On Multivariate Bernstein Polynomials

Article On Multivariate Bernstein Polynomials

Mama Foupouagnigni 1,2,† and Merlin Mouafo Wouodjié 3,*,†

1 African Institute for Mathematical Sciences, P.O. Box 608, Limbe, Cameroon; [email protected] 2 Higher Teacher Training College, University of Yaounde I, P.O. Box 47, Yaounde, Cameroon 3 Institute of Mathematics, University of Kassel, Heinrich-Plett Str. 40, 34132 Kassel, Germany * Correspondence: [email protected] † These authors contributed equally to this work.

Received: 20 June 2020; Accepted: 27 July 2020; Published: 21 August 2020

Abstract: In this paper, we ﬁrst revisit and bring together as a sort of survey, properties of Bernstein polynomials of one variable. Secondly, we extend the results from one variable to several ones, namely—uniform convergence, uniform convergence of the derivatives, order of convergence, monotonicity, ﬁxed sign for the p-th derivative, and deduction of the upper and lower bounds of Bernstein polynomials from those of the corresponding functions.

Keywords: Bernstein polynomials; functions of several variables; derivatives; uniform convergence; order of convergence; monotonicity; convexity; upper and lower bounds

1. Introduction It is not often easy to ﬁnd exact solutions to most equations governing real-life problems. Even when such solutions can be found, for their practical use, there is a need to ﬁnd their approximation by polynomials, when possible. Weierstrass’s theorem asserts the possibility of uniform approximation by polynomials of continuous functions over a closed interval. An analytic function can be expanded in a uniformly convergent power series and a continuous but non-analytic function can be expanded in a uniformly convergent series of general polynomials, with no possibility of rearranging its terms so as to produce a convergent power series [1]. There are many proofs of the Weierstrass theorem. In this paper, we shall present one called “S. Bernstein’s proof” since it also gives an explicit representation of the polynomials for uniform approximation. An example of Bernstein polynomials of one variable is (see Reference [1]):

n ! k n k n−k Bn( f ; x) = ∑ f x (1 − x) . (1) k=0 n k

The polynomials Bn( f ;.) are known to approximate uniformly their corresponding functions f in th [0, 1], provided that f is continuous on the interval [0, 1]. Moreover, the n derivative of Bn( f ;.) converges uniformly to f (n), provided that f ∈ Cn[0, 1] ( f ∈ Cn[0, 1] means f is continuous on the interval [0, 1] and all its derivatives f (i), i = 1, ··· , n, exist and are also continuous on [0, 1]). According to the literature, there are several ways (with different notations) to deﬁne explicitly the Bernstein polynomials of one variable, which lead to the same results as in Reference [1]—uniform convergence as well as uniform convergence of the derivatives (see References [2–7]).

Mathematics 2020, 8, 1397; doi:10.3390/math8091397 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 1397 2 of 34

Bernstein polynomials have been used for approximation of functions in many areas of Mathematics and other ﬁelds such as smoothing in Statistics, and constructing and generalizing Bézier curves (see References [8,9]). We also have some applications:

1. in the quadrature rule on equally spaced knots (see Reference [10]); 2. to numerical differentiation and quadrature and perform numerical experiments showing the effectiveness of the considered technique (see Reference [11]): they perform some numerical tests for approximation by polynomials of a given function, numerical differentiation and numerical quadrature.

“In contrast to other modes of approximation, in particular to Tchebyschev or best uniform approximation, the Bernstein polynomials yield smooth approximants; but there is a price that must be paid for these beautiful approximations properties: the convergence of the Bernstein polynomials is very slow.” P. J. Davis [1].

Several authors like A. Jafarian, S. Measoomy Nia, Alireza K. Golmankhaneh and D. Baleanu [12], D. H. Tucker [13], E. H. Doha, A. H. Bhrawy, M. A. Saker [14], George M. Phillips [15], Henryk Gzyl and José Luis Palacios [16], M. H. T Alshbool, I. Hashim [17], P. J. Davis [1], Richard V. Kadison [18] and others, have worked on Bernstein polynomials for functions with one variable, and have derived some beautiful properties whose most important ones are: the uniform convergence, the uniform convergence of derivatives, the fixed sign for the pth derivative, the deduction of upper and lower bounds of Bernstein polynomials from those of the corresponding functions. The main problem here is to extend these properties of Bernstein polynomials of one variable by generalizing them for functions of several real variables. There are more than five ways with different notations to define Bernstein polynomials in the multi-dimensional case for a function f defined on m m [0, 1] , where m is the dimension of the space. Some used notations are x = (x1, ··· , xm) ∈ [0, 1] , m k k1 km k = (k1, ··· , km) ∈ N , |x| = x1 + ··· + xm, x = x ··· xm , |k| = k1 + ··· + km, k! = k1! ··· km! and 1 k k1 km = , ··· , , n ∈ N \{0}. These notations are used in the following definitions: n n n

(a) the Bernstein polynomials on (m − 1)-dimensional simplex in cube [0, 1]m (see References [19–22]):

k n! bn[ f ](x) := f xk, where x + ··· + x = 1; (2) ∑ n k 1 m |k|=n ! (b) the Bernstein polynomials on a m-dimensional simplex and very useful in stochastic analysis (see References [7,23–26]):

k n! B ( f , x) := f xk(1 − |x|)n−|k|, where x + ··· + x ≤ 1; (3) n,m ∑ n k (n − |k|) 1 m |k|≤n ! !

α (|α|)! B ( f , x) := f xα(1 − |x|)αm+1 , (4) n ∑ n α! |α|=n

m+1 where α = (α1, ··· , αm+1) ∈ N and α = (α1, ··· , αm); Mathematics 2020, 8, 1397 3 of 34

(d) the Bernstein polynomials on (m − 1)-dimensional simplex in cube [0, 1]m used in the ﬁeld of mathematical biology with quadrartic mappings of “standard” (m − 1)-dimensional simplex to self (see Reference [21] ): ! n n k m n ki n−ki Bn( f )(x1, ··· , xm) = ∑ ··· ∑ f ∏ xi (1 − xi) ; (5) n ki k1=0 km=0 i=1

(e) the Bernstein polynomials related to the convergence in the uniform and Lp-norms for polynomials with integral coefﬁcients (see Reference [28] for two variables, References [1,29] for the generalized one):

n1 nm m ! k km n k ( ··· ) = ··· 1 ··· i i ( − )ni−ki Bn1,··· ,nm f ; x1, , xm ∑ ∑ f , , ∏ xi 1 xi , (6) n1 nm ki k1=0 km=0 i=1

where n1, ··· , nm ∈ N.

Using the above mentioned deﬁnitions, many authors did some investigations aimed at generalizing some properties of univariate Bernstein polynomials to Bernstein polynomials with two or more variables. Let us take for example the forms given by (2), (3) and (6).

1. For the multivariate Bernstein polynomials given by Equation (2): they have proven the pointwise and uniform convergence (see References [21,22,30]). They have also presented two variations which are preserved and do coincide with classical convexity in the univariate case (see Reference [31]). Moreover, they showed the monotonicity of successive Bernstein polynomials for classically convex functions (see References [31,32]). 2. For the multivariate Bernstein polynomials given by Equation (3): they have proven two identities for these polynomials on simplex, which are considered on a pointwise (see References [23–25,25]). They also showed that these polynomials converge to their corresponding functions f along with partial derivatives provided that the latter derivatives exist and are continuous (see References [7,26,33]). 3. For the Bernstein polynomials given by Equation (6): they established (but only for two variables) the pointwise and uniform convergence; the same result was established for the partial derivatives of the corresponding function (see References [28,29,34–36]).

According to the last deﬁnition (see Equation (6)) of the generalized Bernstein polynomial

Bn1,··· ,nm ( f ; x1, ··· , xm), there is just a few new results (pointwise and uniform convergence) given by some authors like Clemens Heitzinger (see Reference [29]). That is why we focus, in this paper, on the multivariate Bernstein polynomials given by the form defined by Equation (6). Many authors have made tremendous efforts to prove some properties for multivariate Bernstein polynomials on simplex (see Reference [23]), which are considered on a pointwise convergence (good approximations of Bernstein polynomials for every continuous function on simplex and the higher dimensional q-analogue of Bernstein polynomials on simplex). Some have tried to study the approximation by combination of Bernstein polynomials or operators (the rate of convergence of derivatives and the smoothness of the derivatives of functions) in order to approximate, under definite conditions, the function more closely than the Bernstein polynomials (see References [3,6,37–41]). Moreover, they defined and also tried to study the q-Bernstein polynomials (see References [2,4,42–45]), the generalized Bernstein polynomials (see References [10,26]) and the modified Bernstein polynomials (see Reference [25]). To achieve our aim which is to generalize these beautiful properties for functions with one variable to functions with several variables, we will use the following properties of univariate Bernstein polynomials (with some references where they can be found): Mathematics 2020, 8, 1397 4 of 34

- Property 1 (see Reference [1]): if f is bounded on [0, 1] and continuous at x0 ∈ [0, 1], then

lim Bn( f ; x0) = f (x0); (7) n→∞

- Property 2 (see References [1,3,5–7,10,11,39,46–48]): if f ∈ C[0, 1], the Bernstein polynomial Bn( f ; ·) converges uniformly to f ; - Property 3 (see References [1,7,11]): if f ∈ C p[0, 1] ( f is continuous on the interval [0, 1] and all (i) p its derivatives f , i = 1, ··· , p, exist and are also continuous on [0, 1]), then Bn( f ; ·) converges uniformly to f (p); - Property 4 (see References [1,21]): if f ∈ C p[0, 1], for some 0 ≤ p ≤ n and A ≤ f (p)(x) ≤ B(A and B are constants), 0 ≤ x ≤ 1, then

p n (p) A ≤ B ( f ; x) ≤ B , 0 ≤ x ≤ 1; (8) n(n − 1) ··· (n − p + 1) n

(the consequence is: if f is decreasing (resp. nondecreasing) on [0, 1], then Bn( f ; ·) is decreasing (resp. nondecreasing) on [0, 1].) - Property 5 (see References [1,31]): if f is convex in [0, 1], then Bn( f ; ·) is convex there and for n = 2, 3, ··· Bn−1( f ; x) ≥ Bn( f ; x) , 0 < x < 1; (9) 00 - Property 6 (see References [1,11,39]): if f (x) is bounded on [0, 1] and x0 is a point of [0, 1] at which f exists and is continuous, then

1 00 lim n [Bn( f ; x0) − f (x0)] = x0(1 − x0) f (x0). (10) n→∞ 2

The following intermediate results for the case of univariate Bernstein polynomials are needed to prove all those properties:

(a) see References [1,15]: for x ∈ [0, 1] we have Bn(1; x) = 1; (b) see Reference [1]: for x ∈ [0, 1] and α ∈ R we have

n Bn(exp(αx); x) = (x + exp(α/n) + (1 − x)) ; (11)

n ! n k k Bn( f ; x) = ∑ 4 fn(0)x , (12) k=0 k

1 where f (x) = f ( x ) and 4 is the forward difference operator with step size h = deﬁned as follows n n n (see Reference [15]):  4 f (x ) := f (x + ) − f (x ),  i i 1 i k+1 k k k 4 f (xi) := 4 4 f (xi) = 4 f (xi+1) − 4 f (xi) fork ≥ 1, (13)  1 0 4 f (xi) := 4 f (xi) and 4 f (xi) := f (xi),

where 4k is called the kth difference; Mathematics 2020, 8, 1397 5 of 34

(d) See References [1,15,21,22]: for a given δ > 0, n ∈ N \{0} and x ∈ [0, 1], we have ! n 1 k( − )n−k ≤ ∑ x 1 x 2 ; (14) k 4nδ k

−x ≥δ n

(e) see References [1,15]: for any integer k ≥ 0, the k − th derivative of Bn+k( f ; x) may be expressed in terms of k-th differences of f as

n ! (k) (n + k)! r n B ( f ; x) = 4k f xr(1 − x)n−r, (15) n+k ∑ + n! r=0 n k r

1 for all n ≥ 0, where 4 is applied with step size h = ; n + k (f) see (Reference [1] Lemma 6.3.5): there is a constant C independent of a given positive integer n such that for all x in [0, 1] ! n C xk( − x)n−k ≤ ∑ 1 3/2 . (16) k n k −1/4 −x ≥n n

With all those properties, we will prove the following for our generalized Bernstein polynomials:

m m - Property 1: if f is bounded on [0, 1] and continuous on (a1, ··· , am) ∈ [0, 1] , then the Bernstein

polynomial Bn1,··· ,nm ( f ; a1, ··· , am) converges to f (a1, ··· , am), as ni → ∞ for all i; m - Property 2: if f ∈ C[0, 1] , then the Bernstein polynomial Bn ,··· ,nm ( f ; ·) converges uniformly to f ; p ,··· ,p 1 ∈ C p1···pm [ ]m 1 m ( ·) (p1),··· ,(pm) ∈ - Property 3: if f 0, 1 , then Bn1,··· ,nm f ; converges uniformly to f , where f ∂pi f p ,··· ,pm m m m C 1 [0, 1] means f is continuous on [0, 1] and is continuous on [0, 1] , 0 ≤ pi ≤ ni, ∂xpi i = 1, ··· , m; p ···pm m (p )···(pm) - Property 4: if f ∈ C 1 [0, 1] , 0 ≤ pi ≤ ni, i = 1, ··· , m, and A ≤ f 1 (x1, ··· , xm) ≤ B, m (x1, ··· , xm) ∈ [0, 1] , then

m pi n p ,··· ,p A ≤ i B 1 m ( f ; x , ··· , x ) ≤ B (17) ∏ ( − ) ··· ( − + ) n1,··· ,nm 1 m i=1 ni ni 1 ni pi 1

m for all (x1, ··· , xm) ∈ [0, 1] . As a consequence, if f (x1, ··· , xm) is decreasing (resp. nondecreasing)

on [0, 1] with respect to the variable xi, i ∈ 1, ··· , m, then Bn1,··· ,nm ( f ; x1, ··· , xm) is decreasing (resp. nondecreasing) on [0, 1] with respect to the variable xi there. m - Property 5: if f (x1, ··· , xm) is convex in [0, 1] , then, for ni = 2, 3, ··· with i = 1, ··· , m, we have

( ··· ) ≥ ( ··· ) Bn1−1,··· ,nm−1 f ; x1, , xm Bn1,··· ,nm f ; x1, , xm (18)

m for all (x1, ··· , xm) ∈ [0, 1] ; m m - Property 6: if f (x1, ··· , xm) is bounded in [0, 1] and let (a1, ··· , am) be a point of [0, 1] at which (2)···(2) f (x1, ··· , xm) exists and is continuous, then, for n1 = n2 = ··· = nm = n

1 m ∂2 f lim n [Bn,··· ,n( f ; a1, ··· , am) − f (a1, ··· , am)] = ai(1 − ai) (a1, ··· , am). (19) n→∞ ∑ 2 2 i=1 ∂xi Mathematics 2020, 8, 1397 6 of 34

The following intermediate results for the multivariate case, that we will also prove, are needed in order to prove all these cited properties:

(a) for m ≥ 1, i ∈ {1, ··· , m}, ji and ni integers, ni ≥ 1, and xi ∈ [0, 1]

 ! j j j j n  1 m 1 m m i ki ni−ki  x ··· xm = ∑ ··· ∑ ∏ = σj ,k x (1 − xi) ,  1 k1=0 km=0 i 1 i i k i  i Bn1,··· ,nm (1; x1, ··· , xm) = 1, (20)  m ! m  j j  B x i ; x , ··· , x = B x i ; x ,  n1,··· ,nm ∏ i 1 m ∏ ni i i i=1 i=1

where σji,ki are real numbers; (b) for i ∈ {1, ··· , m}, xi ∈ [0, 1] and αi ∈ R

m ! ! m m ··· = ( ( ) ) = [ ( ) + ( − )]ni Bn1,··· ,nm exp ∑ αi xi ; x1, , xm ∏ Bni exp αi xi ; xi ∏ xi exp αi/ni 1 xi , (21) i=1 i=1 i=1     n1 nm m ji m ni ji ni k ni k i ki ni −ki i ki ni −ki ∑ ··· ∑ ∏   − xi xi (1 − xi) = ∏ ∑   − xi xi (1 − xi) ; (22) ni ni k1=0 km =0 i=1 ki i=1 ki =0 ki

  n n m 1 m ni ( ··· ) = ··· 4t1 · · · 4tm ( ··· ) ti Bn1,··· ,nm f ; x1, , xm ∑ ∑ 1 m fn1,··· ,nm 0, , 0 ∏   xi , (23) t1=0 tm =0 i=1 ti

t t ( ··· ) = 1 ··· m 4ti ∈ { ··· } th where fn1,··· ,nm t1, , tm f , , and i , i 1, , m , is the ti difference(s) related n1 nm th 1 to the i variable xi, and applied with step size hi = . ni m (d) for a given δi > 0, i = 1, ··· , m and (x1, ··· , xm) ∈ [0, 1] , we have ! m n 1 i ki ni−ki ∑ ··· ∑ ∏ xi (1 − xi) ≤ m , (24) ki k k i=1 m 2 1 m 4 ∏niδi −x1 ≥δ1 −xm ≥δm n1 nm i=1

ki summing over those values of ki = 0, ··· , ni, i = 1 ··· m, for which − xi ≥ δi; ni th (e) for p1, ··· , pm ≥ 0 integers with 0 ≤ pi ≤ ni, i = 1, 2, ··· , m, the (p1, ··· , pm) derivative of th Bn1+p1,··· ,nm+pm ( f ; x1, ··· , xm) may be expressed in terms of (p1, ··· , pm) differences of f as

p ,··· ,p B 1 m ( f ; x , ··· , x ) n1+p1,··· ,nm +pm 1 m   m ! n1 nm m (n + p )! p p t tm ni t = i i ··· 4 1 · · · 4 m 1 ··· i ( − )ni −ti ∏ ∑ ∑ 1 m f , , ∏   xi 1 xi , (25) ni! n1 + p1 nm + pm i=1 t1=0 tm =0 i=1 ti

where the ∆ is acting with the mesh h = 1 ; k k nk+pk (f) there is a constant C independent of given positive integers ni such that for all (x1, ··· , xm) ∈ [0, 1]m, i = 1, ··· , m, Mathematics 2020, 8, 1397 7 of 34

! m n C i ki ni−ki ∑ ··· ∑ ∏ xi (1 − xi) ≤ m . (26) i= ki 3/2 k1 km 1 n − ≥ −1/4 − ≥ −1/4 ∏ i x1 n1 xm nm n1 nm i=1

2. Deﬁnition and Properties We shall now derive the properties of the m-dimensional generalization of the Bernstein polynomials. First of all, we give m-dimensional generalization of the Bernstein polynomials.

Deﬁnition 1 (Generalization). (P. J. Davis [1], page 122) m th Let f (x1, ··· , xm) be deﬁned on [0, 1] and ni ≥ 1, i = 1, ··· , m, integers. The (n1, ··· , nm) Bernstein polynomial for f (x1, ··· , xm) is given by

n1 nm k k m B ( f x ··· x ) = ··· f 1 ··· m b (x ) n1,··· ,nm ; 1, , m ∑ ∑ , , ∏ ki,ni i , (27) n1 nm k1=0 km=0 i=1 ! ni k ( ) = i ( − )ni−ki = ··· where bki,ni xi xi 1 xi , i 1, , m. ki

Notice that ( B ··· ( f ; 0, ··· , 0) = f (0, ··· , 0); n1, ,nm (28) Bn1,··· ,nm ( f ; 1, ··· , 1) = f (1, ··· , 1). m We call Bn1,··· ,nm the Bernstein operator; it maps a function f , deﬁned on [0, 1] , to Bn1,··· ,nm f , where the function Bn1,··· ,nm f evaluated at (x1, ··· , xm) is denoted by Bn1,··· ,nm ( f ; x1, ··· , xm). This operator is linear, since it follows from Equation (27) that

Bn1,··· ,nm (λ f + µg) = λBn1,··· ,nm f + µBn1,··· ,nm g, (29) for all functions f and g deﬁned on [0, 1]m, and all real λ and µ. Using the results obtained with one variable, it is easy to obtain some pertinent properties for the generalization of Bernstein polynomials.

Lemma 1.

1 Let m ≥ 1, j1, ··· , jm be integers. Then,

j1 jm m xj1 ··· xjm = ··· b (x ) 1 m ∑ ∑ ∏ σji,ki ki,ji i , (30) k1=0 km=0 i=1

= ··· ≤ ≤ where σji,ki , ki 0, , ji, 1 i m, are real numbers. 2 For i ∈ {1, ··· , m}, let ni ≥ 1, ji be integer and xi ∈ [0, 1]. Then,

 ( ··· ) =  Bn1,··· ,nm 1; x1, , xm 1,  m ! m j j (31) B x i ; x , ··· , x = B x i ; x ,  n1,··· ,nm ∏ i 1 m ∏ ni i i i=1 i=1

(a similar result for one variable can be found in References [1,15] for the ﬁrst equation of the system (31)). Mathematics 2020, 8, 1397 8 of 34

3 For i ∈ {1, ··· , m}, let xi ∈ [0, 1] and αi ∈ R. Then,

m ! ! m m ··· = ( ( ) ) = [ ( ) + ( − )]ni Bn1,··· ,nm exp ∑ αi xi ; x1, , xm ∏ Bni exp αi xi ; xi ∏ xi exp αi/ni 1 xi (32) i=1 i=1 i=1

(a similar result for one variable can be found in Reference [1]). 4 For xi ∈ [0, 1], 1 ≤ i ≤ m, we have

    n1 nm m ji m ni ji ni k ni k i ki ni −ki i ki ni −ki ∑ ··· ∑ ∏   − xi xi (1 − xi) = ∏ ∑   − xi xi (1 − xi) . (33) ni ni k1=0 km =0 i=1 ki i=1 ki =0 ki

Proof of Lemma 1. See References [1,15] for the proofs.

3. Approximation by Bernstein Polynomials with Several Variables We express in this theorem our multivariate Bernstein polynomials in terms of differences. A similar result exists for one variable (see References [1,15]).

m Theorem 1. Let f (x1, ··· , xm) be deﬁned on [0, 1] . We have for positive integers ni, 1 ≤ i ≤ m,

n1 nm m ! t t ni ti B ··· ( f ; x , ··· , x ) = ··· 4 1 · · · 4 m f ··· (0, ··· , 0) x , (34) n1, ,nm 1 m ∑ ∑ 1 m n1, ,nm ∏ t i t1=0 tm=0 i=1 i

t t ( ··· ) = 1 ··· m 4ti ∈ { ··· } th where fn1,··· ,nm t1, , tm f , , and i , i 1, , m , is the ti difference(s) n1 nm th 1 (see References [1,15]) related to the i variable xi, and applied with step size hi = . ni

Proof of Theorem 1. We will sometimes use the square brackets to draw attention on computations of certain terms.

Bn1,··· ,nm ( f ; x1, ··· , xm) ! n1 nm k k m n 1 m i ki ni−ki = ∑ ··· ∑ f , ··· , ∏ xi (1 − xi) (35) n1 nm ki k1=0 km=0 i=1

n2 nm " n1 ! # m ! k k km n k n k = ··· 1 2 ··· 1 1 ( − )n1−k1 i i ( − )ni−ki ∑ ∑ ∑ f , , , x1 1 x1 ∏ xi 1 xi . (36) n1 n2 nm k1 ki k2=0 km=0 k1=0 i=2 k1 km k1 We take the ﬁrst variable of f and ﬁx the others. Hence f , ··· , is seen as a function g1 , n1 nm n1 and we apply the following result from P. J. Davis [1], about Bernstein polynomials with one variable:

n ! t n t Bn(g; x) = ∑ 4 gn(0) x , (37) t=0 t

t where n ∈ N and gn (t) = g . Then we get n Mathematics 2020, 8, 1397 9 of 34

Bn1,··· ,nm ( f ; x1, ··· , xm) !! ! n2 nm n1 n m n t1 t1 1 i ki ni−ki = ··· 4 g1(0)x x (1 − xi) , ∑ ∑ ∑ 1 1 t ∏ k i k2=0 km=0 t1=0 1 i=2 i

n2 nm n1 !! m ! k km n n k = ··· 4t1 2 ··· t1 1 i i ( − )ni−ki ∑ ∑ ∑ 1 f 0, , , x1 ∏ xi 1 xi , n2 nm t k k2=0 km=0 t1=0 1 i=2 i

n3 nm n1 " n2 ! # !! k k km n n = ··· 4t1 2 3 ··· 2 k2 ( − )n2−k2 t1 1 ∑ ∑ ∑ ∑ 1 f 0, , , , x2 1 x2 x1 n2 n3 nm k t k3=0 km=0 t1=0 k2=0 2 1 ! m n i ki ni−ki × ∏ xi (1 − xi) . (38) i=3 ki t1 k2 km We repeat the action with the second variable of the function 41 f 0, , ··· , , to get n2 nm

Bn1,··· ,nm ( f ; x1, ··· , xm)

n3 nm n1 n2 2 ! ! m ! k km n t n k = ··· 4t1 4t2 3 ··· i i i i ( − )ni−ki ∑ ∑ ∑ ∑ 1 2 f 0, 0, , , ∏ xi ∏ xi 1 xi . (39) n3 nm ti ki k3=0 km=0 t1=0 t2=0 i=1 i=3

Then Bn1,··· ,nm ( f ; x1, ··· , xm)

        n4 nm n1 n2 n3 2 k km n3 ni t = ··· 4t1 4t2 3 ··· k3 ( − )n3−k3 i ∑ ∑  ∑ ∑  ∑ 1 2 f 0, 0, , ,   x3 1 x3  ∏   xi  n3 nm k4=0 km =0 t1=0 t2=0 k3=0 k3 i=1 ti   m ni ki ni −ki × ∏   xi (1 − xi) . (40) i=4 ki t1 t2 k3 km Similarly, with the third variable of 41 42 f 0, 0, , ··· , , we get n3 nm

Bn1,··· ,nm ( f ; x1, ··· , xm)       n4 nm n1 n2 n3 3 m t t t k km ni t ni k = ··· 4 1 4 2 4 3 4 ··· i i ( − )ni −ki ∑ ∑  ∑ ∑ ∑ 1 2 3 f 0, 0, 0, , , ∏   xi  ∏   xi 1 xi . (41) n4 nm k4=0 km=0 t1=0 t2=0 t3=0 i=1 ti i=4 ki

By repeating, successively, the actions with the 4th, 5th, ··· , mth variables but with the functions t1 t2 t3 k4 km t1 t2 t3 t4 k5 km t1 t2 tm−1 41 42 43 f 0, 0, 0, , ··· , , 41 42 43 44 f 0, 0, 0, 0, , ··· , , ··· , 41 42 · · · 4m−1 n4 nm n5 nm k f 0, 0, 0, ··· , 0, m respectively, we obtain the result. nm In order to show the pointwise and uniform convergence of our multivariate Bernstein polynomials, let us state and prove the following lemma. A similar result exists for one variable (see References [1,15,21,22]).

m Lemma 2. For a given δi > 0, i = 1, ··· , m and (x1, ··· , xm) ∈ [0, 1] , we have ! m n 1 i ki ni−ki ∑ ··· ∑ ∏ xi (1 − xi) ≤ m , (42) ki k k i=1 m 2 1 m 4 ∏niδi −x1 ≥δ1 −xm ≥δm n1 nm i=1 Mathematics 2020, 8, 1397 10 of 34

ki summing over those values of ki = 0, ··· , ni for which − xi ≥ δi. ni

Proof of Lemma 2. We will sometimes use the square brackets to draw attention to computations of certain terms. ! m n i ki ni−ki ∑ ··· ∑ ∏ xi (1 − xi) ki k k i=1 1 m −x1 ≥δ1 −xm ≥δm n1 nm  

 !  m !  n k  n k = ···  1 1 ( − )n1−k1  i i ( − )ni−ki ∑ ∑  ∑ x1 1 x1  ∏ xi 1 xi k1 ki k k  k  i=2 2 m  1  −x2 ≥δ2 −xm ≥δm −x1 ≥δ1 n2 nm n1    !   n k  =  1 1 ( − )n1−k1  ···  ∑ x1 1 x1  ∑ k1  k  k  1  3 −x1 ≥δ1 −x3 ≥δ3 n1 n3

 

 !  m !  n  n k  2 k2 ( − )n2−k2  i i ( − )ni−ki ∑  ∑ x2 1 x2  ∏ xi 1 xi k2 ki k  k  i=3 m  2  −xm ≥δm −x2 ≥δ2 nm n2  

 2 !  m !  n k  n k =  i i ( − )ni−ki  ··· i i ( − )ni−ki ∏ ∑ xi 1 xi  ∑ ∑ ∏ xi 1 xi . (43) ki ki i=1 k  k k i=3  i  3 m −xi ≥δi −x3 ≥δ3 −xm ≥δm ni n3 nm

By repeating, successively, the actions with k3, ··· , km, we obtain ! ! m n m n i ki ni−ki i ki ni−ki ∑ ··· ∑ ∏ xi (1 − xi) = ∏ ∑ xi (1 − xi) . (44) ki ki k k i=1 i=1 k 1 m i −x1 ≥δ1 −xm ≥δm −xi ≥δi n1 nm ni

Let us use the following result which can be found in References [1,21,22]: ! n 1 k( − )n−k ≤ ∑ x 1 x 2 , (45) k 4nδ k

−x ≥δ n where n, k ∈ N, n 6= 0, k ≤ n, δ > 0 and x ∈ [0, 1]. We then get Mathematics 2020, 8, 1397 11 of 34

m ! m ! n k 1 1 ··· i i ( − )ni−ki ≤ ≤ ∑ ∑ ∏ xi 1 xi ∏ 2 m . (46) ki 4niδ k k i=1 i=1 i m 2 1 m 4 ∏ niδi −x1 ≥δ1 −xm ≥δm n1 nm i=1

Remark 1. Lemma 3 is still valid if the summation is not over all the ki, 1 ≤ i ≤ m but over some of them. In this case, the upper bound will contain only terms involved in the summation.

The following theorem states for the pointwise and uniform convergence of our Bernstein polynomials with several variables. It should be mentionned that similar work has been done but only for two variables (see (Theorem 1.1 in Reference [26]), [30], (Theorems 3, 5 and 8 in [21]), (Theorem 6.7 in [33]), (Lemma 1 in [28]), (Theorem in [32]) and (Theorems 1 and 3 in [7]).

Theorem 2. Let f (x1, ··· , xm) be bounded in the m-dimensional cube 0 ≤ xi ≤ 1, i = 1, ··· , m.

Then Bn1,··· ,nm ( f ; x1, ··· , xm) converges towards f (x1, ··· , xm) at any point of continuity of this function, m m as ni → ∞ for all i. If f ∈ C[0, 1] , the limit holds uniformly in [0, 1] .

m Proof of Theorem 2. The function f (x1, ··· , xm) is assumed bounded in [0, 1] . Hence for some m M > 0, | f (x1, ··· , xm)| ≤ M, ∀(x1, ··· , xm) ∈ [0, 1] , and therefore for any two points (α1, ··· , αm), m (β1, ··· , βm) ∈ [0, 1] | f (α1, ··· , αm) − f (β1, ··· , βm)| ≤ 2M. (47) m Let (x1, ··· , xm) ∈ [0, 1] be a point of continuity of f . Given ε > 0, we can find δ1, ··· , δm depending ε on (x , ··· , x ) and ε such that | f (x , ··· , x ) − f (y , ··· , y )| < , whenever |x − y | < δ , i = 1, ··· , m. 1 m 1 m 1 m 2 i i i Since Bn1,··· ,nm (1; x1, ··· , xm) = 1, we have

f (x1, ··· , xm) = f (x1, ··· , xm)Bn1,··· ,nm (1; x1, ··· , xm) ! n1 nm m n i ki ni−ki = f (x1, ··· , xm) ∑ ··· ∑ ∏ xi (1 − xi) ki k1=0 km=0 i=1 ! n1 nm m n i ki ni−ki = ∑ ··· ∑ f (x1, ··· , xm) ∏ xi (1 − xi) . (48) ki k1=0 km=0 i=1

m Let us consider the set E = ∏i=1{0, ··· , ni}, and for j = 1, ··· , m, we deﬁne the sets ( ) k j Ωj = kj ∈ {0, ··· , nj} : − xj < δj and F = E \ (Ω1 × · · · × Ωm) (49) nj n o Sm m [αik] m Then, F = k=1 Fk, with Fk = ∏i=1 Ωik ∈ F : αik ∈ {0, 1}, ∑i=1 αik = k , where ( [αik] Ωi if αik = 0 c ki Ωik = c and Ωi = ki ∈ {0, ··· , ni} : − xi ≥ δi . Ωi if αik = 1 ni = m [αik] ∈ = ··· = { ∈ { ··· } = } For Ak ∏i=1 Ωik Fk, k 1, , m, let us deﬁne also IAk i 1, , m : αik 1 (that = ≥ means card IAk k 1). We have Mathematics 2020, 8, 1397 12 of 34

| f (x1, ··· , xm) − Bn1,··· ,nm ( f ; x1, ··· , xm)|

n1 nm m ! n k = ··· ( ··· ) i i ( − )ni−ki ∑ ∑ f x1, , xm ∏ xi 1 xi ki k1=0 km=0 i=1

n1 nm m ! k km n k − ··· 1 ··· i i ( − )ni−ki ∑ ∑ f , , ∏ xi 1 xi , n1 nm ki k1=0 km=0 i=1

n1 nm m ! k km n k = ··· ( ··· ) − 1 ··· i i ( − )ni−ki ∑ ∑ f x1, , xm f , , ∏ xi 1 xi , n1 nm ki k1=0 km=0 i=1 ! n1 nm k k m n 1 m i ki ni−ki ≤ ∑ ··· ∑ f (x1, ··· , xm) − f , ··· , ∏ xi (1 − xi) , n1 nm ki k1=0 km=0 i=1 ! k k m n 1 m i ki ni−ki ≤ ∑ ··· ∑ f (x1, ··· , xm) − f , ··· , ∏ xi (1 − xi) n1 nm ki Ω1 Ωm i=1 ! k k m n 1 m i ki ni−ki + ∑ f (x1, ··· , xm) − f , ··· , ∏ xi (1 − xi) . (50) F n1 nm i=1 ki

Using the fact that f is continuous and bounded, we get

f (x1, ··· , xm) − Bn1,··· ,nm ( f ; x1, ··· , xm)     ε m n m n i ki ni −ki i ki ni −ki ≤ ···   x (1 − xi) + 2M   x (1 − xi) , 2 ∑ ∑ ∏ i ∑ ∏ i Ω1 Ωm i=1 ki F i=1 ki     ! ε n1 nm m n m m n m i ki ni −ki i ki ni −ki [ ≤ ···   x (1 − xi) + 2M   x (1 − xi) , since F = F 2 ∑ ∑ ∏ i ∑ ∑ ∏ i l k1=0 km=0 i=1 ki l=1 Al ∈Fl i=1 ki l=1   ε m m n i ki ni −ki ≤ Bn ,··· ,n (1; x , ··· , xm) + 2M   x (1 − xi) , 2 1 m 1 ∑ ∑ ∏ i l=1 Al ∈Fl i=1 ki   ε m m n i ki ni −ki ≤ + 2M   x (1 − xi) (using Equation (31)). (51) 2 ∑ ∑ ∏ i l=1 Al ∈Fl i=1 ki

For l ∈ {1, ··· , m} and Al ∈ Fl,

      m m m ni ni ni ki ni −ki ki ni −ki ki ni −ki ∑ ∏   xi (1 − xi) = ∑ ··· ∑ ∏   xi (1 − xi) = ∏ ∑   xi (1 − xi) , (52) Al i=1 ki [α1l ] [αml ] i=1 ki i=1 [αil ] ki Ω1l Ωml Ωil using a similar proof as in Lemma 1. For i ∈ {1, ··· , m},

= 6∈ - if αil 0, this means i IAl then, ! ! ! n n ni n i ki ni−ki i ki ni−ki i ki ni−ki ∑ xi (1 − xi) = ∑ xi (1 − xi) ≤ ∑ xi (1 − xi) = 1 (53) ki ki ki [αil ] Ωi ki=0 Ωil = ∈ - if αil 1 that means i IAl then ! ! n k n k 1 i i ( − )ni−ki = i i ( − )ni−ki ≤ ∑ xi 1 xi ∑ xi 1 xi 2 (54) [α ] ki c ki 4niδ il Ωi i Ωil Mathematics 2020, 8, 1397 13 of 34

(using Equation (45)).

Hence, for i, l ∈ {1, ··· , m},  ! 1 if i 6∈ IA , n k  l i x i (1 − x )ni−ki ≤ 1 (55) ∑ i i if i ∈ I , [ ] ki 2 Al αil  4n δ Ωil i i and this implies

m ! n k 1 i i ( − )ni−ki ≤ ∑ ∏ xi 1 xi ∏ 2 . (56) A i=1 ki i∈I 4niδi l Al

Therefore,

| f (x1, ··· , xm) − Bn1,··· ,nm ( f ; x1, ··· , xm)| ε m 1 ε m 2M ≤ + 2M = + , 2 ∑ ∑ ∏ 2 2 ∑ ∑ 2 l=1 A ∈F i∈I 4niδi l=1 A ∈F ∏i∈IA 4niδi l l Al l l l ε m 2M ≤ + (because card I = l), ∑ ∑ l 2 Al 2 ∈ n ∏i∈I 4δ l=1 Al Fl Al i ε 1 m 2M ≤ + (with n = min (ni : i = 1, ··· , m)). (57) 2 n ∑ ∑ 4δ2 l=1 Al ∈Fl ∏i∈Il i

1 m 2M ε For n , ··· , nm sufﬁciently large (that means n sufﬁciently large), we have ∑ ∑ < . 1 n l=1 Al ∈Fl 2 2 ∏i∈Il δi Therefore, ε ε | f (x , ··· , x ) − B ··· ( f ; x , ··· , x )| ≤ + = ε. (58) 1 m n1, ,nm 1 m 2 2 Here, it should be noticed that the minimum integer n˜, such that

| f (x1, ··· , xm) − Bn1,··· ,nm ( f ; x1, ··· , xm)| ≤ ε ∀n1, ··· , nm ≥ n ≥ n˜, (59) depends on δ1, ··· , δm which themselves depend on x1, ··· , xm and ε. That is why we only have a pointwise convergence. Suppose now that f ∈ C[0, 1]m, then f is uniformly continuous on [0, 1]m that is, given ε > 0, we can ﬁnd δ1, ··· , δm depending just on ε, but not on any element in [0, 1], such that ε | f (x , ··· , x ) − f (y , ··· , y )| < , for all (x , ··· , x ), (y , ··· , y ) ∈ [0, 1]m satisfying |x − y | < δ , 1 m 1 m 2 1 m 1 m i i i i = 1, ··· , m. By following the same process, we obtain that

ε 1 m 2M | f (x , ··· , x ) − B ··· ( f ; x , ··· , x )| ≤ + (60) 1 m n1, ,nm 1 m 2 n ∑ 2 l=1 ∏i∈Il δi for all x1, ··· , xm ∈ [0, 1] and n = min (ni : i = 1, ··· , m) . Hence, there exist n˜ depending only on ε and not on x1, ··· , xm such that

| f (x1, ··· , xm) − Bn1,··· ,nm ( f ; x1, ··· , xm)| < ε, ∀x1, ··· , xm ∈ [0, 1] (61)

m and n1, ··· , nm ≥ n ≥ n˜. Therefore, the convergence is uniform in [0, 1] . Mathematics 2020, 8, 1397 14 of 34

We express now, in Lemma 3, the (p1, ··· , pm)-th derivative of our multivariate Bernstein polynomials of order (n1 + p1, ··· , nm + pm)-th in terms of (p1, ··· , pm)-th differences of the corresponding functions. A similar result exists for one variable (see References [1,15]).

Lemma 3. For p1, ··· , pm ≥ 0 integers with 0 ≤ pi ≤ ni, i = 1, 2, ··· , m, the (p1, ··· , pm)-th derivative of

Bn1+p1,··· ,nm+pm ( f ; x1, ··· , xm) may be expressed in terms of (p1, ··· , pm)-th differences of f as

p ,··· ,p B 1 m ( f ; x , ··· , x ) n1+p1,··· ,nm+pm 1 m " m # n1 nm m ! (n + p )! p p t tm n t = i i ··· 4 1 · · · 4 m 1 ··· i i ( − )ni−ti ∏ ∑ ∑ 1 m f , , ∏ xi 1 xi . (62) ni! n1 + p1 nm + pm t i=1 t1=0 tm=0 i=1 i

Proof of Lemma 3. We will sometimes use the square brackets to draw attention on computations of certain terms. We write

Bn1+p1,··· ,nm+pm ( f ; x1, ··· , xm) + + ! n1 p1 nm pm k k m n 1 m i ki ni−ki = ∑ ··· ∑ f , ··· , ∏ xi (1 − xi) , n1 + p1 nm + pm ki k1=0 km=0 i=1 n2+p2 nm+pm "n1+p1 ! # k km n k = ··· 1 ··· 1 1 ( − )n1−k1 ∑ ∑ ∑ f , , x1 1 x1 n1 + p1 nm + pm k1 k2=0 km=0 k1=0 ! m n i ki ni−ki × ∏ xi (1 − xi) . (63) i=2 ki

k k We take the ﬁrst variable of f and ﬁx the others. Hence, f 1 , ··· , m is seen as a n + p nm + pm 1 1 k1 function g1 : n1 + p1

Bn1+p1,··· ,nm+pm ( f ; x1, ··· , xm) n2+p2 nm+pm "n1+p1 ! # m ! k n k n k = ··· 1 1 1 ( − )n1−k1 i i ( − )ni−ki ∑ ∑ ∑ g1 x1 1 x1 ∏ xi 1 xi , n1 + p1 k1 ki k2=0 km=0 k1=0 i=2

n2+p2 nm+pm m ! n k = ··· ( ) i i ( − )ni−ki ∑ ∑ Bn1+p1 g1, x1 ∏ xi 1 xi . (64) ki k2=0 km=0 i=2

Hence

n2+p2 nm+pm m ! p p n k B 1 ( f ; x , ··· , x ) = ··· B 1 (g , x ) i x i (1 − x )ni−ki . (65) n1+p1,··· ,nm+pm 1 m ∑ ∑ n1+p1 1 1 ∏ i i ki k2=0 km=0 i=2

Let us use the following result from P. J. Davis [1]:

n ! (p) (n + p)! p n t n−t Bn+p( f ; x) = ∑ 4 fn+p (t) x (1 − x) , (66) n! t=0 t

t where n, p ∈ N and fn+p (t) = f . We then get n + p Mathematics 2020, 8, 1397 15 of 34

p B 1 ( f ; x , ··· , x ) n1+p1,··· ,nm+pm 1 m       n2+p2 nm+pm n1 m (n + p )! p t n1 t ni k = ··· 1 1 4 1 1 1 ( − )n1−t1 i ( − )ni −ki ∑ ∑  ∑ 1 g1   x1 1 x1  ∏   xi 1 xi , n! n1 + p1 k2=0 km=0 t1=0 t1 i=2 ki     n2+p2 nm+pm n1 (n + p )! p t k km n1 t = ··· 1 1 4 1 1 2 ··· 1 ( − )n1−t1 ∑ ∑  ∑ 1 f , , ,   x1 1 x1  n! n1 + p1 n2 + p2 nm + pm k2=0 km=0 t1=0 t1   m n i ki ni −ki × ∏   xi (1 − xi) , i=2 ki + + " + n3 p3 nm pm (n + p )! n1 n2 p2 t k k = ··· 1 1 4p1 1 2 ··· m ∑ ∑ ∑ ∑ 1 f , , , n! n1 + p1 n2 + p2 nm + pm k3=0 km=0 t1=0 k2=0

        m n2 k n1 t ni k × 2 ( − )n2−k2 1 ( − )n1−t1 i ( − )ni −ki   x2 1 x2    x1 1 x1  ∏   xi 1 xi . (67) k2 t1 i=3 ki

p1 t1 k2 km We repeat the action with the second variable of the function 41 f , , ··· , , n1 + p1 n2 + p2 nm + pm we get

p ,p B 1 2 ( f ; x , ··· , x ) n1+p1,··· ,nm +pm 1 m + + ! n3 p3 nm pm 2 (n + p )! n1 n2 t t = ··· i i 4p1 4p2 1 2 ∑ ∑ ∏ ∑ ∑ 1 2 f , , ni! n1 + p1 n2 + p2 k3=0 km =0 i=1 t1=0 t2=0      2 m k3 km ni t − ni k − , ··· ,   x i (1 − x )ni ti    x i (1 − x )ni ki , n + p n + p ∏ i i ∏ i i 3 3 m m i=1 ti i=3 ki + + ! " + n3 p3 nm pm 2 (n + p )! n1 n2 n3 p3 t t = ··· i i 4p1 4p2 1 2 ∑ ∑ ∏ ∑ ∑ ∑ 1 2 f , , ni! n1 + p1 n2 + p2 k3=0 km =0 i=1 t1=0 t2=0 k3=0         2 m k3 km n3 k − ni t − ni k − , ··· ,   x 3 (1 − x )n3 k3    x i (1 − x )ni ti    x i (1 − x )ni ki . (68) n + p n + p 3 3 ∏ i i ∏ i i 3 3 m m k3 i=1 ti i=4 ki

t1 t2 t1 t2 k3 After, we continue the same action with the third variable of 41 42 f ( , , , ··· , n1 + p1 n2 + p2 n3 + p3 k m ) and get nm + pm

p ,p ,p B 1 2 3 ( f ; x , ··· , x ) n1+p1,··· ,nm+pm 1 m + + ! n4 p4 nm pm 3 (n + p )! n2 t t t = ··· i i 4p1 4p2 1 2 3 ∑ ∑ ∏ ∑ 1 2 f , , , ni! n1 + p1 n2 + p2 n3 + p3 k4=0 km=0 i=1 t2=0 3 ! ! m ! k km n t n k 4 , ··· , i x i (1 − x )ni−ki i x i (1 − x )ni−ki , (69) + + ∏ i i ∏ i i n4 p4 nm pm i=1 ti i=4 ki which can be written as Mathematics 2020, 8, 1397 16 of 34

p ,p ,p B 1 2 3 ( f ; x , ··· , x ) n1+p1,··· ,nm+pm 1 m + + ! " + n5 p5 nm pm 3 (n + p )! n1 n2 n3 n4 p4 t t = ··· i i 4p1 4p2 4p3 1 2 ∑ ∑ ∏ ∑ ∑ ∑ ∑ 1 2 3 f , , ni! n1 + p1 n2 + p2 k5=0 km=0 i=1 t1=0 t2=0 t3=0 k4=0 ! # 3 ! ! t k km n t n t 3 , 4 , ··· , 4 x 4 (1 − x )n4−k4 i x i (1 − x )ni−ki + + + 4 4 ∏ i i n3 p3 n4 p4 nm pm t4 i=1 ti ! m n i ki ni−ki × ∏ xi (1 − xi) . (70) i=5 ki

By repeating successively the actions with the 4 − th, 5 − th, ··· , m − th variables but with the functions t1 t2 t3 t1 t2 t3 k4 km 41 42 43 f , , , , ··· , , n + p n2 + p2 n3 + p3 n + p nm + pm 1 1 4 4 t1 t2 t3 t4 t1 t2 t3 t4 k5 km 41 42 43 44 f , , , , , ··· , , ··· , n + p n2 + p2 n3 + p3 n + p n5 + p5 nm + pm 1 1 4 4 t1 t2 tm−1 t1 t2 tm−1 km 41 42 · · · 4m−1 f , , ··· , , respectively, we obtain the result. n1 + p1 n2 + p2 nm−1 + pm−1 nm + pm

Let us see how to deduce the monotonicity, the ﬁxed sign of the (p1, ··· , pm)-th derivative, the convexity, and the upper and lower bounds of our multivariate Bernstein polynomials from those of the corresponding function. Similar results exist for one variable (see References [1,21,31]).

Theorem 3. Let 0 ≤ pi ≤ ni, i = 1, ··· , m, be ﬁxed integers and the restriction of f to each of its variables i belong to C pi [0, 1], i = 1, ··· , m. If

(p1)···(pm) m A ≤ f (x1, ··· , xm) ≤ B, (x1, ··· , xm) ∈ [0, 1] , (71) m pi n p ,··· ,p then A ≤ i B 1 m ( f ; x , ··· , x ) ≤ B (72) ∏ ( − ) ··· ( − + ) n1,··· ,nm 1 m i=1 ni ni 1 ni pi 1

( ··· ) ∈ [ ]m = = ··· p1,··· ,pm with x1, , xm 0, 1 . For all pi 0, i 1, , m, the multiplier of Bn1,··· ,nm is to be interpreted as 1.

(p )···(pm) m - If f 1 (x1, ··· , xm) ≥ 0, (x1, ··· , xm) ∈ [0, 1] , then

p1,··· ,pm ( ··· ) ≥ ( ··· ) ∈ [ ]m Bn1,··· ,nm f ; x1, , xm 0, x1, , xm 0, 1 (73)

m m - If f (x1, ··· , xm) is convex on [0, 1] , then Bn1,··· ,nm ( f ; x1, ··· , xm) is convex on [0, 1] .

Proof of Theorem 3. We begin with Equation (62) and replace ni by ni − pi, i = 1, ··· , m. We have

m ! n1−p1 nm−pm ··· n ! t t p1, ,pm ( ··· ) = i ··· 4p1 · · · 4pm 1 ··· m Bn1,··· ,nm f ; x1, , xm ∏ ∑ ∑ 1 m f , , (ni − pi)! n1 nm i=1 t1=0 tm=0 ! m n i ti ni−pi−ti × ∏ xi (1 − xi) . (74) i=1 ti

p Let us use the following result from P. J. Davis [1]: for g ∈ C [a, b] and a = e0 < e1 < ··· < en = b with ei = e0 + ih ∈ R, h = ei+1 − ei, a subdivision of [a, b],

4pg(x ) ∃ε ∈]e , e [, 0 = g(p)(ε ). (75) 0 0 p hp 0 Mathematics 2020, 8, 1397 17 of 34

1 Then we get for any variable of f with step size hi = , i = 1, ··· , m, ni

(p )···(p ) p p t t f 1 m (εt , ··· , εt ) 4 1 · · · 4 m f 1 , ··· , m = 1 m , (76) 1 m n n p1 pm 1 m n1 ··· nm

ti ti + pi where < εti < , i = 1, ··· , m. Thus, ni ni

p1,··· ,pm ( ··· ) Bn1,··· ,nm f ; x1, , xm ! ! m n1−p1 nm−pm (p )···(pm) m n ! f 1 (εt , ··· , εt ) n − p t = i ··· 1 m i i x i (1 − x )ni−pi−ti ∏ ∑ ∑ p1 pm ∏ i i (ni − pi)! n ··· n t i=1 t1=0 tm=0 1 m i=1 i ! m ( − ) ··· ( − + ) n1−p1 nm−pm ni ni 1 ni pi 1 (p )···(pm) = ··· f 1 (εt , ··· , εt ) ∏ npi ∑ ∑ 1 m i=1 i t1=0 tm=0 ! m n − p i i ti ni−pi−ti × ∏ xi (1 − xi) . (77) i=1 ti

(p1)···(pm) m A ≤ f (x1, ··· , xm) ≤ B, (x1, ··· , xm) ∈ [0, 1] (78)

(p1)···(pm) =⇒ A ≤ f (εt1 , ··· , εtm ) ≤ B (79)

n1−p1 nm−pm m ! n − p t =⇒ A ≤ f (p1)···(pm)(ε , ··· , ε ) ··· i i x i (1 − x )ni−pi−ti ≤ B (80) t1 tm ∑ ∑ ∏ t i i t1=0 tm=0 i=1 i

n1−p1 nm−pm m ! ! n − p t since ··· i i x i (1 − x )ni−pi−ti = 1 (81) ∑ ∑ ∏ t i i t1=0 tm=0 i=1 i

n1−p1 nm−pm m ! n − p t =⇒ A ≤ ··· f (p1)···(pm)(ε , ··· , ε ) i i x i (1 − x )ni−pi−ti ≤ B (82) ∑ ∑ t1 tm ∏ t i i t1=0 tm=0 i=1 i m pi ! n p ,··· ,p =⇒ A ≤ i B 1 m ( f ; x , ··· , x ) ≤ B for all (83) ∏ ( − ) ··· ( − + ) n1,··· ,nm 1 m i=1 ni ni 1 ni pi 1 m (x1, ··· , xm) ∈ [0, 1] (using (77)). (84)

= p1,··· ,pm When all pi 0, the multiplier of Bn1,··· ,nm is

p m n i m (n − p )!npi i = i i = 1 (p = 0 ∀i = 1, ··· , m) . ∏ ( − ) ··· ( − + ) ∏ i i=1 ni ni 1 ni pi 1 i=1 ni!

(p )···(pm) m * If f 1 (x1, ··· , xm) ≥ 0, (x1, ··· , xm) ∈ [0, 1] , we set A = 0 and obtain

m pi ! n p ,··· ,p 0 ≤ i B 1 m ( f ; x , ··· , x ) (85) ∏ ( − ) ··· ( − + ) n1,··· ,nm 1 m i=1 ni ni 1 ni pi 1

( ··· ) ∈ [ ]m p1,··· ,pm ( ··· ) ≥ ( ··· ) ∈ [ ]m for all x1, , xm 0, 1 . This means Bn1,··· ,nm f ; x1, , xm 0, x1, , xm 0, 1 . Mathematics 2020, 8, 1397 18 of 34

(2)···(2) t t f (εt , ··· , εt ) * If f is convex, 42 · · · 42 f 1 , ··· , m ≥ 0 and hence we have 1 m ≥ 0 since 1 m 2 2 n1 nm n1 ··· nm (p1)···(pm)( ··· ) f εt1 , , εtm p1 pm t1 tm (2)···(2) = 4 · · · 4 f , ··· , , this means f (εt , ··· , εt ) ≥ 0. p1 pm 1 m n n 1 m n1 ··· nm 1 m Thus, from (77) with pi = 2 ∀i = 1, ··· , m,

B2,··· ,2 ( f x ··· x ) ≥ (x ··· x ) ∈ [ ]m n1,··· ,nm ; 1, , m 0, 1, , m 0, 1 . (86)

m This implies that Bn1,··· ,nm is convex in every closed interval of (0, 1) . Since Bn1,··· ,nm is continuous, it is convex in [0, 1]m.

Corollary 1. If f (x1, ··· , xm) is decreasing (resp. nondecreasing) on [0, 1] with respect to the variable xi, i ∈

{1, ··· , m}, then Bn1,··· ,nm ( f ; x1, ··· , xm) is decreasing (resp. nondecreasing) with respect to the variable xi, i ∈ {1, ··· , m} on [−1, 1].

Proof of Corollary 1. We just use Theorem 3, and the rest follows easily.

th The following theorem states for the pointwise and uniform convergence of the (p1, ··· , pm) derivative of our multivariate Bernstein polynomials, compare ([35] Theorem of Kingsley) and ([28] Theorem 1) (just for two variables), and ([25] Theorem 4) and ([7] Theorem 2).

m Theorem 4. Let 0 ≤ pi ≤ ni, i = 0, ··· , m, be ﬁxed integers, f uniformly continuous on [0, 1] and p ∂ i f ··· also uniformly continuous on [ ]m i = ··· m Then Bp1, ,pm ( f x ··· x ) converges towards pi 0, 1 , 1, , . n1,··· ,nm ; 1, , m ∂xi (p )···(pm) m f 1 (x1, ··· , xm) uniformly on [0, 1] .

Proof of Theorem 4. We have already shown that the the above result holds for pi = 0, ∀i = 0, ··· , m. We have to look at the case where pi are not all zero. p ,··· ,p We begin with the expression for B 1 m ( f ; x , ··· , x ) given in Equation (62) and write n1+p1,··· ,nm+pm 1 m 1 using Equation (75) with hi = , i = 1, ··· , m ni + pi

(p1)···(pm) p1 pm t1 tm f (εt1 , ··· , εtm ) 41 · · · 4m f , ··· , = m , (87) n + p nm + pm 1 1 pi ∏(ni + pi) i=1

ti ti + pi where < εti < , i = 1, ··· , m. We get ni + pi ni + pi

p ,··· ,p B 1 m ( f ; x , ··· , x ) n1+p1,··· ,nm+pm 1 m m ! n1 nm m ! (n + p )! p p t tm n t = i i ··· 4 1 · · · 4 m 1 ··· i i ( − )ni−ti ∏ ∑ ∑ 1 m f , , ∏ xi 1 xi ni! n1 + p1 nm + pm t i=1 t1=0 tm=0 i=1 i

m ! n1 nm m ! (n + p )! n t = i i ··· (p1)···(pm)( ··· ) i i ( − )ni−ti ∏ p ∑ ∑ f εt1 , , εtm ∏ xi 1 xi , (88) ni!(ni + pi) i t i=1 t1=0 tm=0 i=1 i which gives us Mathematics 2020, 8, 1397 19 of 34

! m pi ni!(ni + pi) p1,··· ,pm B ( f ; x , ··· , xm) ∏ ( + ) n1+p1,··· ,nm+pm 1 i=1 ni pi !

n1 nm m ! n t = ··· f (p1)···(pm)(ε , ··· , ε ) i x i (1 − x )ni−ti . (89) ∑ ∑ t1 tm ∏ t i i t1=0 tm=0 i=1 i

(p1)···(pm) We then approximate f (εt1 , ··· , εtm ) writing

(p1)···(pm) f (εt1 , ··· , εtm ) t t t t (p1)···(pm) (p1)···(pm) 1 m (p1)···(pm) 1 m = f (εt1 , ··· , εtm ) − f , ··· , + f , ··· , . (90) n1 nm n1 nm

We thus obtain ! m pi ni!(ni + pi) p1,··· ,pm B ( f ; x , ··· , xm) ∏ ( + ) n1+p1,··· ,nm+pm 1 i=1 ni pi ! n1 nm t tm = ··· (p1)···(pm) 1 ··· + (p1)···(pm)( ··· ) ∑ ∑ f , , f εt1 , , εtm n1 nm t1=0 tm=0 ! t t m n (p1)···(pm) 1 m i ti ni−ti − f , ··· , ∏ xi (1 − xi) n1 nm i=1 ti

= T1(x1, ··· , xm) + T2(x1, ··· , xm) where (91)  n1 nm t t (p1)···(pm) 1 m  T1(x1, ··· , xm) = ∑ ··· ∑ f , ··· ,  n1 nm  t1=0 tm=0  !  ni t  × m i ( − )ni−ti  ∏i=1 xi 1 xi ,  ti (92) n1 nm h  (p1)···(pm)  T2(x1, ··· , xm) = ··· f (εt1 , ··· , εtm )  ∑ ∑  t1=0 tm=0  !  t1 tm ni t  − (p1)···(pm) ··· m i ( − )ni−ti  f , , ∏i=1 xi 1 xi . n1 nm ti

ti ti ti + pi Since < < , i = 1, ··· , m, ti = 0, ··· , ni, it follows from the bounds on εti , ni + pi ni ni + pi i = 1, ··· , m, that

pi ti ti + pi ti ti + pi ti pi − = − < εti − < − = (93) ni + pi ni + pi ni + pi ni ni + pi ni + pi ni + pi

ti pi for i = 1, ··· , m. This means εti − < , i = 1, ··· , m. From the uniform continuity of ni ni + pi (p )···(pm) f 1 (x1, ··· , xm), given ε > 0, we can ﬁnd ni0 such that for all ni ≥ ni0 and all ti,

t t (p1)···(pm) (p1)···(pm) 1 m f (εt1 , ··· , εtm ) − f , ··· , < ε. (94) n1 nm

m Thus, T2(x1, ··· , xm) converges uniformly to zero on [0, 1] . We have Mathematics 2020, 8, 1397 20 of 34

m n !(n + p )pi i i i −→ 1 as n → ∞, ∀i = 1, ··· , m , (95) ∏ ( + ) i i=1 ni pi !

(p )···(pm) and we see from Theorem 2) with f 1 in place of f that T1(x1, ··· , xm) converges uniformly to (p )···(pm) f 1 (x1, ··· , xm). This completes the proof.

The following theorem, in the case where f is a convex function, is the generalization of the inequality related to the decreasing of the sequence of Bernstein’s polynomials with one variable (see References [1,31]).

m Theorem 5. Let f (x1, ··· , xm) be convex in [0, 1] . Then, for ni = 2, 3, ··· , i = 1, ··· , m, we have

( ··· ) ≥ ( ··· ) ( ··· ) ∈ [ ]m Bn1−1,··· ,nm−1 f ; x1, , xm Bn1,··· ,nm f ; x1, , xm , x1, , xm 0, 1 . (96)

Proof of Theorem 5. The case of one variable is already solved (see P. J. Davis [1]). Because of long calculations and space limitations, we will just show how this theorem works for three variables (we did not choose the case of two variables because with three variables, our strategy is well illustrated and we xi can easily use it for any large number of variables). Set ti = , i = 1, 2, 3. Then, 1 − xi

( k k ( k − k i −ki i i ( − ) ki = i xi (1 − xi) = ti xi 1 xi ti − that is, k k (97) + = ( − ) 1 i −ki−1 i 1 ti 1 xi , xi (1 − xi) = ti (1 + ti), and

( − )−n1 ( − )−n2 ( − )−n3 ( ) − ( ) 1 x1 1 x2 1 x3 Bn1−1,n2−1,n3−1 f ; x1, x2, x3 Bn1,n2,n3 f ; x1, x2, x3   n1−1 n2−1 n3−1 k k k 3 n − 1 1 2 3 i ki −ki −1 = ∑ ∑ ∑ f , , ∏   xi (1 − xi) n1 − 1 n2 − 1 n3 − 1 k1=0 k2=0 k3=0 i=1 ki   n1 n2 n3 k k k 3 n 1 2 3 i ki −ki − ∑ ∑ ∑ f , , ∏   xi (1 − xi) n1 n2 n3 k1=0 k2=0 k3=0 i=1 ki   n1−1 n2−1 n3−1 3 − k1 k2 k3 ni 1 ki = ∑ ∑ ∑ f , , ∏   ti (1 + ti) n1 − 1 n2 − 1 n3 − 1 k1=0 k2=0 k3=0 i=1 ki   n1 n2 n3 3 k1 k2 k3 ni ki − ∑ ∑ ∑ f , , ∏   ti n1 n2 n3 k1=0 k2=0 k3=0 i=1 ki       n1−1+i1 n2−1+i2 n3−1+i3 k − i k − i k − i n − 1 n − 1 n − 1 = 1 1 2 2 3 3 1 2 3 k1 k2 k3 ∑ ∑ ∑ ∑ f , ,       t1 t2 t3 n1 − 1 n2 − 1 n3 − 1 − − − i1,i2,i3∈{0,1} k1=i1 k2=i2 k3=i3 k1 i1 k2 i2 k3 i3       n1 n2 n3 k k k n n n − 1 2 3 1 2 3 k1 k2 k3 ∑ ∑ ∑ f , ,       t1 t2 t3 n1 n2 n3 k1=0 k2=0 k3=0 k1 k2 k3       n1−1+i1 n2−1+i2 n3−1+i3 k − i k − i k − i n − 1 n − 1 n − 1 = 1 1 2 2 3 3 1 2 3 k1 k2 k3 ∑ ∑ ∑ ∑ f , ,       t1 t2 t3 n1 − 1 n2 − 1 n3 − 1 − − − i1,i2,i3∈{0,1} k1=i1 k2=i2 k3=i3 k1 i1 k2 i2 k3 i3       n1 n2 n3 k k k n n n − 1 2 3 1 2 3 k1 k2 k3 ∑ ∑ ∑ f , ,       t1 t2 t3 n1 n2 n3 k1=0 k2=0 k3=0 k1 k2 k3

= Fn1,n2,n3 (t1, t2, t3). (98)

Fn1,n2,n3 (t1, t2, t3) is a polynomial of the variables t1, t2 and t3. It can be written as Mathematics 2020, 8, 1397 21 of 34

n1 n2 n3 ( ) = k1,k2,k3 k1 k2 k3 Fn1,n2,n3 t1, t2, t3 ∑ ∑ ∑ Fn1,n2,n3 t1 t2 t3 . (99) k1=0 k2=0 k3=0

To have our result, we just need to prove that Fn1,n2,n3 (t1, t2, t3) is positive. For ie ∈ {0, 1} with e ∈ {1, 2, 3}, we deﬁne ( 0 if ie = 1; ie = (100) 1 if ie = 0.

k1,k2,k3 k1 k2 k3 Now, we will analyse all the possibilities of the summand Fn1,n2,n3 t1 t2 t3 of our polynomial

Fn1,n2,n3 (t1, t2, t3). For simplicity and depending of the upper and lower bounds of (99), we will write

Fn1,n2,n3 (t1, t2, t3) as

(t t t ) = 1 (t t t ) + 2 (t t t ) + 3 (t t t ) + 4 (t t t ) Fn1,n2,n3 1, 2, 3 Fn1,n2,n3 1, 2, 3 Fn1,n2,n3 1, 2, 3 Fn1,n2,n3 1, 2, 3 Fn1,n2,n3 1, 2, 3 , (101) where

1 (t t t ) ≤ k ≤ n − ≤ k ≤ n − ≤ k ≤ n − *F n1,n2,n3 1, 2, 3 is the sum with index 1 1 1 1, 1 2 2 1 and 1 3 3 1; 2 (t t t ) k i ∈ { } * Fn1,n2,n3 1, 2, 3 is the sum where all the index i, 1, 2, 3 start at 1, but with just one index reaching the upper bound ni, i ∈ {1, 2, 3}; 3 (t t t ) k n − i ∈ { } * Fn1,n2,n3 1, 2, 3 is the sum where all the index i are less or equal to i 1, 1, 2, 3 , but just one index ki, i ∈ {1, 2, 3} reaching the lower bound 0; 4 (t t t ) ( ) 1 (t t t ) * Fn1,n2,n3 1, 2, 3 is the rest of the sum in Equation 99 after removing the terms Fn1,n2,n3 1, 2, 3 , 2 (t t t ) 3 (t t t ) Fn1,n2,n3 1, 2, 3 and Fn1,n2,n3 1, 2, 3 .

- Case 1: First, we take the sum with index 1 ≤ k1 ≤ n1 − 1, 1 ≤ k2 ≤ n2 − 1 and 1 ≤ k3 ≤ n3 − 1 in Equation (99). We obtain

1 (t t t ) Fn1,n2,n3 1, 2, 3        n1−1 n2−1 n3−1 k1 − i1 k2 − i2 k3 − i3 n1 − 1 n2 − 1 n3 − 1 = ∑ ∑ ∑  ∑ f , ,       n1 − 1 n2 − 1 n3 − 1 k1=1 k2=1 k3=1 i1,i2,i3∈{0,1} k1 − i1 k2 − i2 k3 − i3       k1 k2 k3 n1 n2 n3 k k k − f , ,       t 1 t 2 t 3 . (102) n n n 1 2 3 1 2 3 k1 k2 k3

1 (t t t ) Next, we add and subtract the following expression in Fn1,n2,n3 1, 2, 3 to obtain:

       n1−1 n2−1 n3−1 k1 − i1 k2 − i2 k3 n1 − 1 n2 − 1 n3 ∑ ∑ ∑  ∑ f , ,       n1 − 1 n2 − 1 n3 k1=1 k2=1 k3=1 i1,i2∈{0,1} k1 − i1 k2 − i2 k3       k − i k k n1 − 1 n2 n3 + 1 1 2 3 k1 k2 k3 ∑ f , ,       t1 t2 t3 . (103) n1 − 1 n2 n3 i1∈{0,1} k1 − i1 k2 k3

1 (t t t ) Next, we group now the terms in Fn1,n2,n3 1, 2, 3 given by the previous expression in the following 1 (t t t ) = B + B + B three blocks: Fn1,n2,n3 1, 2, 3 1 2 3 with Mathematics 2020, 8, 1397 22 of 34

       n1−1 n2−1 n3−1 k1 − i1 k2 − i2 k3 − i3 n1 − 1 n2 − 1 n3 − 1 B1 = ∑ ∑ ∑  ∑ f , ,       n1 − 1 n2 − 1 n3 − 1 − − − k1=1 k2=1 k3=1 i1,i2,i3∈{0,1} k1 i1 k2 i2 k3 i3 !       k1 − i1 k2 − i2 k3 − i3 n1 − 1 n2 − 1 n3 − 1 + f , ,       n − n − n − 1 1 2 1 3 1 k1 − i1 k2 − i2 k3 − i3       k1 − i1 k2 − i2 k3 n1 − 1 n2 − 1 n3 k k k − f , ,       t 1 t 2 t 3 , (104) n − n − n 1 2 3 1 1 2 1 3 k1 − i1 k2 − i2 k3        n1−1 n2−1 n3−1 k1 − i1 k2 − i2 k3 n1 − 1 n2 − 1 n3 B2 = ∑ ∑ ∑  ∑ f , ,       n1 − 1 n2 − 1 n3 − − k1=1 k2=1 k3=1 i1,i2∈{0,1} k1 i1 k2 i2 k3 !       k1 − i1 k2 − i2 k3 n1 − 1 n2 − 1 n3 + f , ,       n − n − n 1 1 2 1 3 k1 − i1 k2 − i2 k3       k1 − i1 k2 k3 n1 − 1 n2 n3 k k k − f , ,       t 1 t 2 t 3 , (105) n − n n 1 2 3 1 1 2 3 k1 − i1 k2 k3        n1−1 n2−1 n3−1 k1 − i1 k2 k3 n1 − 1 n2 n3 B3 = ∑ ∑ ∑  ∑ f , ,       n1 − 1 n2 n3 − k1=1 k2=1 k3=1 i1∈{0,1} k1 i1 k2 k3 !       k1 − i1 k2 k3 n1 − 1 n2 n3 + f , ,       n − n n 1 1 2 3 k1 − i1 k2 k3       k1 k2 k3 n1 n2 n3 k k k − f , ,       t 1 t 2 t 3 . (106) n n n 1 2 3 1 2 3 k1 k2 k3

We are going to show that Bi ≥ 0 for i ∈ {1, 2, 3}. It is obvious that

        n − 1 n n − k n − 1 n k   =   (n ≥ k + 1) and   =   (n ≥ k) with k, n positive integers. (107) k k n k − 1 k n

Now, using Equation (107) in each Bi we get

 n1−1 n2−1 n3−1 ! ! ! n1 − 1 n2 − 1 n3 n3 − k3 k1 − i1 k2 − i2 k3 B1 = ∑ ∑ ∑  ∑ f , , − − n3 n1 − 1 n2 − 1 n3 − 1 k1=1 k2=1 k3=1 i1,i2∈{0,1} k1 i1 k2 i2 k3 k k − i k − i k − 1 k − i k − i k + 3 f 1 1 , 2 2 , 3 − f 1 1 , 2 2 , 3 n3 n1 − 1 n2 − 1 n3 − 1 n1 − 1 n2 − 1 n3 ! ! ! n1 − 1 n2 − 1 n3 k k − i k − i k − 1 + ∑ 3 f 1 1 , 2 2 , 3 − − n3 n1 − 1 n2 − 1 n3 − 1 i1,i2∈{0,1} k1 i1 k2 i2 k3 n3 − k3 k1 − i1 k2 − i2 k3 k1 − i1 k2 − i2 k3 k1 k2 k3 + f , , − f , , t1 t2 t3 , (108) n3 n1 − 1 n2 − 1 n3 − 1 n1 − 1 n2 − 1 n3 Mathematics 2020, 8, 1397 23 of 34

 n1−1 n2−1 n3−1 ! ! ! n1 − 1 n2 n3 n2 − k2 k1 − i1 k2 k3 B2 = ∑ ∑ ∑  ∑ f , , k1 − i1 k2 k3 n2 n1 − 1 n2 − 1 n3 k1=1 k2=1 k3=1 i1∈{0,1} k k − i k − 1 k k − i k k + 2 f 1 1 , 2 , 3 − f 1 1 , 2 , 3 n2 n1 − 1 n2 − 1 n3 n1 − 1 n2 n3 ! ! ! n − 1 n n k k − i k − 1 k + ∑ 1 2 3 2 f 1 1 , 2 , 3 k1 − i1 k2 k3 n2 n1 − 1 n2 − 1 n3 i1∈{0,1} n2 − k2 k1 − i1 k2 k3 k1 − i1 k2 k3 k1 k2 k3 + f , , − f , , t1 t2 t3 , (109) n2 n1 − 1 n2 − 1 n3 n1 − 1 n2 n3

n1−1 n2−1 n3−1 ! ! ! n1 n2 n3 n1 − k1 k1 k2 k3 B3 = ∑ ∑ ∑ f , , k1 k2 k3 n1 n1 − 1 n2 n3 k1=1 k2=1 k3=1 k k − 1 k k k k k + 1 f 1 , 2 , 3 − f 1 , 2 , 3 n1 n1 − 1 n2 n3 n1 n2 n3 ! ! ! n n n k k − 1 k k + 1 2 3 1 f 1 , 2 , 3 k1 k2 k3 n1 n1 − 1 n2 n3 n1 − k1 k1 k2 k3 k1 k2 k3 k1 k2 k3 + f , , − f , , t1 t2 t3 , (110) n1 n1 − 1 n2 n3 n1 n2 n3

which implies that Bi ≥ 0, i ∈ {1, 2, 3} by the convexity of f and the fact that t1, t2 and t3 are also positive. - Case 2: We take the sum in Equation (99), where all the index start at 1, but just one index ki, i ∈ { } n i ∈ { } 2 (t t t ) 1, 2, 3 reaches the upper bound i, 1, 2, 3 . This corresponds to Fn1,n2,n3 1, 2, 3 which can be written as

2 (t t t ) = 2,1 (t t t ) + 2,2 (t t t ) + 2,3 (t t t ) Fn1,n2,n3 1, 2, 3 Fn1,n2,n3 1, 2, 3 Fn1,n2,n3 1, 2, 3 Fn1,n2,n3 1, 2, 3 (111)

2,i ( ) ∈ { } where Fn1,n2,n3 t1, t2, t3 is the sum where the index ki reaches the upper bound ni for i 1, 2, 3 . 2,i ( ) ∈ { } 2,1 ( ) We have to show that Fn1,n2,n3 t1, t2, t3 , i 1, 2, 3 , are positive. Let us start with Fn1,n2,n3 t1, t2, t3 :  − − ! ! n2 1 n3 1 k − i k − i n − 1 n − 1 F2,1 (t , t , t ) = f 1, 2 2 , 3 3 2 3 n1,n2,n3 1 2 3 ∑ ∑  ∑ n2 − 1 n3 − 1 k2 − i2 k3 − i3 k2=1 k3=1 i2,i3∈{0,1} ! !! k2 k3 n2 n3 n1 k2 k3 − f 1, , t1 t2 t3 . (112) n2 n3 k2 k3

2,1 ( ) We add and subtract the following expression in Fn1,n2,n3 t1, t2, t3   n2−1 n3−1 ! ! k2 − i2 k3 n2 − 1 n3 n1 k2 k3 ∑ ∑  ∑ f 1, ,  t1 t2 t3 . (113) n2 − 1 n3 k2 − i2 k3 k2=1 k3=1 i2∈{0,1}

2,1 ( ) We now group the terms in Fn1,n2,n3 t1, t2, t3 in the following two blocks: 2,1 ( ) = + Fn1,n2,n3 t1, t2, t3 B1 B2 with Mathematics 2020, 8, 1397 24 of 34

 n2−1 n3−1 ! ! k2 − i2 k3 − i3 n2 − 1 n3 − 1 B1 = ∑ ∑  ∑ f 1, , n2 − 1 n3 − 1 k2 − i2 k3 − i3 k2=1 k3=1 i2,i3∈{0,1} ! ! ! k − i k − i n − 1 n − 1 + f 1, 2 2 , 3 3 2 3 n2 − 1 n3 − 1 k2 − i2 k3 − i3 ! !!! k2 − i2 k3 n2 − 1 n3 n1 k2 k3 − f 1, , t1 t2 t3 , (114) n2 − 1 n3 k2 − i2 k3  n2−1 n3−1 ! ! k2 − i2 k3 n2 − 1 n3 B2 = ∑ ∑  ∑ f 1, , n2 − 1 n3 k2 − i2 k3 k2=1 k3=1 i2∈{0,1} ! ! ! ! !!! k2 − i2 k3 n2 − 1 n3 k2 k3 n2 n3 n1 k2 k3 + f 1, , − f 1, , t1 t2 t3 . (115) n2 − 1 n3 k2 − i2 k3 n2 n3 k2 k3

We are going to show that Bi ≥ 0 for i ∈ {1, 2}. Using (107) in each Bi, we get  n2−1 n3−1 ! ! n2 − 1 n3 n3 − k3 k2 − i2 k3 B1 = ∑ ∑  ∑ f 1, , k2 − i2 k3 n3 n2 − 1 n3 − 1 k2=1 k3=1 i2∈{0,1} k k − i k − 1 k − i k + 3 f 1, 2 2 , 3 − f 1, 2 2 , 3 n3 n2 − 1 n3 − 1 n2 − 1 n3 ! ! n − 1 n k k − i k − 1 + ∑ 2 3 3 f 1, 2 2 , 3 k2 − i2 k3 n3 n2 − 1 n3 − 1 i2∈{0,1} n3 − k3 k2 − i2 k3 k2 − i2 k3 n1 k2 k3 + f 1, , − f 1, , t1 t2 t3 , (116) n3 n2 − 1 n3 − 1 n2 − 1 n3

n2−1 n3−1 ! ! n2 n3 n2 − k2 k2 k3 k2 k2 − 1 k3 B2 = ∑ ∑ f 1, , + f 1, , k2 k3 n2 n2 − 1 n3 n2 n2 − 1 n3 k2=1 k3=1 k2 k3 n1 k2 k3 − f 1, , t1 t2 t3 , (117) n2 n3

which implies that Bi ≥ 0, i ∈ {1, 2} by the convexity of f and the fact that t1, t2 and t3 are also positive. 2,1 ( ) 2,2 ( ) Using a similar reasoning as for Fn1,n2,n3 t1, t2, t3 , one shows that Fn1,n2,n3 t1, t2, t3 and 2,3 ( ) Fn1,n2,n3 t1, t2, t3 are also positive. The main difference appears in the expression that we add and subtract:

2,2 ( ) (i) for Fn1,n2,n3 t1, t2, t3 , the added and subtracted expression is   n1−1 n3−1 ! ! k1 − i1 k3 n1 − 1 n3 k1 n2 k3 ∑ ∑  ∑ f , 1,  t1 t2 t3 ; (118) n1 − 1 n3 k1 − i1 k3 k1=1 k3=1 i1∈{0,1}

2,3 ( ) (ii) for Fn1,n2,n3 t1, t2, t3 , the added and subtracted expression is   n1−1 n2−1 ! ! k1 − i1 k2 n1 − 1 n2 k1 k2 n3 ∑ ∑  ∑ f , , 1  t1 t2 t3 . (119) n1 − 1 n2 k1 − i1 k2 k1=1 k2=1 i1∈{0,1} Mathematics 2020, 8, 1397 25 of 34

- Case 3: We take the sum in Equation (99), where all the index ki are less or equal to ni − 1, i ∈ {1, 2, 3}, k i ∈ { } 3 (t t t ) but just one index i, 1, 2, 3 reaching the lower bound 0. This corresponds to Fn1,n2,n3 1, 2, 3 which can be written as

3 (t t t ) = 3,1 (t t t ) + 3,2 (t t t ) + 3,3 (t t t ) Fn1,n2,n3 1, 2, 3 Fn1,n2,n3 1, 2, 3 Fn1,n2,n3 1, 2, 3 Fn1,n2,n3 1, 2, 3 , (120)

3,i ( ) ∈ { } where Fn1,n2,n3 t1, t2, t3 is the sum where the index ki reaches the lower bound 0 for i 1, 2, 3 . We have  n2−1 n3−1  F3,1 (t , t , t ) = F0,k2,k3 tk2 tk3 ;  n1,n2,n3 1 2 3 ∑ ∑ n1,n2,n3 2 3  k2=1 k3=1   n1−1 n3−1 3,1 ( ) = k1,0,k3 k1 k3 Fn1,n2,n3 t1, t2, t3 ∑ ∑ Fn1,n2,n3 t1 t3 ; (121)  k1=1 k3=1  n − −  1 1 n2 1  F3,1 (t , t , t ) = Fk1,k2,0 tk1 tk2 .  n1,n2,n3 1 2 3 ∑ ∑ n1,n2,n3 1 2 k1=1 k2=1

3,i ( ) ∈ { } Using a similar reasoning as in case 2, one shows that Fn1,n2,n3 t1, t2, t3 , i 1, 2, 3 , are positive. The main difference appears in the expression that we add and subtract:

3,1 ( ) (i) for Fn1,n2,n3 t1, t2, t3 , the added and subtracted expression is   n2−1 n3−1 ! ! k2 − i2 k3 n2 − 1 n3 k2 k3 ∑ ∑  ∑ f 0, ,  t2 t3 ; (122) n2 − 1 n3 k2 − i2 k3 k2=1 k3=1 i2∈{0,1}

3,2 ( ) (ii) for Fn1,n2,n3 t1, t2, t3 , the added and subtracted expression is   n1−1 n3−1 ! ! k1 − i1 k3 n1 − 1 n3 k1 k3 ∑ ∑  ∑ f , 0,  t1 t3 ; (123) n2 − 1 n3 k1 − i1 k3 k1=1 k3=1 i1∈{0,1}

3,3 ( ) (iii) for Fn1,n2,n3 t1, t2, t3 , the added and subtracted expression is   n1−1 n2−1 ! ! k1 − i1 k2 n1 − 1 n2 k1 k2 ∑ ∑  ∑ f , , 0  t1 t2 . (124) n2 − 1 n2 k1 − i1 k2 k1=1 k2=1 i3∈{0,1}

Case 4 ( ) 1 (t t t ) - : We take in Equation 99 the rest of the sum after removing Fn1,n2,n3 1, 2, 3 , 2 (t t t ) 3 (t t t ) 4 (t t t ) Fn1,n2,n3 1, 2, 3 and Fn1,n2,n3 1, 2, 3 . This corresponds to Fn1,n2,n3 1, 2, 3 and has all its terms n n n k ke j np ne j np 0 0 0 ke j j related with the following monomials: te tj tp , te tj tp , tj te tp, te tj , tj in (99) where e, j, p ∈ {1, 2, 3}, e 6= j, e 6= p, j 6= p, and ke ∈ {1, ··· , ne − 1}. In this case we do not need to add and 4 (t t t ) subtract some expressions in Fn1,n2,n3 1, 2, 3 which can be written as Mathematics 2020, 8, 1397 26 of 34

F4 (t , t , t ) n1,n2,n3 1 2 3 −   n1 1 n n − k k k k − 1 k = 1 1 1 1 + 1 1 − 1 k1 ∑   f , 0, 0 f , 0, 0 f , 0, 0 t1 n1 n1 − 1 n1 n1 − 1 n1 k1=1 k1   n2−1 n 2 n2 − k2 k2 k2 k2 − 1 k2 k2 + ∑   f 0, , 0 + f 0, , 0 − f 0, , 0 t2 n2 n2 − 1 n2 n2 − 1 n2 k2=1 k2   n1−1 n 3 n3 − k3 k3 k3 k3 − 1 k3 k3 + ∑   f 0, 0, + f 0, 0, − f 0, 0, t3 n3 n3 − 1 n3 n3 − 1 n3 k1=1 k3

−   n1 1 n n − k k k k − 1 k + 1 1 1 1 + 1 1 − 1 k1 n2 n3 ∑   f , 1, 1 f , 1, 1 f , 1, 1 t1 t2 t3 n1 n1 − 1 n1 n1 − 1 n1 k1=1 k1   − n2 1 n n − k k k k − 1 k + 2 2 2 2 + 2 2 − 2 n1 k2 n3 ∑   f 1, , 1 f 1, , 1 f 1, , 1 t1 t2 t3 n2 n2 − 1 n2 n2 − 1 n2 k2=1 k2

−   n1 1 n n − k k k k − 1 k + 3 3 3 3 + 3 3 − 3 n1 n2 k3 ∑   f 1, 1, f 1, 1, f 1, 1, t1 t2 t3 n3 n3 − 1 n3 n3 − 1 n3 k1=1 k3

−   n1 1 n n − k k k k − 1 k + 1 1 1 1 + 1 1 − 1 k1 n2 ∑   f , 1, 0 f , 1, 0 f , 1, 0 t1 t2 n1 n1 − 1 n1 n1 − 1 n1 k1=1 k1

−   n1 1 n n − k k k k − 1 k + 1 1 1 1 + 1 1 − 1 k1 n3 ∑   f , 0, 1 f , 0, 1 f , 0, 1 t1 t3 n1 n1 − 1 n1 n1 − 1 n1 k1=1 k1   − n2 1 n n − k k k k − 1 k + 2 2 2 2 + 2 2 − 2 n1 k2 ∑   f 1, , 0 f 1, , 0 f 1, , 0 t1 t2 n2 n2 − 1 n2 n2 − 1 n2 k2=1 k2   n2−1 n 2 n2 − k2 k2 k2 k2 − 1 k2 k2 n3 + ∑   f 0, , 1 + f 0, , 1 − f 0, , 1 t2 t3 n2 n2 − 1 n2 n2 − 1 n2 k2=1 k2

−   n1 1 n n − k k k k − 1 k + 3 3 3 3 + 3 3 − 3 n1 k3 ∑   f 1, 0, f 1, 0, f 1, 0, t1 t3 n3 n3 − 1 n3 n3 − 1 n3 k1=1 k3   n1−1 n 3 n3 − k3 k3 k3 k3 − 1 k3 n2 k3 + ∑   f 0, 1, + f 0, 1, − f 0, 1, t2 t3 . (125) n3 n3 − 1 n3 n3 − 1 n3 k1=1 k3

4 (t t t ) f t i ∈ { } Then Fn1,n2,n3 1, 2, 3 is positive since is convex and all the i, 1, 2, 3 , are positive.

4. Order of Convergence To provide information about the speed of convergence of multivariate Bernstein polynomials, we need the following intermediate result. This result also exists for one variable (see (Reference [1] Lemma 6.3.5)).

Lemma 4. There is a constant C independent of ni such that, for all m (x1, ··· , xm) ∈ [0, 1] , i = 1, ··· , m, ! m n C i ki ni−ki ∑ ··· ∑ ∏ xi (1 − xi) ≤ m . (126) i= ki 3/2 k1 km 1 n − ≥ −1/4 − ≥ −1/4 ∏ i x1 n1 xm nm n1 nm i=1 Mathematics 2020, 8, 1397 27 of 34

Proof of Lemma 4. We will sometimes use the square brackets to draw attention to computations of certain terms. ! m n i ki ni−ki ∑ ··· ∑ ∏ xi (1 − xi) i= ki k1 km 1 − ≥ −1/4 − ≥ −1/4 x1 n1 xm nm n1 nm    !   n k  = ···  1 1 ( − )n1−k1  ∑ ∑  ∑ x1 1 x1   k1  k2 km k1 − ≥ −1/4 − ≥ −1/4  − ≥ −1/4  x2 n2 xm nm x1 n1 n2 nm n1 ! m n i ki ni−ki × ∏ xi (1 − xi) , i=2 ki    !   n k  =  1 1 ( − )n1−k1  ···  ∑ x1 1 x1  ∑ ∑  k1  k1 k3 km  − ≥ −1/4  − ≥ −1/4 − ≥ −1/4 x1 n1 x3 n3 xm nm n1 n3 nm  

 !  m !  n  n k  2 k2 ( − )n2−k2  i i ( − )ni−ki  ∑ x2 1 x2  ∏ xi 1 xi  k2  = ki k2 i 3  − ≥ −1/4  x2 n2 n2  

2  !   n k  =  i i ( − )ni−ki  ··· ∏  ∑ xi 1 xi  ∑ ∑ i=  ki  1 ki k3 km  − ≥ −1/4  − ≥ −1/4 − ≥ −1/4 xi ni x3 n3 xm nm ni n3 nm ! m n i ki ni−ki ∏ xi (1 − xi) . (127) i=3 ki

By repeating successively the actions with k3, k4, ··· , km, we obtain

    m m ni ni ki ni −ki ki ni −ki ∑ ··· ∑ ∏   xi (1 − xi) = ∏ ∑   xi (1 − xi) . (128) i= k i= k k1 km 1 i 1 ki i − ≥ −1/4 − ≥ −1/4 − ≥ −1/4 x1 n1 xm nm xi ni n1 nm ni

Let us use the following result which can be found in References [1,21,22]: There is a constant C independent of a given positive integer n such that for all x in [0, 1], ! n C xk( − x)n−k ≤ ∑ 1 3/2 . (129) k n k −1/4 −x ≥n n

By applying this to any xi, i = {1, ··· , m}, we have Mathematics 2020, 8, 1397 28 of 34

! m n i ki ni−ki ∑ ··· ∑ ∏ xi (1 − xi) i= ki k1 km 1 − ≥ −1/4 − ≥ −1/4 x1 n1 xm nm n1 nm ! m C ≤ i C ∏ 3/2 with i constants (130) i=1 ni C = m with C constant (C = C1 ··· Cm). (131) 3/2 ∏ ni i=1

Remark 2. Lemma 4 is still valid if the summation is not on all the ki, 1 ≤ i ≤ m but on some of them. In this case, the upper bound will contain only terms involved in the summation.

The following Theorem will help us to conclude on the order of convergence of our Bernstein polynomials with several variables, see References [22,36].

m m Theorem 6. Let f (x1, ··· , xm) be bounded in [0, 1] and let (a1, ··· , am) be a point of [0, 1] at which (2)···(2) f (a1, ··· , am) exists and is continuous. Then,

" m 2 # 1 aj(1 − aj) ∂ f lim Bn ,··· ,nm ( f ; a1, ··· , am) − f (a1, ··· , am) − (a1, ··· , am) = 0, n →∞, 1≤j≤m 1 ∑ 2 j 2 j=1 nj ∂xj and

1 m ∂2 f lim N [BN,··· ,N( f ; a1, ··· , am) − f (a1, ··· , am)] = ai(1 − ai) (a1, ··· , am). (132) N→∞ ∑ 2 2 i=1 ∂xi

(2)···(2) Proof of Theorem 6. Since f (x1, ··· , xm) exists and is continuous at (a1, ··· , am), then using Taylor expansion of order two

f (x1, ··· , xm)

m ∂ f 1 m ∂2 f = ( ··· ) + ( ··· )( − ) + ( ··· )( − )2 f a1, , am ∑ a1, , am xj aj ∑ 2 a1, , am xj aj j=1 ∂xj 2 j=1 ∂xj m 2 ! m ∂ f 2 + ∑ (a1, ··· , am)(xi − ai)(xj − aj) + s(x1, ··· , xm) ∑(xj − aj) , (133) i,j=1,i6=j ∂xi∂xj j=1

k k ( ··· ) = ( ··· ) = 1 ··· m where lim(x ,··· ,xm)→(a ,··· ,am) s x1, , xm 0. Set x1, , xm , , , multiplying both sides 1 1 n nm ! 1 n m i ki ni−ki by ∏i=1 ai (1 − ai) and sum from ki = 0 to ki = ni, for all i = 1, ··· , m, we obtain ki Mathematics 2020, 8, 1397 29 of 34

Bn1,··· ,nm ( f ; a1, ··· , am) m n n ! m ! ∂ f 1 m kj n i ki ni −ki = f (a1, ··· , am) + ∑ (a1, ··· , am) ∑ ··· ∑ − aj ∏ ai (1 − ai) ∂xj nj j=1 k1=0 km=0 i=1 ki 2 m 2 n1 nm ! m ! 1 ∂ f kj ni k + ( ··· ) ··· − i ( − )ni −ki ∑ 2 a1, , am ∑ ∑ aj ∏ ai 1 ai 2 ∂x nj j=1 j k1=0 km=0 i=1 ki

m 2 n1 nm ! m ! 1 ∂ f k kj nl k + ( ··· ) ··· i − − l ( − )nl −kl ∑ a1, , am ∑ ∑ ai aj ∏ al 1 al 2 ∂xi∂xj ni nj i,j=1,i6=j k1=0 km=0 l=1 kl 2 n1 nm m ! m ! k k kj n 1 m i ki ni −ki + ∑ ··· ∑ s , ··· , ∑ − aj ∏ ai (1 − ai) n1 nm nj k1=0 km=0 j=1 i=1 ki  n ! !  m j k ∂ f j nj kj nj−kj = f (a1, ··· , am) + ∑ (a1, ··· , am)  ∑ − aj aj 1 − aj  ∂xj nj j=1 kj=0 kj ! n1 nm m n 1 m ∂2 f i ki ni −ki × ··· a (1 − ai) + (a1, ··· , am) ∑ ∑ ∏ i 2 ∑ ∂x2 k1=0,16=j km=0,m6=j i=1,i6=j ki j=1 j  2  nj ! ! n n m ! kj n k 1 m n j j nj−kj i ki ni −ki ×  ∑ − aj aj (1 − aj)  ∑ ··· ∑ ∏ ai (1 − ai) nj kj=0 kj k1=0,16=j km=0,m6=j i=1,i6=j ki  m 2 ni ! ! nj ! ! 1 ∂ f k n kj n i i ki ni −ki j + ∑ (a1, ··· , am) ∑ − ai ai (1 − ai)  ∑ − aj 2 ∂xi∂xj ni nj i,j=1,i6=j ki =0 ki kj=0 kj

n1 nm m ! kj − nl k × ( − )nj kj ··· l ( − )nl −kl aj 1 aj ∑ ∑ ∏ al 1 al k1=0,16=i,j km=0,m6=i,j l=1,l6=i,j kl  2  m n1 nm ! m ! k k kj n 1 m i ki ni −ki + ∑  ∑ ··· ∑ s , ··· , − aj ∏ ai (1 − ai)  . (134) n1 nm nj j=1 k1=0 km=0 i=1 ki

Let us use the following results from P. J. Davis [1] which are valid for all x ∈ [0, 1]:

 n !  n k k n−k  ∑ − x x (1 − x) = 0;  = k n  k 0 !  n 2 ( − )  n k k n−k x 1 x  ∑ − x x (1 − x) = ;  = k n n k 0 ! (135) n n k 3 −2x + 1  − k( − )n−k = ( − )  ∑ x x 1 x x 1 x 2 ;  = k n n  k 0 !  n n k 4 (3n − 6)x(1 − x) + 1  − k( − )n−k = ( − )  ∑ x x 1 x x 1 x 3 .  k=0 k n n

The second equation in system (135) can also be found in (Reference [33] Corollary 5.3). Hence, with Equations (31) and (135), we have

Bn1,··· ,nm ( f ; a1, ··· , am) m 2 aj(1 − aj) ∂ f = ( ··· ) + ( ··· ) f a1, , am ∑ 2 a1, , am j=1 2nj ∂xj  2  m n1 nm ! m ! k k kj n 1 m i ki ni−ki + ∑  ∑ ··· ∑ s , ··· , − aj ∏ ai (1 − ai)  . (136) n1 nm nj ki j=1 k1=0 km=0 i=1 Mathematics 2020, 8, 1397 30 of 34

Let  2  m n1 nm ! m ! k k kj n 1 m i ki ni−ki S = ∑  ∑ ··· ∑ s , ··· , − aj ∏ ai (1 − ai)  (137) n1 nm nj ki j=1 k1=0 km=0 i=1 1 ε > n i = ··· m |x − a | < and 0. We can find i, 1, , sufficiently large such that i i 1/4 implies ni |s(x ··· x )| ≤ ε s(x ··· x ) = 1, , m since lim(x1,··· ,xm)→(a1,··· ,am) 1, , m 0. Let us consider the set m E = ∏i=1{0, ··· , ni}, and for j = 1, ··· , m, define the sets    kj 1  Ω = k ∈ {0, ··· , n } : − a < and F = E \ (Ω × · · · × Ω ). (138) j j j n j 1/4 1 m  j nj  n o Sm m [αik] m [αik] Then F = k=1 Fk, with Fk = ∏i=1 Ωik ∈ F : αik ∈ {0, 1}, ∑i=1 αik = k , where Ωik = ( ( ) Ωi if α = 0 ki 1 [α ] ik and Ωc = k ∈ {0, ··· , n } : − x ≥ . For A = ∏m Ω ik ∈ F , Ωc if α = 1 i i i n i 1/4 k i=1 ik k i ik i ni = ··· = { ∈ { ··· } = } = ··· k 1, , m, let us also define IAk i 1, , m : αik 1 for k 1, , m (that means = ≥ card IAk k 1). We have  m !2 m ! k k kj n 1 m i ki ni−ki |S| ≤ ∑ ∑ ··· ∑ s , ··· , − aj ∏ ai (1 − ai) n1 nm nj ki j=1 Ω1 Ωm i=1  !2 m ! k k kj n 1 m i ki ni−ki + ∑ s , ··· , − aj ∏ ai (1 − ai)  F n1 nm nj i=1 ki

  m n n !2 m ! 1 m kj n i ki ni−ki ≤ ε ∑  ∑ ··· ∑ − aj ∏ ai (1 − ai)  nj ki j=1 k1=0 km=0 i=1 ! # ! m m n m i ki ni−ki [ + M ∑ ∑ ∏ ai (1 − ai) ) since F = Fl ki l=1 Al ∈Fl i=1 l=1 m 2 where M = sup m |s(x , ··· , x )| (x − a ) (x1,··· ,xm)∈[0,1] 1 m ∑ j j j=1  2  m nj ! ! n k k 1 j nj j nj−kj ≤ ε ∑  ∑ − aj aj (1 − aj)  ∑ ··· nj kj j=1 kj=0 k1=0,16=j

nm m ! ! m n k C i a i ( − a )ni−ki + M i ∑ ∏ i 1 i ∑ ∑ ∏ 3/2 , (139) k =0,m6=j i=1,i6=j ki l=1 A ∈F i∈I n m l l Al i with Ci constants (we bounded the summation on F using Equation (129) and the same procedure as we have used to show (55) in Theorem 2. Hence, Mathematics 2020, 8, 1397 31 of 34

 2  m nj ! ! m kj n k C |S| ≤ ε − a j a j ( − a )nj−kj + M ∑  ∑ j j 1 j  ∑ ∑ ∏ 3/2 j=1 k =0 nj kj l=1 A ∈F i∈I n j l l Al using (31), and n = min{nj, 1 ≤ j ≤ m} ≤ ni, i = 1, ··· , m and C = max(Ci : i = 1, ··· , m) m ! m l aj(1 − aj) C ≤ ε + M using Equation (135) and card I = l ∑ ∑ ∑ 3l/2 Al nj n j=1 l=1 Al ∈Fl ε m 1 m M × Cl ≤ a (1 − a ) + since n3l/2 > n3/2 . (140) n ∑ j j n ∑ ∑ n1/2 j=1 l=1 Al ∈Fl

l m M × C For n , ··· , nm sufﬁciently large (that means n sufﬁciently large), we have ∑ ∑ < ε. 1 l=1 Al ∈Fl n1/2 This implies ! ε m |S| ≤ 1 + ∑ aj(1 − aj) . (141) n j=1 Therefore,

m 2 aj(1 − aj) ∂ f B ( f a ··· a ) − f (a ··· a ) − (a ··· a ) n1,··· ,nm ; 1, , m 1, , m ∑ 2 1, , m j=1 2nj ∂xj ! ε m = |S| ≤ 1 + ∑ aj(1 − aj) . (142) n j=1

Since ε is arbitrary, we have

" m 2 # aj(1 − aj) ∂ f lim Bn ,··· ,nm ( f ; a1, ··· , am) − f (a1, ··· , am) − (a1, ··· , am) = 0. n →∞, 1≤j≤m 1 ∑ 2 j j=1 2nj ∂xj

Remark 3. By taking ni = N, 1 ≤ i ≤ m, in Theorem 6, we have

1 m ∂2 f lim N (BN,··· ,N( f ; a1, ··· , am) − f (a1, ··· , am)) = ai(1 − ai) (a1, ··· , am). (143) N→∞ ∑ 2 2 i=1 ∂xi

From the previous Remark, one can deduce that the order of convergence of Bernstein polynomials m with several variables for a bounded function f (x1, ··· , xm), (x1, ··· , xm) ∈ [0, 1] , in some neighborhood m m (2) of (a1, ··· , am) ∈ [0, 1] included in [0, 1] , is one, provided that f (a1, ··· , am) exists. That is less than Taylor’s order of convergence.

5. Conclusions Our concern here was the multivariate Bernstein polynomials given by Equation (6) which we recall here

n1 nm m ! k km n k ( ··· ) = ··· 1 ··· i i ( − )ni−ki Bn1,··· ,nm f ; x1, , xm ∑ ∑ f , , ∏ xi 1 xi , (144) n1 nm ki k1=0 km=0 i=1 and which is a generalization in several variables of the univariate Bernstein polynomials given by (1) Mathematics 2020, 8, 1397 32 of 34

n ! k n k n−k Bn( f ; x) = ∑ f x (1 − x) . (145) k=0 n k We extended the beautiful properties of the above mentioned univariate Bernstein polynomials by generalizing them for functions of several real variables: uniform convergence, uniform convergence of the derivatives, order of convergence, monotonicity, ﬁxed sign for the p-th derivative, and deduction of the upper and lower bounds of the Bernstein polynomials from those of the corresponding functions. To achieve that, we ﬁrst recalled not only these properties for univariate Bernstein polynomials, but also the intermediate properties used to prove them (see Equations (11)–(16)). Then, we extended these intermediate properties for the univariate case to obtain the generalised intermediate properties for the multivariate case given by (6) (see (20)–(26)). Finally, using those generalized intermediate properties in combination with some others tools related to functions with several variables, we got the generalization of these beautiful properties of the univariate Bernstein polynomials to multivariate Bernstein polynomials given by Equation (6). More precisely, our contributions in this paper related to this form of multivariate Bernstein polynomials can be summarized as follows:

1. deriving the expression of these multivariate Bernstein polynomials in terms of differences (similar to the case of one variable in References [1,15]); 2. establishing the pointwise and uniform convergence of these Bernstein polynomials with several variables (a generalization of the case of two variables treated in (Reference [28] Lemma 1)); 3. deriving the expression of the (p1, ··· , pm)-th derivative of these multivariate Bernstein polynomials of order (n1 + p1, ··· , nm + pm)-th in terms of (p1, ··· , pm)-th differences of the corresponding functions (similar to the case of one variable in References [1,15]); 4. proving the monotonicity, the ﬁxed sign of the (p1, ··· , pm)-th derivative, the convexity, and the upper and lower bounds of these multivariate Bernstein polynomials from those of the corresponding functions (similar to the case of one variable in References [1,31]); 5. establishing the pointwise and uniform convergence of the (p1, ··· , pm)-th derivative of these multivariate Bernstein polynomials (a generalization of the case of two variables treated in (Reference [35] Theorem of Kingsley) and (Reference [28] Theorem 1)); 6. generalizing the inequality related to the decreasing of the sequence of Bernstein’s polynomials with one variable (see References [1,31]) when f is a convex function; 7. proving that the order of convergence of these Bernstein polynomials with several variables is one.

With these new results, we have proved that the conclusion made by P.J. Davis in Reference [1] for Bernstein polynomials of one variable is true for the case of several variables, that is, the Bernstein polynomials of several variables yield smooth approximants, but their convergence is very slow.

Author Contributions: The conceptualization and the supervision were done by the first author (M.F.). The formal analysis and the investigation were done by the second author (M.M.W.). All authors have read and agreed to the published version of the manuscript. Funding: The work of these two authors was initially funded by the African Institute for Mathematical Sciences- Cameroon, during the academic year 2013/2014. Acknowledgments: The authors would like to acknowledge the support of the two institutions where this work has been carried out: The Mathematics Institute of the University of Kassel in Germany and the African Institute for Mathematical Sciences in Cameroon. Conflicts of Interest: The authors declare that they have no known competing financial interests or personal relation-ships that could have appeared to influence the work reported in this paper. Mathematics 2020, 8, 1397 33 of 34

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