Iterates of multivariate Cheney-Sharma operators

Diana Otrocol
(original), paper

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)

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

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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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[21] D.D. Stancu, L.A. Cabulea, D. Pop, Approximation of bivariate functions by means of the operator S α,β;a,b m,n , Stud. Univ. Babes–Bolyai Math., 47(2002), No. 4, 105-113.

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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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