also

fα :=
∂αf
∂xα

and we say it is order l.

We denote

∥fα∥
max
∞,m := max

|α|=m
{∥fα∥∞}, (3)

where ∥·∥
∞

is the supremum norm.

We assume here that ∥fα∥
max
∞,m < ∞.

Next we follow [3], [4].

We consider here the hyperbolic tangent function tanh x, x ∈ R :

tanh x :=
ex − e−x

ex + e−x
.

It has the properties tanh 0 = 0, −1 < tanh x < 1, ∀ x ∈ R, and tanh (−x) = − tanh x. Furthermore

tanh x → 1 as x → ∞, and tanh x → −1, as x → −∞, and it is strictly increasing on R.

This function plays the role of an activation function in the hidden layer of neural networks.

We further consider

Ψ (x) :=
1

4
(tanh (x + 1) − tanh (x − 1)) > 0, ∀ x ∈ R.

We easily see that Ψ (−x) = Ψ (x), that is Ψ is even on R. Obviously Ψ is differentiable, thus

continuous.

Proposition 1. ([3]) Ψ (x) for x ≥ 0 is strictly decreasing.

Obviously Ψ (x) is strictly increasing for x ≤ 0. Also it holds lim
x→−∞

Ψ (x) = 0 = lim
x→∞

Ψ (x) .

Infact Ψ has the bell shape with horizontal asymptote the x-axis. So the maximum of Ψ is

zero, Ψ (0) = 0.3809297.

Theorem 2. ([3]) We have that
∑∞

i=−∞ Ψ (x − i) = 1, ∀ x ∈ R.

Thus
∞∑

i=−∞

Ψ (nx − i) = 1, ∀ n ∈ N, ∀ x ∈ R.

Also it holds
∞∑

i=−∞

Ψ (x + i) = 1, ∀x ∈ R.

Theorem 3. ([3]) It holds
∫∞
−∞

Ψ (x) dx = 1.

So Ψ (x) is a density function on R.



CUBO
16, 2 (2014)

Voronovskaya type asymptotic expansions for multivariate . . . 39

Theorem 4. ([3]) Let 0 < α < 1 and n ∈ N. It holds

∞∑

⎧
⎪⎨

⎪⎩

k = −∞

: |nx − k| ≥ n1−α

Ψ (nx − k) ≤ e4 · e−2n
(1−α)

.

Theorem 5. ([3]) Let x ∈ [a, b] ⊂ R and n ∈ N so that ⌈na⌉ ≤ ⌊nb⌋. It holds

1
∑⌊nb⌋

k=⌈na⌉
Ψ (nx − k)

<
1

Ψ (1)
= 4.1488766.

Also by [3] we get that

lim
n→∞

⌊nb⌋∑

k=⌈na⌉

Ψ (nx − k) ̸= 1,

for at least some x ∈ [a, b].

In this article we will use

Θ (x1, ..., xN) := Θ (x) :=
N∏

i=1

Ψ (xi) , x = (x1, ..., xN) ∈ R
N, N ∈ N. (4)

It has the properties:

(i)∗ Θ (x) > 0, ∀ x ∈ RN,

(ii)∗
∞∑

k=−∞

Θ (x − k) :=
∞∑

k1=−∞

∞∑

k2=−∞

...
∞∑

kN=−∞

Θ (x1 − k1, ..., xN − kN) = 1,

where k := (k1, ..., kN), ∀ x ∈ R
N.

(iii)∗
∞∑

k=−∞

Θ (nx − k) :=

∞∑

k1=−∞

∞∑

k2=−∞

...
∞∑

kN=−∞

Θ (nx1 − k1, ..., nxN − kN) = 1,

∀ x ∈ RN; n ∈ N.

(iv)∗ ∫

RN

Θ (x) dx = 1,

that is Θ is a multivariate density function.



40 George A. Anastassiou CUBO
16, 2 (2014)

We obviously see that

⌊nb⌋∑

k=⌈na⌉

Θ (nx − k) =

⌊nb⌋∑

k=⌈na⌉

N∏

i=1

Ψ (nxi − ki) =

⌊nb1⌋∑

k1=⌈na1⌉

...

⌊nbN⌋∑

kN=⌈naN⌉

N∏

i=1

Ψ (nxi − ki) =
N∏

i=1

⎛

⎝

⌊nbi⌋∑

ki=⌈nai⌉

Ψ (nxi − ki)

⎞

⎠ .

For 0 < β < 1 and n ∈ N, fixed x ∈ RN, we have that

⌊nb⌋∑

k=⌈na⌉

Θ (nx − k) =

⌊nb⌋∑

⎧
⎪⎨

⎪⎩

k = ⌈na⌉
∥

∥

k
n
− x
∥

∥

∞
≤ 1

nβ

Θ (nx − k) +

⌊nb⌋∑

⎧
⎪⎨

⎪⎩

k = ⌈na⌉
∥

∥

k
n
− x
∥

∥

∞
> 1

nβ

Θ (nx − k) .

In the last two sums the counting is over disjoint vector of k’s, because the condition
∥

∥

k
n
− x
∥

∥

∞
> 1

nβ
implies that there exists at least one

∣

∣

kr
n

− xr
∣

∣ > 1
nβ

, r ∈ {1, ..., N}.

Il holds

(v)∗

⌊nb⌋∑

⎧
⎪⎨

⎪⎩

k = ⌈na⌉
∥

∥

k
n
− x
∥

∥

∞
> 1

nβ

Θ (nx − k) ≤ e4 · e−2n
(1−β)

,

0 < β < 1, n ∈ N, x ∈
(∏N

i=1 [ai, bi]
)

.

Also it holds

(vi)∗

0 <
1

∑⌊nb⌋
k=⌈na⌉

Θ (nx − k)
<

1

(Ψ (1))
N

= (4.1488766)
N
,

∀ x ∈
(∏N

i=1 [ai, bi]
)

, n ∈ N.

It is clear that

(vii)∗
∞∑

⎧
⎪⎨

⎪⎩

k = −∞
∥

∥

k
n
− x
∥

∥

∞
> 1

nβ

Θ (nx − k) ≤ e4 · e−2n
(1−β)

,

0 < β < 1, n ∈ N, x ∈ RN.



CUBO
16, 2 (2014)

Voronovskaya type asymptotic expansions for multivariate . . . 41

Also we get

lim
n→∞

⌊nb⌋∑

k=⌈na⌉

Θ (nx − k) ̸= 1,

for at least some x ∈
(∏N

i=1 [ai, bi]
)

.

Let f ∈ C
(∏N

i=1 [ai, bi]
)

and n ∈ N such that ⌈nai⌉ ≤ ⌊nbi⌋, i = 1, ..., N.

We define the multivariate positive linear neural network operator (x := (x1, ..., xN) ∈
(∏N

i=1 [ai, bi]
)

)

Fn (f, x1, ..., xN) := Fn (f, x) :=

∑⌊nb⌋
k=⌈na⌉

f
(

k
n

)

Θ (nx − k)
∑⌊nb⌋

k=⌈na⌉
Θ (nx − k)

(5)

:=

∑⌊nb1⌋
k1=⌈na1⌉

∑⌊nb2⌋
k2=⌈na2⌉

...
∑⌊nbN⌋

kN=⌈naN⌉
f
(

k1
n
, ..., kN

n

)

(∏N
i=1 Ψ (nxi − ki)

)

∏N
i=1

(∑⌊nbi⌋
ki=⌈nai⌉

Ψ (nxi − ki)
) .

Our considered neural networks here are of one hidden layer.

In this article we find Voronovskaya type asymptotic expansions for the above described neural

networks quasi-interpolation normalized operators Gn (f, x), Fn (f, x), where x ∈
(∏N

i=1 [ai, bi]
)

is fixed but arbitrary. For other neural networks related work, see [2], [3], [4], [5], [6] and [7]. For

neural networks in general, see [8], [9] and [10].

Next we follow [1], pp. 284-286.

About Taylor formula -Multivariate Case and Estimates

Let Q be a compact convex subset of RN; N ≥ 2; z := (z1, ..., zN) , x0 := (x01, ..., x0N) ∈ Q.

Let f : Q → R be such that all partial derivatives of order (m − 1) are coordinatewise abso-

lutely continuous functions, m ∈ N. Also f ∈ Cm−1 (Q). That is f ∈ ACm (Q). Each mth order

partial derivative is denoted by fα :=
∂αf
∂xα

, where α := (α1, ..., αN), αi ∈ Z
+, i = 1, ..., N and

|α| :=
∑N

i=1 αi = m. Consider gz (t) := f (x0 + t (z − x0)), t ≥ 0. Then

g(j)z (t) =

⎡

⎣

(

N∑

i=1

(zi − x0i)
∂

∂xi

)j

f

⎤

⎦(x01 + t (z1 − x01) , ..., x0N + t (zN − x0N)) , (6)

for all j = 0, 1, 2, ..., m.

Example 6. Let m = N = 2. Then

gz (t) = f (x01 + t (z1 − x01) , x02 + t (z2 − x02)) , t ∈ R,

and

g′z (t) = (z1 − x01)
∂f

∂x1
(x0 + t (z − x0)) + (z2 − x02)

∂f

∂x2
(x0 + t (z − x0)) . (7)



42 George A. Anastassiou CUBO
16, 2 (2014)

Setting

(∗) = (x01 + t (z1 − x01) , x02 + t (z2 − x02)) = (x0 + t (z − x0)) ,

we get

g′′z (t) = (z1 − x01)
2 ∂f

2

∂x21
(∗) + (z1 − x01) (z2 − x02)

∂f2

∂x2∂x1
(∗) +

(z1 − x01) (z2 − x02)
∂f2

∂x1∂x2
(∗) + (z2 − x02)

2 ∂f
2

∂x22
(∗) . (8)

Similarly, we have the general case of m, N ∈ N for g
(m)
z (t) .

We mention the following multivariate Taylor theorem.

Theorem 7. Under the above assumptions we have

f (z1, ..., zN) = gz (1) =
m−1∑

j=0

g
(j)
z (0)

j!
+ Rm (z, 0) , (9)

where

Rm (z, 0) :=

∫1

0

(∫t1

0

...

(∫tm−1

0

g(m)z (tm) dtm

)

...

)

dt1, (10)

or

Rm (z, 0) =
1

(m − 1) !

∫1

0

(1 − θ)
m−1

g(m)z (θ) dθ. (11)

Notice that gz (0) = f (x0) .

We make

Remark 8. Assume here that

∥fα∥
max
∞,Q,m := max

|α|=m
∥fα∥∞,Q < ∞.

Then
∥

∥

∥
g(m)z

∥

∥

∥

∞,[0,1]
=

∥

∥

∥

∥

∥

[(

N∑

i=1

(zi − x0i)
∂

∂xi

)m

f

]

(x0 + t (z − x0))

∥

∥

∥

∥

∥

∞,[0,1]

≤ (12)

(

N∑

i=1

|zi − x0i|

)m

∥fα∥
max
∞,Q,m ,

that is
∥

∥

∥
g(m)z

∥

∥

∥

∞,[0,1]
≤
(

∥z − x0∥l1
)m

∥fα∥
max
∞,Q,m < ∞. (13)

Hence we get by (11) that

|Rm (z, 0)| ≤

∥

∥

∥
g
(m)
z

∥

∥

∥

∞,[0,1]

m!
< ∞. (14)



CUBO
16, 2 (2014)

Voronovskaya type asymptotic expansions for multivariate . . . 43

And it holds

|Rm (z, 0)| ≤

(

∥z − x0∥l1
)m

m!
∥fα∥

max
∞,Q,m , (15)

∀ z, x0 ∈ Q.

Inequality (15) will be an important tool in proving our main results.

2 Main Results

We present our first main result

Theorem 9. Let 0 < β < 1, x ∈
∏N

i=1 [ai, bi], n ∈ N large enough, f ∈ AC
m
(∏N

i=1 [ai, bi]
)

,

m, N ∈ N. Assume further that ∥fα∥
max
∞,m < ∞. Then

Gn (f, x) − f (x) =

m−1∑

j=1

⎛

⎝

∑

|α|=j

(

fα (x)
∏N

i=1 αi!

)

Gn

(

N∏

i=1

(· − xi)
αi , x

)

⎞

⎠ + o

(

1

nβ(m−ε)

)

, (16)

where 0 < ε ≤ m.

If m = 1, the sum in (16) collapses.

The last (16) implies that

nβ(m−ε)

⎡

⎣Gn (f, x) − f (x) −
m−1∑

j=1

⎛

⎝

∑

|α|=j

(

fα (x)
∏N

i=1 αi!

)

Gn

(

N∏

i=1

(· − xi)
αi , x

)

⎞

⎠

⎤

⎦ (17)

→ 0, as n → ∞, 0 < ε ≤ m.

When m = 1, or fα (x) = 0, for |α| = j, j = 1, ..., m − 1, then we derive that

nβ(m−ε) [Gn (f, x) − f (x)] → 0,

as n → ∞, 0 < ε ≤ m.

Proof. Consider gz (t) := f (x0 + t (z − x0)), t ≥ 0; x0, z ∈
∏N

i=1 [ai, bi]. Then

g(j)z (t) =

⎡

⎣

(

N∑

i=1

(zi − x0i)
∂

∂xi

)j

f

⎤

⎦(x01 + t (z1 − x01) , ..., x0N + t (zN − x0N)) , (18)

for all j = 0, 1, ..., m.

By (9) we have the multivariate Taylor’s formula

f (z1, ..., zN) = gz (1) =
m−1∑

j=0

g
(j)
z (0)

j!
+

1

(m − 1) !

∫1

0

(1 − θ)
m−1

g(m)z (θ) dθ. (19)



44 George A. Anastassiou CUBO
16, 2 (2014)

Notice gz (0) = f (x0). Also for j = 0, 1, ..., m − 1, we have

g(j)z (0) =
∑

α:=(α1,...,αN), αi∈Z
+,

i=1,...,N, |α|:=
∑

N
i=1

αi=j

(

j!
∏N

i=1 αi!

)(

N∏

i=1

(zi − x0i)
αi

)

fα (x0) . (20)

Furthermore

g(m)z (θ) =

∑

α:=(α1,...,αN), αi∈Z
+,

i=1,...,N, |α|:=
∑N

i=1
αi=m

(

m!
∏N

i=1 αi!

)(

N∏

i=1

(zi − x0i)
αi

)

fα (x0 + θ (z − x0)) , (21)

0 ≤ θ ≤ 1.

So we treat f ∈ ACm
(∏N

i=1 [ai, bi]
)

with ∥fα∥
max
∞,m < ∞.

Thus, by (19) we have for k
n
, x ∈

(∏N
i=1 [ai, bi]

)

that

f

(

k1
n
, ...,

kN
n

)

− f (x) =

m−1∑

j=1

∑

α:=(α1,...,αN), αi∈Z
+,

i=1,...,N, |α|:=
∑N

i=1
αi=j

(

1
∏N

i=1 αi!

)(

N∏

i=1

(

ki
n

− xi

)αi
)

fα (x) + R, (22)

where

R := m

∫1

0

(1 − θ)
m−1

∑

α:=(α1,...,αN), αi∈Z
+,

i=1,...,N, |α|:=
∑N

i=1
αi=m

(

1
∏N

i=1 αi!

)

·

(

N∏

i=1

(

ki
n

− xi

)αi
)

fα

(

x + θ

(

k

n
− x

))

dθ. (23)

By (15) we obtain

|R| ≤

(

∥

∥x − k
n

∥

∥

l1

)m

m!
∥fα∥

max
∞,m . (24)

Notice here that
∥

∥

∥

∥

k

n
− x

∥

∥

∥

∥

∞

≤
1

nβ
⇔

∣

∣

∣

∣

ki
n

− xi

∣

∣

∣

∣

≤
1

nβ
, i = 1, ..., N. (25)

So, if
∥

∥

k
n
− x
∥

∥

∞
≤ 1

nβ
we get that

∥

∥x − k
n

∥

∥

l1
≤ N

nβ
, and

|R| ≤
Nm

nmβm!
∥fα∥

max
∞,m . (26)

Also we see that
∥

∥

∥

∥

x −
k

n

∥

∥

∥

∥

l1

=

N∑

i=1

∣

∣

∣

∣

xi −
ki
n

∣

∣

∣

∣

≤
N∑

i=1

(bi − ai) = ∥b − a∥l1 ,



CUBO
16, 2 (2014)

Voronovskaya type asymptotic expansions for multivariate . . . 45

therefore in general it holds

|R| ≤

(

∥b − a∥l1
)m

m!
∥fα∥

max
∞,m . (27)

Call

V (x) :=

⌊nb⌋∑

k=⌈na⌉

Φ (nx − k) .

Hence we have

Un (x) :=

∑⌊nb⌋
k=⌈na⌉

Φ (nx − k) R

V (x)
= (28)

∑⌊nb⌋
⎧
⎪⎨

⎪⎩

k = ⌈na⌉

:
∥

∥

k
n
− x
∥

∥

∞
≤ 1

nβ

Φ (nx − k) R

V (x)
+

∑⌊nb⌋
⎧
⎪⎨

⎪⎩

k = ⌈na⌉

:
∥

∥

k
n
− x
∥

∥

∞
> 1

nβ

Φ (nx − k) R

V (x)
.

Consequently we obtain

|Un (x)| ≤

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

∑⌊nb⌋
⎧
⎪⎨

⎪⎩

k = ⌈na⌉

:
∥

∥

k
n
− x
∥

∥

∞
≤ 1

nβ

Φ (nx − k)

V (x)

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

(

Nm

nmβm!
∥fα∥

max
∞,m

)

+

1

V (x)

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⌊nb⌋∑

⎧
⎪⎨

⎪⎩

k = ⌈na⌉

:
∥

∥

k
n
− x
∥

∥

∞
> 1

nβ

Φ (nx − k)

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

(

∥b − a∥l1
)m

m!
∥fα∥

max
∞,m

(by (v)’, (vi)’)
≤

Nm

nmβm!
∥fα∥

max
∞,m + (5.250312578)

N
(3.1992) e−n

(1−β)

(

∥b − a∥l1
)m

m!
∥fα∥

max
∞,m . (29)

Therefore we have found

|Un (x)| ≤
∥fα∥

max
∞,m

m!

{
Nm

nmβ
+ (5.250312578)

N
(3.1992) e−n

(1−β) (

∥b − a∥l1
)m

}

. (30)

For large enough n ∈ N we get

|Un (x)| ≤

(

2 ∥fα∥
max
∞,m N

m

m!

)(

1

nmβ

)

. (31)

That is

|Un (x)| = O

(

1

nmβ

)

, (32)



46 George A. Anastassiou CUBO
16, 2 (2014)

and

|Un (x)| = o (1) . (33)

And, letting 0 < ε ≤ m, we derive

|Un (x)|
(

1
nβ(m−ε)

) ≤

(

2 ∥fα∥
max
∞,m N

m

m!

)

1

nβε
→ 0, (34)

as n → ∞.

I.e.

|Un (x)| = o

(

1

nβ(m−ε)

)

. (35)

By (22) we observe that
∑⌊nb⌋

k=⌈na⌉
f
(

k
n

)

Φ (nx − k)

V (x)
− f (x) =

m−1∑

j=1

⎛

⎝

∑

|α|=j

(

fα (x)
∏N

i=1 αi!

)

⎞

⎠

(∑⌊nb⌋
k=⌈na⌉

Φ (nx − k)
(∏N

i=1

(

ki
n

− xi
)αi
))

V (x)
+ (36)

∑⌊nb⌋
k=⌈na⌉

Φ (nx − k) R

V (x)
.

The last says

Gn (f, x) − f (x) −
m−1∑

j=1

⎛

⎝

∑

|α|=j

(

fα (x)
∏N

i=1 αi!

)

Gn

(

N∏

i=1

(· − xi)
αi , x

)

⎞

⎠ = Un (x) . (37)

The proof of the theorem is complete.

We present our second main result

Theorem 10. Let 0 < β < 1, x ∈
∏N

i=1 [ai, bi], n ∈ N large enough, f ∈ AC
m
(∏N

i=1 [ai, bi]
)

,

m, N ∈ N. Assume further that ∥fα∥
max
∞,m < ∞. Then

Fn (f, x) − f (x) =

m−1∑

j=1

⎛

⎝

∑

|α|=j

(

fα (x)
∏N

i=1 αi!

)

Fn

(

N∏

i=1

(· − xi)
αi , x

)

⎞

⎠ + o

(

1

nβ(m−ε)

)

, (38)

where 0 < ε ≤ m.

If m = 1, the sum in (38) collapses.

The last (38) implies that

nβ(m−ε)

⎡

⎣Fn (f, x) − f (x) −
m−1∑

j=1

⎛

⎝

∑

|α|=j

(

fα (x)
∏N

i=1 αi!

)

Fn

(

N∏

i=1

(· − xi)
αi , x

)

⎞

⎠

⎤

⎦ (39)



CUBO
16, 2 (2014)

Voronovskaya type asymptotic expansions for multivariate . . . 47

→ 0, as n → ∞, 0 < ε ≤ m.

When m = 1, or fα (x) = 0, for |α| = j, j = 1, ..., m − 1, then we derive that

nβ(m−ε) [Fn (f, x) − f (x)] → 0,

as n → ∞, 0 < ε ≤ m.

Proof. Similar to Theorem 9, using the properties of Θ (x), see (4), (i)∗-(vii)∗ and (5).

Received: December 2011. Revised: May 2012.

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1998.

[9] W. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity,

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