Iterates of multivariate Cheney-Sharma operators

Diana Otrocol
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Abstract

Using the weakly Picard operators technique, we study the convergence of the iterates of some hivariate and trivariate Cheney-Sharma operators. Also, we generalize the procedure for the multivariate case.

Authors

T. Catinas
(Babes Bolyai Univ.)

D. Otrocol
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Diana Otrocol

Keywords

Cheney-Sharma operators; contraction principle; weakly Picard operators

Cite this paper as:

T. Catinas, D.  Otrocol,  Iterates of multivariate Cheney-Sharma operators, J. Comput. Anal. Appl., Vol. 15 (2013), no. 7, pp. 1240-1246

PDF

https://ictp.acad.ro/otrocol/papers/2013-CatinasOtrocol-JOCAA-Iterates%20of%20multivariate%20Cheney-Sharma%20operators.pdf

About this paper

Journal

Journal of Computational Analysis and Applications

Publisher Name

Eudoxus Press, Cordova, USA

DOI
Print ISSN

1521-1398

Online ISSN
MR

MR3075657

ZBL

Google Scholar

https://scholar.google.ro/scholar?oi=bibs&hl=en&cites=15905832510231040605

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Iterates of multivariate Cheney-Sharma operators

Teodora Cătinaş and Diana Otrocol

Abstract. Using the weakly Picard operators technique, we study the convergence of the iterates of some bivariate and trivariate Cheney-Sharma operators. Also, we generalize the procedure for the multivariate case.

Keywords: Cheney-Sharma operators, contraction principle, weakly Picard operators.

2000 Mathematics Subject Classification: 41A36, 41A05, 41A25, 39B12, 47H10.

1. Preliminaries

We recall some results regarding weakly Picard operators that will be used in the sequel (see, e.g., [17], [20]).

Let (X,d)(X,d) be a metric space and A:X→XA:X\rightarrow X an operator. We denote by

FA\displaystyle F_{A} :={x∈X|A​(x)=x}​-the fixed point set of ​A​;\displaystyle:=\{x\in X~|~A(x)=x\}\text{-the fixed point set of }A\text{;}
I​(A)\displaystyle I(A) :={Y⊂X|A​(Y)⊂Y,Y≠∅}​-the family of the nonempty invariant\displaystyle:=\{Y\subset X~|~A(Y)\subset Y,\ Y\neq\emptyset\}\text{-the family of the nonempty invariant }
subset of ​A\displaystyle\text{subset of }A
A0\displaystyle A^{0} :=1X,A1:=A,…,An+1:=A∘An,n∈ℕ​.\displaystyle:=1_{X},\ A^{1}:=A,\ ...,\ A^{n+1}:=A\circ A^{n},\ \ n\in\mathbb{N}\text{.}
Definition 1.1.

The operator A:X→XA:X\rightarrow X is a Picard operator if there exists x∗∈Xx^{\ast}\in X such that:

(i) FA={x∗};F_{A}=\{x^{*}\};

(ii) the sequence (An​(x0))n∈ℕ(A^{n}(x_{0}))_{n\in\mathbb{N}} converges to x∗x^{*} for all x0∈Xx_{0}\in X.

Definition 1.2.

The operator AA is a weakly Picard operator if the sequence (An​(x))n∈ℕ(A^{n}(x))_{n\in\mathbb{N}} converges, for all x∈Xx\in X, and the limit (which may depend on xx) is a fixed point of AA.

Definition 1.3.

We define the operator A∞,A∞:X→XA^{\infty},\;A^{\infty}:X\rightarrow X, by

A∞​(x):=limn→∞​An​(x).A^{\infty}(x):=\underset{n\rightarrow\infty}A^{n}(x).
Theorem 1.4.

[17] An operator AA is a weakly Picard operator if and only if there exists a partition of X,X, X=⋃λ∈ΛXλ,X={\textstyle\bigcup\limits_{\lambda\in\Lambda}}X_{\lambda}, such that

  • (a)

    Xλ∈I​(A),X_{\lambda}\in I(A), ∀λ∈Λ;\forall\lambda\in\Lambda;

  • (b)

    A|Xλ:Xλ→Xλ\left.A\right|_{X_{\lambda}}:X_{\lambda}\rightarrow X_{\lambda} is a Picard operator, ∀λ∈Λ.\forall\lambda\in\Lambda.

2. Cheney-Sharma operator

In [21] there was given an extension to two variables of the second univariate operator of Cheney-Sharma introduced in [5].

Let ff be a real-valued function defined on D=[0,1]×[0,1].D=[0,1]\times[0,1]. The bivariate Cheney-Sharma operator is defined by

(Sm,n​f)​(x,y;β,b)=∑i=0m∑j=0npm,i​(x;β)​qn,j​(y;b)​f​(im,jn),\left(S_{m,n}f\right)\left(x,y;\beta,b\right)=\sum_{i=0}^{m}\sum_{j=0}^{n}p_{m,i}\left(x;\beta\right)q_{n,j}(y;b)f\left(\tfrac{i}{m},\tfrac{j}{n}\right), (1)

with

pm,i​(x;β)=(mi)​x​(x+i​β)i−1​(1−x)​[1−x+(m−i)​β]m−i−1(1+m​β)m−1,p_{m,i}\left(x;\beta\right)=\frac{\binom{m}{i}x(x+i\beta)^{i-1}(1-x)\left[1-x+(m-i)\beta\right]^{m-i-1}}{(1+m\beta)^{m-1}},

and

qn,j​(y;b)=(nj)​y​(y+j​b)j−1​(1−y)​[1−y+(n−j)​b]n−j−1(1+n​b)n−1,q_{n,j}\left(y;b\right)=\frac{\binom{n}{j}y(y+jb)^{j-1}(1-y)\left[1-y+(n-j)b\right]^{n-j-1}}{(1+nb)^{n-1}},

where β\beta and bb are nonnegative parameters.

For a function ff defined on D1=[0,1]×[0,1]×[0,1],D_{1}=[0,1]\times[0,1]\times[0,1], the trivariate operator Cheney-Sharma is defined by [22]

(Sm,n,l​f)​(x,y,z;β,γ,δ)=∑i=0m∑j=0n∑k=0lpm,i​(x;β)​qn,j​(y;γ)​rl,k​(z;δ)​f​(im,jn,kr),\left(S_{m,n,l}f\right)\left(x,y,z;\beta,\gamma,\delta\right)=\sum_{i=0}^{m}\sum_{j=0}^{n}\sum_{k=0}^{l}p_{m,i}\left(x;\beta\right)q_{n,j}(y;\gamma)r_{l,k}(z;\delta)f\left(\tfrac{i}{m},\tfrac{j}{n},\tfrac{k}{r}\right), (2)

with

pm,i​(x;β)=(mi)​x​(x+i​β)i−1​(1−x)​[1−x+(m−i)​β]m−i−1(1+m​β)m−1,p_{m,i}\left(x;\beta\right)=\frac{\binom{m}{i}x(x+i\beta)^{i-1}(1-x)\left[1-x+(m-i)\beta\right]^{m-i-1}}{(1+m\beta)^{m-1}},
qn,j​(y;γ)=(nj)​y​(y+j​γ)j−1​(1−y)​[1−y+(n−j)​γ]n−j−1(1+n​γ)n−1,q_{n,j}\left(y;\gamma\right)=\frac{\binom{n}{j}y(y+j\gamma)^{j-1}(1-y)\left[1-y+(n-j)\gamma\right]^{n-j-1}}{(1+n\gamma)^{n-1}},

and

rl,k​(z;δ)=(lk)​z​(z+k​δ)k−1​(1−z)​[1−z+(l−k)​δ]l−k−1(1+l​δ)l−1r_{l,k}\left(z;\delta\right)=\frac{\binom{l}{k}z(z+k\delta)^{k-1}(1-z)\left[1-z+(l-k)\delta\right]^{l-k-1}}{(1+l\delta)^{l-1}}

where β,γ\beta,\gamma and δ\delta are nonnegative parameters. This operator represents an extension to three variables of the second univariate operator of Cheney-Sharma [5].

Theorem 2.1.

[21] If ff is a real-valued function defined on DD\ then we have

(Sm,n​ei​j)​(x,y)=xi​yj,i,j=0,1,\left(S_{m,n}e_{ij}\right)\left(x,y\right)=x^{i}y^{j},\ \ i,j=0,1,

and therefore, span{e00,e10,e01,e11}⊂FSm,n,\operatorname*{span}\{e_{00},e_{10},e_{01},e_{11}\}\subset F_{S_{m,n}}, where FSm,nF_{S_{m,n}} denotes the fixed points set of Sm,n.S_{m,n}.

Theorem 2.2.

[22] If ff is a real-valued function defined on D1D_{1}\ then we have

(Sm,n,l​ei​j​k)​(x,y,z)=xi​yj​zk,i,j,k∈{0,1},\left(S_{m,n,l}e_{ijk}\right)\left(x,y,z\right)=x^{i}y^{j}z^{k},\ \ i,j,k\in\{0,1\},

and therefore, span{e000,e100,e001,e001,e110,e011,e101,e111}⊂FSm,n,l,\operatorname*{span}\{e_{000},e_{100},e_{001},e_{001},e_{110},e_{011},e_{101},e_{111}\}\subset F_{S_{m,n,l}}, where FSm,n,lF_{S_{m,n,l}} denotes the fixed points set of Sm,n,l.S_{m,n,l}.

3. Iterates of Cheney-Sharma operator

Using the weakly Picard operators technique and the contraction principle, we study the convergence of the iterates of the bivariate Cheney-Sharma operator given in (1).

A similar approach for the univariate case was given in [4]. Some other linear and positive operators lead to similar results in [1], [2], [7], [18] and [19]. The limit behavior for the iterates of some classes of positive linear operators were also studied, for example, in [3], [8]-[16].

Let ff be a real-valued function defined on D.D.

Theorem 3.1.

The operator Sm,nS_{m,n} is a weakly Picard operator and

(Sm,n∞​f)​(x,y;β,b)=\displaystyle\left(\mathit{S_{m,n}^{\infty}}f\right)\left(x,y;\beta,b\right)= (1−x)​(1−y)​f​(0,0)+(1−x)​y​f​(1,0)\displaystyle(1-x)(1-y)f\left(0,0\right)+(1-x)yf(1,0) (3)
+x​(1−y)​f​(0,1)+x​y​f​(1,1).\displaystyle+x(1-y)f(0,1)+xyf(1,1).
Proof.

Taking into account the interpolation properties (Theorem 2.1), of Sm,nS_{m,n}, consider

Xα1,α2,α3,α4={f∈C​(D)|f​(0,0)=α1,f​(1,0)=α2,f​(0,1)=α3,f​(1,1)=α4},X_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}=\{f\in C(D)\ |\ f\left(0,0\right)=\alpha_{1},f(1,0)=\alpha_{2},f(0,1)=\alpha_{3},f(1,1)=\alpha_{4}\},\ (4)

 and denote by

fα1,α2,α3,α4∗​(x,y):=(1−x)​(1−y)​α1+(1−x)​y​α2+x​(1−y)​α3+x​y​α4,f_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}^{\ast}(x,y):=(1-x)(1-y)\alpha_{1}+(1-x)y\alpha_{2}+x(1-y)\alpha_{3}+xy\alpha_{4},

with α1,α2,α3,α4∈ℝ.\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\in\mathbb{R}.

We have the following properties:

  • (i)

    Xα1,α2,α3,α4X_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}} is closed subset of C​(D)C(D);

  • (ii)

    Xα1,α2,α3,α4X_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}} is an invariant subset of Sm,n\mathit{S_{m,n}}, for α1,α2,α3,α4∈ℝ,m,n∈ℕ+;\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\in\mathbb{R},\ m,n\in\mathbb{N}_{+};

  • (iii)

    C​(D)=∪α1,α2,α3,α4∈ℝ​Xα1,α2,α3,α4C(D)=\underset{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\in\mathbb{R}}{\cup}X_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}} is a partition of C​(D)C(D);

  • (iv)

    Xα1,α2,α3,α4∩FSm,n={fα1,α2,α3,α4∗}.X_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}\cap F_{S_{m,n}}=\{f_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}^{\ast}\}.

The statements (i)(i) and (i​i​i)(iii) are obvious.

(i​i)(ii) By interpolation properties of Sm,nS_{m,n} we have that Xα1,α2,α3,α4X_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}} is an invariant subset of Sm,n,S_{m,n},\ for any α1,α2,α3,α4∈ℝ,m,n∈ℕ+;\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\in\mathbb{R},\ m,n\in\mathbb{N}_{+};

(i​v)(iv) We prove that

Sm,n|Xα1,α2,α3,α4:Xα1,α2,α3,α4→Xα1,α2,α3,α4\left.S_{m,n}\right|_{X_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}}\!:\!X_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}\!\rightarrow X_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}

is a contraction for α1,α2,α3,α4∈ℝ,m,n∈ℕ+.\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\in\mathbb{R},\ m,n\in\mathbb{N}_{+}.

Let f,g∈Xα1,α2,α3,α4f,g\in X_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}. From (1) and (4) we obtain

|Sm,n​(f)​(x,y)−Sm,n​(g)​(x,y)|=\displaystyle\left|S_{m,n}(f)(x,y)-S_{m,n}(g)(x,y)\right|=
=|Sm,n​(f−g)​(x,y)|≤\displaystyle=\left|S_{m,n}(f-g)(x,y)\right|\leq
≤|pm,0​(x;β)​qn,0​(y;b)​[f​(0,0)−g​(0,0)]|\displaystyle\leq\left|p_{m,0}\left(x;\beta\right)q_{n,0}(y;b)\left[f\left(0,0\right)-g(0,0)\right]\right|
+|∑i=1m∑j=1npm,i​(x;β)​qn,j​(y;b)​[f​(im,jn)−g​(im,jn)]|\displaystyle\quad+\left|\sum_{i=1}^{m}\sum_{j=1}^{n}p_{m,i}\left(x;\beta\right)q_{n,j}(y;b)\left[f\left(\tfrac{i}{m},\tfrac{j}{n}\right)-g\left(\tfrac{i}{m},\tfrac{j}{n}\right)\right]\right|
=∑i=1m∑j=1npm,i​(x;β)​qn,j​(y;b)​|f​(im,jn)−g​(im,jn)|\displaystyle=\sum_{i=1}^{m}\sum_{j=1}^{n}p_{m,i}\left(x;\beta\right)q_{n,j}(y;b)\left|f\left(\tfrac{i}{m},\tfrac{j}{n}\right)-g\left(\tfrac{i}{m},\tfrac{j}{n}\right)\right|
≤∑i=1mpm,i​(x;β)​∑j=1nqn,j​(y;b)​‖f−g‖∞\displaystyle\leq\sum_{i=1}^{m}p_{m,i}\left(x;\beta\right)\sum_{j=1}^{n}q_{n,j}(y;b)\left\|f-g\right\|_{\infty}
=[∑i=0mpm,i​(x;β)−pm,0​(x;β)]​[∑j=0nqn,j​(y;b)−qn,0​(y;b)]​‖f−g‖∞\displaystyle=\left[\sum_{i=0}^{m}p_{m,i}\left(x;\beta\right)-p_{m,0}\left(x;\beta\right)\right]\left[\sum_{j=0}^{n}q_{n,j}(y;b)-q_{n,0}(y;b)\right]\left\|f-g\right\|_{\infty}
=[1−(1−x1+m​β)m−1]​[1−(1−y1+n​b)n−1]​‖f−g‖∞\displaystyle=\left[1-\left(1-\tfrac{x}{1+m\beta}\right)^{m-1}\right]\left[1-\left(1-\tfrac{y}{1+nb}\right)^{n-1}\right]\left\|f-g\right\|_{\infty}
≤[1−(1−11+m​β)m−1]​[1−(1−11+n​b)n−1]​‖f−g‖∞.\displaystyle\leq\left[1-\left(1-\tfrac{1}{1+m\beta}\right)^{m-1}\right]\left[1-\left(1-\tfrac{1}{1+nb}\right)^{n-1}\right]\left\|f-g\right\|_{\infty}.

where ∥⋅∥∞\left\|\cdot\right\|_{\infty} denotes the Chebyshev norm.

From [2, Lemma 8] it follows that

|Sm,n​(f)​(x,y)−Sm,n​(g)​(x,y)|=\displaystyle\left|S_{m,n}(f)(x,y)-S_{m,n}(g)(x,y)\right|=
≤[1−(1−11+m​β)m−1​(1−11+n​b)n−1]​‖f−g‖∞.\displaystyle\leq\left[1-\left(1-\tfrac{1}{1+m\beta}\right)^{m-1}\left(1-\tfrac{1}{1+nb}\right)^{n-1}\right]\left\|f-g\right\|_{\infty}.

So,

‖Sm,n​(f)​(x,y)−Sm,n​(g)​(x,y)‖∞\displaystyle\left\|S_{m,n}(f)(x,y)-S_{m,n}(g)(x,y)\right\|_{\infty}
≤[1−(1−11+m​β)m−1​(1−11+n​b)n−1]​‖f−g‖∞,∀f,g∈Xα1,α2,α3,α4,\displaystyle\leq\left[1-\left(1-\tfrac{1}{1+m\beta}\right)^{m-1}\left(1-\tfrac{1}{1+nb}\right)^{n-1}\right]\left\|f-g\right\|_{\infty},\ \forall f,g\in X_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}},

i.e., Sm​n|Xα1,α2,α3,α4\left.S_{mn}\right|_{X_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}} is a contraction for α1,α2,α3,α4∈ℝ\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\in\mathbb{R}.

On the other hand, we have that

fα1,α2,α3,α4∗​(x,y):=(1−x)​(1−y)​α1+(1−x)​y​α2+x​(1−y)​α3+x​y​α4f_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}^{\ast}(x,y):=(1-x)(1-y)\alpha_{1}+(1-x)y\alpha_{2}+x(1-y)\alpha_{3}+xy\alpha_{4}

and

Sm,n\displaystyle S_{m,n} ((1−x)​(1−y)​α1+(1−x)​y​α2+x​(1−y)​α3+x​y​α4)=\displaystyle\left((1-x)(1-y)\alpha_{1}+(1-x)y\alpha_{2}+x(1-y)\alpha_{3}+xy\alpha_{4}\right)=
=(1−x)​(1−y)​α1+(1−x)​y​α2+x​(1−y)​α3+x​y​α4.\displaystyle=(1-x)(1-y)\alpha_{1}+(1-x)y\alpha_{2}+x(1-y)\alpha_{3}+xy\alpha_{4}.

From the contraction principle we have that fα1,α2,α3,α4∗f_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}^{\ast} is the unique fixed point of Sm,nS_{m,n} in Xα1,α2,α3,α4X_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}} and Sm,n|Xα1,α2,α3,α4\left.S_{m,n}\right|_{X_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}} is a Picard operator, so (3) holds. Consequently, taking into account (i​i)(ii), by Theorem 1.4 it follows that the operator Sm,nS_{m,n} is a weakly Picard operator. We remark that FSm,n=span{e00,e10,e01,e11}.F_{S_{m,n}}=\operatorname*{span}\{e_{00},e_{10},e_{01},e_{11}\}. ∎

Next, we study the convergence of the iterates of the trivariate Cheney-Sharma operator given in (2).

Let ff be a real-valued function defined on D1.D_{1}.

Theorem 3.2.

The operator Sm,n,lS_{m,n,l} is a weakly Picard operator and

(Sm,n,l∞​f)​(x,y,z;β,γ,δ)=\displaystyle\left(\mathit{S_{m,n,l}^{\infty}}f\right)\left(x,y,z;\beta,\gamma,\delta\right)= (5)
=(1−x)​(1−y)​(1−z)​f​(0,0,0)+x​(1−y)​(1−z)​f​(1,0,0)\displaystyle=(1-x)(1-y)(1-z)f\left(0,0,0\right)+x(1-y)(1-z)f(1,0,0)
+(1−x)​y​(1−z)​f​(0,1,0)+(1−x)​(1−y)​z​f​(0,0,1)+x​y​(1−z)​f​(1,1,0)\displaystyle\quad+(1-x)y(1-z)f(0,1,0)+(1-x)(1-y)zf(0,0,1)+xy(1-z)f(1,1,0)
+x​(1−y)​z​f​(1,0,1)+(1−x)​y​z​f​(0,1,1)+x​y​z​f​(1,1,1).\displaystyle\quad+x(1-y)zf(1,0,1)+(1-x)yzf(0,1,1)+xyzf(1,1,1).
Proof.

The proof follows the same steps as in Theorem 3.1. Using the following inequality

|Sm,n,l​(f)​(x,y,z)−Sm,n,l​(g)​(x,y,z)|≤\displaystyle\left|S_{m,n,l}(f)(x,y,z)-S_{m,n,l}(g)(x,y,z)\right|\leq
≤[1−(1−11+m​β)m−1]​[1−(1−11+n​γ)n−1]​[1−(1−11+l​δ)l−1]​‖f−g‖∞,\displaystyle\leq\left[1-\left(1-\tfrac{1}{1+m\beta}\right)^{m-1}\right]\left[1-\left(1-\tfrac{1}{1+n\gamma}\right)^{n-1}\right]\left[1-\left(1-\tfrac{1}{1+l\delta}\right)^{l-1}\right]\left\|f-g\right\|_{\infty},

and further [2, Lemma 8]

‖Sm,n,l​(f)​(x,y,z)−Sm,n,l​(g)​(x,y,z)‖∞≤\displaystyle\left\|S_{m,n,l}(f)(x,y,z)-S_{m,n,l}(g)(x,y,z)\right\|_{\infty}\leq
≤[1−(1−11+m​β)m−1​(1−11+n​γ)n−1​(1−11+l​δ)l−1]​‖f−g‖∞,\displaystyle\leq\left[1-\left(1-\tfrac{1}{1+m\beta}\right)^{m-1}\left(1-\tfrac{1}{1+n\gamma}\right)^{n-1}\left(1-\tfrac{1}{1+l\delta}\right)^{l-1}\right]\left\|f-g\right\|_{\infty},\

∀f,g∈Xα1,α2,α3,α4,\forall f,g\in X_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}, we prove that Sm,n,l​ S_{m,n,l\text{ }}is a contraction. ∎

We generalize these results to multivariate case.

Theorem 3.3.

Consider a function f∈C​(Dp),f\in C(D_{p}), with Dp=[0,1]×…p​t​i​m​e​s×[0,1].D_{p}=[0,1]\times\underset{p\ times}{...}\times[0,1]. The pp-variate Cheney-Sharma operator, denoted by Si1,…,ip,S_{i_{1},...,i_{p}}, is a weakly Picard operator and

(Si1,…,ip∞​f)​(x1,…,xp)=∑αi∈{0,1},i=1,p¯si1,…,ip∞​(x1,…,xp)​f​(α1,…,αp),\left(S_{i_{1},...,i_{p}}^{\infty}f\right)\left(x_{1},...,x_{p}\right)=\sum_{\alpha_{i}\in\{0,1\},i=\overline{1,p}}s_{i_{1},...,i_{p}}^{\infty}\left(x_{1},...,x_{p}\right)f(\alpha_{1},...,\alpha_{p}), (6)

where αi∈{0,1},\alpha_{i}\in\{0,1\}, i=1,…,pi=1,...,p and

si1,…,ip∞(x1,…,xp)=x1α1⋅…⋅xpαp(1−x1)(1−α1)⋅...⋅(1−xp)(1−αp).s_{i_{1},...,i_{p}}^{\infty}\left(x_{1},...,x_{p}\right)=x_{1}^{\alpha_{1}}\cdot...\cdot x_{p}^{\alpha_{p}}(1-x_{1})^{(1-\alpha_{1})}\cdot...\cdot(1-x_{p})^{(1-\alpha_{p})}.
Proof.

The proof follows the same steps as in Theorem 3.1. ∎

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2013

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