On the iterates of uni- and multidimensional operators

Abstract

The problem of the convergence of the iterates of operators or of a sequence of operators is discussed in a general framework related to the fixed point theory, but with a permanent look towards the theory of linear approximation operators. The results are obtained for operators not necessarily linear. Some examples including the class of approximation operators defined by H. Brass are given.

Authors

Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Keywords

Bernstein type operator; contraction principle; Perov theorem.

Paper coordinates

R. Precup, On the iterates of uni-and multidimensional operators, Bulletin of the Transilvania University of Brasov Series III: Mathematics and Computer Science, 3(65) (2023) no. 2, pp. 143-152, https://doi.org/10.31926/but.mif.2023.3.65.2.12

PDF

About this paper

Journal

Bulletin of the Transilvania University of Brasov

Publisher Name

Transilvania University Press

Print ISSN

2810-2029

Online ISSN

2971-9763

google scholar link

Paper (preprint) in HTML form

1 Iterates of unidimensional operators

Bulletin of the Transilvania University of Braşov

Series III: Mathematics and Computer Science, Vol. 3(65), No. 1 - 2023, xx-xx

https://doi.org/10.31926/but.mif.2023.3.65.1.x


ON THE ITERATES OF UNI- AND MULTIDIMENSIONAL OPERATORS

Radu PRECUPβˆ—,111βˆ— Babeş-Bolyai University, Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, 400084 Cluj-Napoca, Romania & Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 400110 Cluj-Napoca, e-mail: r.precup@math.ubbcluj.ro


Abstract

The problem of the convergence of the iterates of operators or of a sequence of operators is discussed in a general framework related to the fixed point theory, but with a permanent look towards the theory of linear approximation operators. The results are obtained for operators not necessarily linear. Some examples including the class of approximation operators defined by H. Brass are given.

2000 Mathematics Subject Classification: 41A36, 47H10.

Key words: Bernstein type operator, contraction principle, Perov theorem.

Dedicated to Professor Radu Păltănea on his 70th anniversary

1 Iterates of unidimensional operators

1.1 Iterates of a single operator

Any discussion about the iterates of an operator should start with the notion of a contraction operator and Banach’s contraction principle.

Let (X,d) be a metric space and let L:X→X an operator. One says that L is Lipschitz continuous with Lipschitz constant λ if

d⁒(L⁒(x),L⁒(y))≀λ⁒d⁒(x,y)for all ⁒x,y∈X.

If Ξ»<1, then L is a contraction operator. Observe that if L is Lipschitz continuous with Lipschitz constant Ξ», then any iterate Lk of L, where Lk=L⁒(Lkβˆ’1)⁒(kβ‰₯1,L0⁒ being the identity operator ⁒I), is also Lipschitz continuous with Lipschitz constant Ξ»k. With this remark, we have the following proposition.

Theorem 1.

Let L:X→X be such that:

(i)

L is a contraction operator with Lipschitz constant Ξ»<1;

(ii)

L has the (unique) fixed point xβˆ—.

Then for every x∈X, the sequence (Lk⁒(x)) of the iterates of L calculated at x is convergent to xβˆ—.

Proof.

Clearly Lk⁒(xβˆ—)=xβˆ— for all k. Then

d⁒(Lk⁒(x),xβˆ—)=d⁒(Lk⁒(x),Lk⁒(xβˆ—))≀λk⁒d⁒(x,xβˆ—).

Since Ξ»<1, one has Ξ»kβ†’0 and then d⁒(Lk⁒(x),xβˆ—)β†’0 as kβ†’βˆž. Thus Lk⁒(x) converges to xβˆ— as claimed. ∎

This proposition is useful in case that the fixed point xβˆ— is known for a contraction operator L. This happens for all many linear approximation operators L:C⁒[0,1]β†’C⁒[0,1] leaving invariant the functions 1 and x, that is with L⁒(1)=1 and L⁒(x)=x. Then, restricted to a set

Xα,β:={f∈C⁒[0,1]:f⁒(0)=α,f⁒(1)=β},

the operator L maps XΞ±,Ξ² into itself and the function

fβˆ—β’(x)=Ξ±+(Ξ²βˆ’Ξ±)⁒x

belongs to XΞ±,Ξ² and is a fixed point of L as can easily be seen. It remains to be clarified the contraction property of L. This aspect in the context of the theory of linear approximation operators was first highlighted by Rus [15] for Bernstein’s operators. The contraction property was later highlighted for other well-known classes of operators (see, e.g., [1], [2], [3], [6], [7]).

The proposition above is given in the idea of a priori knowledge of the fixed point. For a contraction operator on a complete metric space, the existence and uniqueness of the fixed point is guaranteed by Banach’s contraction principle which in addition gives an estimate of the approximation error, namely

d⁒(Lk⁒(x),xβˆ—)≀λk1βˆ’Ξ»β’d⁒(L⁒(x),x). (1)

In case of linear approximation operators, when metric d is given by the uniform norm βˆ₯.βˆ₯ of the space C⁒[0,1], this estimate reads as

β€–Lk⁒(f)βˆ’fβˆ—β€–β‰€Ξ»k1βˆ’Ξ»β’β€–L⁒(f)βˆ’fβ€– (2)

for all kβ‰₯1 and f∈XΞ±,Ξ². Thus the error β€–Lk⁒(f)βˆ’fβˆ—β€– of the approximation of fβˆ— by the k-th iterate is expressed by the error β€–L⁒(f)βˆ’fβ€– with which the arbitrary start function f is approximated by the operator L. This error is known for many operators beginning with Bernstein’s ones, in terms of some smoothness modules (see [13]).

1.2 Iterates of a sequence of operators

Assume now that (Ln)nβ‰₯1 is an approximation process in X, in the sense that

Ln⁒(x)β†’xas ⁒nβ†’βˆžfor every ⁒x∈X.

For such a sequence of operators, one has

Theorem 2.

Assume that every Ln is a contraction operator with a Lipschitz constant Ξ»n<1 and xβˆ— is the unique fixed point of all operators Ln. Then

(a)

Ξ»nβ†’1 as nβ†’βˆž;

(b)

For each x∈X, a sequence of iterates (Lnkn⁒(x))nβ‰₯1 converges to xβˆ— if (kn)nβ‰₯1 is such

Ξ»nkn1βˆ’Ξ»n≀C,for all ⁒nβ‰₯1 (3)

and some constant C.

Proof.

(a) Let Ξ» be any limit point of the sequence (Ξ»n)nβ‰₯1. Obviously λ≀1. Passing eventually to a subsequence, we may assume that Ξ»nβ†’Ξ». Choose two different element x,y∈X. From

d⁒(Ln⁒(x),Ln⁒(y))≀λn⁒d⁒(x,y),

letting nβ†’βˆž and using Ln⁒(x)β†’x and Ln⁒(y)β†’y, we deduce that d⁒(x,y)≀λ⁒d⁒(x,y). Since d⁒(x,y)>0, we find 1≀λ. Then Ξ»=1 and the proof of statement (a) is finished.

(b) Using (1) one has

d⁒(Lnkn⁒(x),xβˆ—)≀λnkn1βˆ’Ξ»n⁒d⁒(Ln⁒(x),x).

Here d⁒(Ln⁒(x),x)β†’0 as nβ†’βˆž, while from (3), the front coefficients are uniformly bounded. As a result d⁒(Lnkn⁒(x),xβˆ—)β†’0 as nβ†’βˆž, which is our statement. ∎

Remark 1.

(10) By virtue of (a), if knβ†’βˆž, then the limit of coefficient Ξ»nkn1βˆ’Ξ»n is 1∞+0=+βˆžβ‹…1∞. Hence a necessary condition for (3) to hold is that 1∞=0, i.e., Ξ»nknβ†’0.

(20) Condition (3) is only a sufficient condition for the convergence of the iterates Lnkn, not necessarily the best for particular sequences of operators. For example, in case of Bernstein’s operators, when Ξ»n=1βˆ’21βˆ’n, condition (3) returns to kn/2nβ†’βˆž as nβ†’βˆž. However, by the classical result of Kelinsky-Rivlin [11], the convergence is guaranteed if kn/n β†’βˆž.

1.3 Example: Iterates of Brass’s operators

Let nβˆˆβ„•βˆ–{0} and let ΞΌ1,ΞΌ2,…,ΞΌnβˆˆβ„• such that

ΞΌ1+2⁒μ2+…+n⁒μn=n.

Consider the polynomials

pn⁒ν⁒(x)=βˆ‘(ΞΌ1Ξ·1)⁒(ΞΌ2Ξ·2)⁒⋯⁒(ΞΌnΞ·n)⁒xη⁒(1βˆ’x)ΞΌβˆ’Ξ·,Ξ½=0,1,…,n,

where ΞΌ=βˆ‘i=1nΞΌi,Ξ·=βˆ‘i=1nΞ·i, and the summation is done in relation to all systems of nonnegative integer numbers (Ξ·1,Ξ·2,…,Ξ·n) for which

Ξ·1+2⁒η2+…+n⁒ηn=Ξ½.

Define the operator

Pn(ΞΌ1,ΞΌ2,…,ΞΌn)⁒(f)⁒(x)=βˆ‘Ξ½=0nf⁒(Ξ½n)⁒pn⁒ν⁒(x)(f∈C⁒[0,1],x∈[0,1]). (4)

and consider the class β„³ of all operators Ln which are convex combinations of operators of the form (4) which leave invariant the functions 1 and x. Hence

Ln=βˆ‘Ξ³β’(ΞΌ1,ΞΌ2,…,ΞΌn)⁒Pn(ΞΌ1,ΞΌ2,…,ΞΌn),

where the coefficients γ⁒(ΞΌ1,ΞΌ2,…,ΞΌn), in finite number, are nonnegative and with the sum equal to one, and Ln⁒(1)=1,Ln⁒(x)=x.

We note that Bernstein’s operators are of type (4), more exactly

Bn=Pn(n,0,…,0),B1=Pn(0,…,0,1).

The class β„³ also contains other classical operators such as the operators of Cheney-Sharma [8], MΓΌhlbach [12] and Stancu [16].

Lemma 1.

Any operator L:=Pn(ΞΌ1,ΞΌ2,…,ΞΌn) from β„³ is a contraction operator on XΞ±,Ξ² with the Lipschitz constant 1βˆ’21βˆ’ΞΌ and has the fixed point fβˆ—β’(x)=Ξ±+(Ξ²βˆ’Ξ±)⁒x.

Proof.

Indeed, for any f,g∈Xα,β, since L⁒(1)=1, one has

β€–L⁒(f)βˆ’L⁒(g)β€– ≀ β€–fβˆ’g‖⁒(L⁒(1)⁒(x)βˆ’pn⁒0⁒(x)βˆ’pn⁒n⁒(x))
= β€–fβˆ’g‖⁒(1βˆ’pn⁒0⁒(x)βˆ’pn⁒n⁒(x))
≀ (1βˆ’minx∈[0,1]⁑(pn⁒0⁒(x)+pn⁒n⁒(x)))⁒‖fβˆ’gβ€–.

Also, for Ξ½=0, one has Ξ·=0 and so pn⁒0⁒(x)=(1βˆ’x)ΞΌ, while if Ξ½=n, then Ξ·=Ξ½ and so pn⁒n⁒(x)=xΞΌ. Hence

minx∈[0,1]⁑(pn⁒0⁒(x)+pn⁒n⁒(x))=minx∈[0,1]⁑((1βˆ’x)ΞΌ+xΞΌ)=12ΞΌβˆ’1.

Then

β€–L⁒(f)βˆ’L⁒(g)‖≀(1βˆ’12ΞΌβˆ’1)⁒‖fβˆ’gβ€–.

The fact that fβˆ— is the fixed point of L in XΞ±,Ξ² is a simple consequence of the two properties L⁒(1)=1,L⁒(x)=x. ∎

The above lemma guarantees that any operator from the class β„³ is a contraction operator and in XΞ±,Ξ² has the unique fixed point fβˆ—. Thus Theorem 1 applies for any operator from the class β„³, and we have

Theorem 3.

If Lβˆˆβ„³, then Lk⁒(f)β†’fβˆ— as kβ†’βˆž for every f∈XΞ±,Ξ².

Additionally, formula (2) is true and

β€–Lk⁒(f)βˆ’fβˆ—β€–β‰€2ΞΌβˆ’1⁒(1βˆ’12ΞΌβˆ’1)⁒‖L⁒(f)βˆ’fβ€–(kβ‰₯1,f∈XΞ±,Ξ²).

For a subclass of β„³, namely that of the so called operators of Bernstein type, the result given by Theorem 3 was proved by using completely different techniques, by Albu [4] and myself [14].

Obviously, Theorem 2 also applies to sequences (Ln)nβ‰₯1 of operators in β„³, which are assumed to be approximation processes. A better result was obtained in [14] for the subclass of Bernstein type operators by using the extremality property of the classical Bernstein operators in that subclass, namely the relations

Bn⁒(f)≀Ln⁒(f)≀B1⁒(f),

for every nonnegative and convex function f∈C⁒[0,1].

2 Iterates of multidimensional operators

We begin this section with the matrix version of Theorem 1.

Theorem 4.

Let (Xi,di),i=1,2,…,p be metric spaces and let 𝐋:X=X1Γ—X2×…×Xpβ†’X, 𝐋=(𝐋1,𝐋2,…,𝐋p), where 𝐋i:Xβ†’Xi, be such that

(i)

For each i,

di⁒(𝐋i⁒(x),𝐋i⁒(y))β‰€βˆ‘j=1pΞ³i⁒j⁒dj⁒(xj,yj) (5)

for all x=(x1,x2,…,xp),y=(y1,y2,…,yp)∈X and some nonnegative numbers Ξ³i⁒j for which the matrix Ξ“=[Ξ³i⁒j]1≀i,j≀p is convergent to zero in the sense that its power Ξ“k tends to the zero matrix as kβ†’βˆž.

(ii)

𝐋 has the (unique) fixed point xβˆ—.

Then for each x∈X, the sequence (𝐋k⁒(x)) of the iterates of 𝐋 calculated at x is convergent in X to xβˆ—.

Proof.

Denoting

𝐝⁒(x,y)=[d1⁒(x1,y1)…dp⁒(xp,yp)](x,y∈X),

we may rewrite (5) in the matrix way as

𝐝⁒(𝐋⁒(x),𝐋⁒(y))≀Γ⁒𝐝⁒(x,y). (6)

Then

𝐝⁒(𝐋k⁒(x),xβˆ—)=𝐝⁒(𝐋k⁒(x),𝐋k⁒(xβˆ—))≀Γk⁒𝐝⁒(x,xβˆ—)

and the result follows since Ξ“kβ†’0 as kβ†’βˆž. ∎

If Xi,i=1,2,…,p, are complete metric spaces, then inequality (6) showing that 𝐋 is a Perov contraction on X implies the existence and uniqueness of the fixed point xβˆ— of 𝐋 and moreover that the following estimate holds for all kβ‰₯1 and x∈X:

𝐝⁒(𝐋k⁒(x),xβˆ—)≀(Iβˆ’Ξ“)βˆ’1⁒Γk⁒𝐝⁒(𝐋⁒(x),x). (7)

Example 1. Let Xi=D,i=1,2,…,p, where D is a closed convex subset of a Banach space, and Li⁒j:Dβ†’D (1≀i,j≀p) be Ξ»i⁒j-Lipschitz continuous, i.e.,

β€–Li⁒j⁒(xj)βˆ’Li⁒j⁒(yj)‖≀λi⁒j⁒‖xjβˆ’yjβ€–(xj,yj∈D).

Let Οƒi⁒jβ‰₯0⁒(1≀i,j≀p) be such that βˆ‘j=1pΟƒi⁒j=1 for i=1,2,…,p. Define the operator 𝐋:Dpβ†’Dp, 𝐋=(𝐋1,𝐋2,…,𝐋p) by

𝐋i⁒(x)=βˆ‘j=1pΟƒi⁒j⁒Li⁒j⁒(xj),x=(x1,x2,…,xp),i=1,2,…,p.

If the matrix Ξ“:=[Οƒi⁒j⁒λi⁒j]1≀i,j≀p is convergent to zero, then the operator 𝐋 is a Perov contraction on Dp. Consequently, it has a unique fixed point xβˆ—, its iterates 𝐋k⁒(x) converge to xβˆ— and estimate (7) holds. In particular, if Β Ξ»i⁒j<1 for all i and j, matrix Ξ“ is convergent to zero as shown in [1]. Indeed, letting Ξ»:=max{Ξ»i⁒j: 1≀i,j≀p}, one has Ξ»<1 and Γ≀λ⁒M, where M:=[Οƒi⁒j]1≀i,j≀p and the powers of M are dominated by the matrix U having all entries equal to 1 (this is proved by induction). Then

Ξ“k=Ξ»k⁒Mk≀λk⁒Uβ†’0askβ†’βˆž,

as claimed. In this case, each operator Li⁒j has a unique fixed point xi⁒jβˆ—. In case that for each i, one has that xi⁒jβˆ—=:xiβˆ— for all j, then xβˆ—=(x1βˆ—,x2βˆ—,…,xpβˆ—). Such a case occurs if the operators Li⁒j are classical Bernstein operators (of different degrees), or other operators of Bernstein type, and D={f∈C⁒[0,1]:f⁒(0)=Ξ±,f⁒(1)=Ξ²}, when the fixed point of 𝐋 is πŸβˆ—:=(fβˆ—,fβˆ—,…,fβˆ—).

Theorem 2 extends to multidimensional operators as follows.

Theorem 5.

Let (𝐋n)nβ‰₯1 be a sequence of Perov contraction operators with Lipschitz matrices Ξ“n convergent to zero, and let xβˆ— is the unique fixed point of all operators 𝐋n. Assume that

𝐋n⁒(x)β†’x⁒ as ⁒nβ†’βˆžβ’Β for every ⁒x∈X. (8)

Then

(a)

The element from the diagonal of Ξ“n tend to 1 as nβ†’βˆž;

(b)

For each x∈X, a sequence of iterates (𝐋nkn⁒(x))nβ‰₯1 converges to xβˆ— if (kn)nβ‰₯1 is such that the matrices

(Iβˆ’Ξ“n)βˆ’1⁒Γnkn,nβ‰₯1

are bounded (componentwise).

Proof.

(a) It is known that the elements from the diagonal of a matrix which is convergent to zero are strictly less than one. Letting Ξ“n=[(Ξ»n)i⁒j]1≀i,j≀p one has (Ξ»n)i⁒i<1 for i=1,2,…,p. For any i, let Ξ»i⁒i be a limit point of the sequence ((Ξ»n)i⁒i). Then Ξ»i⁒i≀1. Passing eventually to a subsequence, we may assume that (Ξ»n)i⁒iβ†’Ξ»i⁒i. Now we take two element x,y∈X having the components equal to zero except xi and yi which are chosen different. From

𝐝⁒(𝐋n⁒(x),𝐋n⁒(y))≀Γn⁒𝐝⁒(x,y),

looking at the i-th component, one has

di⁒((𝐋n)i⁒(x),(𝐋n)i⁒(y))≀(Ξ»n)i⁒i⁒di⁒(xi,yi).

Here we let nβ†’βˆž and use (𝐋n)i⁒(x)β†’xi and (𝐋n)i⁒(y)β†’yi, to deduce that di⁒(xi,yi)≀λi⁒i⁒di⁒(xi,yi). Since di⁒(xi,yi)>0, we find 1≀λi⁒i. Then Ξ»i⁒i=1 and the proof of statement (a) is finished.

(b) Using (7) one has

𝐝⁒(𝐋nkn⁒(x),xβˆ—)≀(Iβˆ’Ξ“n)βˆ’1⁒Γnkn⁒𝐝⁒(𝐋n⁒(x),x),

whence the conclusion of (b) is immediate. ∎

The next example gives a scheme of construction of sequences of multidimensional operators which are approximation processes in the sense of (8).

Example 2. Let (𝐋n)nβ‰₯1 be a sequence of operators of the type of those from the previous example, that is

(𝐋n)i⁒(x)=βˆ‘j=1p(Οƒn)i⁒j⁒(Ln)i⁒j⁒(xj),i=1,2,…,p,nβ‰₯1.

It is easily seen that the sequence (𝐋n)nβ‰₯1 is an approximation process on Dp, i.e., 𝐋n⁒(x)β†’xas nβ†’βˆžfor every x∈Dp, if the following conditions are satisfied:

(i)

for every i, the sequence ((Ln)i⁒i)nβ‰₯1is an approximation process on D;

(ii)

for every (i,j) with iβ‰ j, and for each xj∈D, the sequence ((Ln)i⁒j⁒(xj))nβ‰₯1 is bounded;

(ii)

for every (i,j) with iβ‰ j, (Οƒn)i⁒jβ†’0 as nβ†’βˆž.

References

  • [1] Agratini, O., and Precup, R., Iterates of multidimensional operators via Perov theorem, Carpathian J. Math. 38 (2022), 539-546.
  • [2] Agratini, O., and Precup, R., Estimates related to the iterates of positive linear operators and their multidimensional analogues, submitted.
  • [3] Agratini, O., and Rus, I.A., Iterates of a class of discrete operators via contraction principle, Comment. Math. Univ. Carolinae 44 (2003), 555-563.
  • [4] Albu, M., Asupra convergenΕ£ei iteratelor unor operatori liniari şi mΔƒrginiΕ£i, In ”Sem. Itin. Ec. Func. Aprox. Convex., Cluj-Napoca, 1979, pp 6-16.
  • [5] Brass, H., Eine verallgemeinerung der Bernsteinschen operatoren, Abh. Math. Sem. Univ. Hamburg 36 (1971), 111-122.
  • [6] CΔƒtinaş, T., Iterates of a modified Bernstein type operator, J. Numer. Anal. Approx. Theory 48 (2019), no. 2, 144-147.
  • [7] CΔƒtinaş, T., and Otrocol, D., Iterates of Bernstein type operators on a square with one curved side via contraction principle, Fixed Point Theory 14 (2013), 97-106.
  • [8] Cheney, E.W., and Sharma, A., On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma (2) 5 (1964), 77-84.
  • [9] Gavrea, I., and Ivan, M., On the iterates of positive linear operators preserving the affine functions, J. Math. Anal. Appl. 372 (2010), 366-368.
  • [10] Karlin, S., and Ziegler, Z., Iteration of positive approximation operators, J. Approx. Th. 3 (1970), 310-339.
  • [11] Kelisky, R.P., and Rivlin, T.J., Iterates of Bernstein polynomials, Pacific J. Math. 21 (1967), 511-520.
  • [12] MΓΌhlbach, G., Über das approximationsverhalten gewisser positiver linearer operatoren, Dissertation Technische UniversitΓ€t Hannover, 1969.
  • [13] PΔƒltΔƒnea, R., Optimal constant in approximation by Bernstein operators, J. Comput. Analysis Appl. 5 (2003), 195-235.
  • [14] Precup, R., ProprietΔƒΕ£i de alurΔƒ şi unele aplicaΕ£ii ale lor, Ph.D. Thesis, Babeş-Bolyai University, Cluj-Napoca, 1985.
  • [15] Rus, I.A., Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl. 292 (2004), 259-261.
  • [16] Stancu, D.D., Asupra unei generalizΔƒri a polinoamelor lui Bernstein, Studia Univ. Babeş-Bolyai Math. 14 (1969), no. 2, 31-46.

[1] Agratini, O., and Precup, R., Iterates of multidimensional operators via Perov theorem, Carpathian J. Math. 38 (2022), 539-546.
[2] Agratini, O., and Precup, R., Estimates related to the iterates of positive linear operators and their multidimensional analogues, submitted.
[3] Agratini, O., and Rus, I.A., Iterates of a class of discrete operators via contraction principle, Comment. Math. Univ. Carolinae 44 (2003), 555-563.
[4] Albu, M., Asupra convergentei iteratelor unor operatori liniari si marginiti, In ”Sem. Itin. Ec. Func. Aprox. Convex., Cluj-Napoca, 1979, pp 6-16.
[5] Brass, H., Eine verallgemeinerung der Bernsteinschen operatoren, Abh. Math. Sem. Univ. Hamburg 36 (1971), 111-122.
[6] Catinas, T., Iterates of a modified Bernstein type operator, J. Numer. Anal. Approx. Theory 48 (2019), no. 2, 144-147.
[7] Catinas, T., and Otrocol, D., Iterates of Bernstein type operators on a square with one curved side via contraction principle, Fixed Point Theory 14 (2013), 97-106.
[8] Cheney, E.W., and Sharma, A., On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma (2) 5 (1964), 77-84.
[9] Gavrea, I., and Ivan, M., On the iterates of positive linear operators preserving the affine functions, J. Math. Anal. Appl. 372 (2010), 366-368.
[10] Karlin, S., and Ziegler, Z., Iteration of positive approximation operators, J. Approx. Th. 3 (1970), 310-339.
[11] Kelisky, R.P., and Rivlin, T.J., Iterates of Bernstein polynomials, Pacific J. Math. 21 (1967), 511-520.
[12] Muhlbach, G., Uber das approximationsverhalten gewisser positiver linearer operatoren, Dissertation Technische Universitat Hannover, 1969.
[13] Paltanea, R., Optimal constant in approximation by Bernstein operators, J. Comput. Analysis Appl. 5 (2003), 195-235.
[14] Precup, R., Proprietati de alura si unele aplicatii ale lor, Ph.D. Thesis, Babes-Bolyai University, Cluj-Napoca, 1985.
[15] Rus, I.A., Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl. 292 (2004), 259-261.
[16] Stancu, D.D., Asupra unei generalizari a polinoamelor lui Bernstein, Studia Univ. Babes-Bolyai Math. 14 (1969), no. 2, 31-46.

2023

Related Posts