Iterates of multidimensional approximation operators via Perov theorem

Abstract

The starting point is an approximation process consisting of linear and positive operators. The purpose of this note is to establish the limit of the iterates of some multidimensional approximation operators. The main tool is a Perov’s result which represents a generalization of Banach fixed point theorem. In order to support the theoretical aspects, we present three applications targeting respectively the operators Bernstein, Cheney-Sharma and those of binomial type. The last class involves an incursion into umbral calculus.

Authors

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

Radu Precup
Institute of Advanced Studies in Science and Technology, Babes-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Keywords

Linear positive operator, contraction principle, Perov theorem, Bernstein operator, CheneySharma operator, umbral calculus, binomial operator.

Paper coordinates

O. Agratini, R. Precup, Iterates of multidimensional approximation operators via Perov theorem, Carpathian J. Math., 38 (2022) no. 3, pp. 539-546, https://doi.org/10.37193/CJM.2022.03.02

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Carpathian Journal Math.

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1584 – 2851

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1843 – 4401

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Iterates of multidimensional approximation operators via Perov theorem

Iterates of multidimensional approximation operators via Perov theorem

Octavian Agratini 1  and  Radu Precup 2 1 Babeş-Bolyai University
Faculty of Mathematics and Computer Science
Street M. Kogălniceanu, 1, 400084 Cluj-Napoca, Romania
AND
Tiberiu Popoviciu Institute of Numerical Analysis
Romanian Academy
P.O. Box 68-1, 400110 Cluj-Napoca, Romania
agratini@math.ubbcluj.ro 2 Babeş-Bolyai University
Faculty of Mathematics and Computer Science
Street M. Kogălniceanu, 1, 400084 Cluj-Napoca, Romania
AND
Tiberiu Popoviciu Institute of Numerical Analysis
Romanian Academy
P.O. Box 68-1, 400110 Cluj-Napoca, Romania
r.precup@math.ubbcluj.ro
(Date: ??)
Abstract.

The starting point is an approximation process consisting of linear and positive operators. The purpose of this note is to establish the limit of the iterates of some multidimensional approximation operators. The main tool is a Perov’s result which represents a generalization of Banach fixed point theorem. In order to support the theoretical aspects, we present three applications targeting respectively the operators Bernstein, Cheney-Sharma and those of binomial type. The last class involves an incursion into umbral calculus.

Key words and phrases:
Linear positive operator, contraction principle, Perov theorem, Bernstein operator, Cheney-Sharma operator, umbral calculus, binomial operator.
2020 Mathematics Subject Classification:
41A36, 47H10

1. Introduction

It is acknowledged that linear positive operators are a useful tool in approximation signals. Referring to classical operators of discrete or continuous type, a constant concern is to highlight their properties such as the rate of convergence for functions belonging to various spaces, preservation of the properties of functions that have been approximated, extensions in q-Calculus, replacement of the classical convergence with statistical convergence.

For a linear positive approximation process (Ln)n1, a distinct direction of study is to investigate the convergence behavior of the sequence (Lnj)j1, where

Ln1f=Lnf,Lnjf=Ln(Lnj1f),j>1,

assuming that n is fixed and j does not depend on n.

In the following we consider that the space C([a,b]) is endowed with the Chebyshev norm , f=supx[a,b]|f(x)|.

Best of our knowledge, the first result was obtained in 1967 by Kelisky and Rivlin [11]. It targeted the well-known Bernstein operators

(1) Bn:C([0,1])C([0,1]),(Bnf)(x)=k=0npn,k(x)f(kn),

where

pn,k(x)=(nk)xk(1x)nk,k=0,n¯.

The result established [11, Eq. (2.4)] is read as follows

(2) limj(Bnjf)(x)=f(0)+(f(1)f(0))x,0x1,

fC([0,1]), saying that ((Bnjf)(x))j1 converges uniformly to the line segment joining (0,f(0)) to (1,f(1)).

The above result was reobtained in 2004 by I.A. Rus [17] using another technique based on fixed point theory, more exactly on Banach contraction principle. The idea was taken over and developed in other papers, for example we mention [1], [2], [9].

The purpose of this note is to provide an approach for multidimensional operators. To achieve this, we rely on a specific generalization of the concept of metric space. Perov [13] used the notion of vector-valued metric space and obtained a Banach type fixed theorem on such a complete generalized metric space by using matrices instead of Lipschitz constants. Perov’s result have been exploited in various works, see, e.g., [3], [7], [8], [12], [15].

2. Iterates of multidimensional linear positive operators

For a wider information of the reader and the accomplishment of an independent exposition, we will briefly present the method used for the study of the limit of iterations for one-dimensional operators via fixed point theory, as, for example, it appears in [1].

Set

(3) Xα,β:={fC([a,b]):f(a)=α,f(b)=β},(α,β)×.

Every Xα,β is a closed subset of C([a,b]) and, clearly, the system Xα,β, (α,β)×, constitutes a partition of this space. We define the linear positive operators Ln:C([a,b])C([a,b]), n, that enjoy the following properties:

(i) they reproduce the Korovkin test function e0 and e1, where e0(x)=1, e1(x)=x, x[a,b];

(ii) each Ln|Xα,β, (α,β)×, is a contraction allowing us to say that λn[0,1) exists such that

(4) LnfLngλnfg, where f and g belong to Xα,β;

(iii)

(5) (Lnf)(a)=f(a) and (Lnf)(b)=f(b).

Since Ln is linear, the first condition implies that it reproduces affine functions. Such operators are also called Markov type. Also, the last requirement indicates the interpolation property of the operators at the end of the domain. This condition guarantees that each Xα,β is an invariant subset of the operators Ln.

Considering the function pα,β defined on [a,b]

(6) pα,β=α+βαba(e1a),

it is observed that pα,βXα,β. Since Ln reproduces the affine functions, pα,β is a fixed point of Ln. For any f belonging to the space C([a,b]) one has fXf(a),f(b) and, by applying the contraction principle on Xf(a),f(b) we deduce

limmLnmf=pf(a),f(b),

uniformly on [a,b].

With these preparations and using Perov’s theorem we will extend the above result to the multidimensional case.

In the first step we consider a net on [a,b] named Δn(a=xn,0<xn,1<<xn,n=b) and a system ϕn(φn,k)k=0,n¯ where each function belongs to C([a,b]). Defining

(7) (Lnf)(x)=k=0nφn,k(x)f(xn,k),x[a,b],

we assume

φn,k0,k=0,n¯,k=0nφn,k=e0,k=0nxn,kφn,k=e1,

i.e., our system ϕn is a blending one related to Δn, n. Also we suppose that conditions (4) and (5) are fulfilled. In order to construct a convex combination of p such operators, let

(8) γi,j[0,1], 1i,jp, such that j=1pγi,j=1.

In the second step, let p and (α,β)×. Define the operator

(9) 𝐋:Xα,βpXα,βp,𝐋=(𝐋1,,𝐋p),
(10) 𝐋i(f1,,fp)=j=1pγi,jLni(fj),i=1,,p,

where Xα,β is given by (3). The above combination uses p operators of the form (7) having different orders, namely ni, i=1,,p. Our main result will be read as follows.

{theorem}

For any vector-valued function 𝐟Xα,βp, 𝐟=(f1,,fp), one has

(11) limk𝐋k(𝐟)=𝐟0,

where 𝐋 is given by (9) and the components of the vector-valued function 𝐟0 are all equal with the affine function pα,β defined by (6).

Proof.

We will use the vector version of Banach contraction principle. We only need to show that 𝐋 is a Perov contraction on the space Xα,βp. This involves proving that there is a quadratic matrix, say M, of size p which converges to zero (i.e. Mk tends to the null matrix as k tends to infinity) such that

(12) 𝐋(𝐟)𝐋(𝐠)CM𝐟𝐠C

for all 𝐟,𝐠Xα,βp, where by C we mean the vector-valued norm given as follows

𝐟C=[f1fp],

indicating the Chebyshev norm on C([a,b]).

According to our hypothesis, any Ln operator defined by (7) is a contraction on Xα,β with Lipschitz constant λn. Taking in view relation (10), for any i{1,,p} we get

(13) 𝐋i(𝐟)𝐋i(𝐠)Cj=1pγi,jλnifjgj.

Noting λ:=max{λni: 1ip}, we have λ[0,1). We define the matrix M as follows M=λΓ, where Γ=[γi,j]1i,jp. Clearly, relation (13) yields (12).

By induction we prove

(14) ΓkU,

where all entries of the matrix U are equal with 1. The key relation used is (8). For k=2 we have

j=1pγi,jγj,kj=1pγi,j=1 for all 1i,kp,

that is Γ2U. Further, assuming that ΓkU and observing that ΓU=U, we obtain that Γk+1=ΓΓkΓU=U and the induction is completed.

Based on definition of M and relation (14), we can write

Mk=λkΓkλkU,

which implies limkMk=Θp, null matrix of the order p.

Our statement (11) follows from Perov theorem since 𝐟0 is the unique fixed point of the operator 𝐋. ∎

3. Applications

3.1. Application 1

Let Ln be Bernstein operator Bn, see (1). It is known that Bn is a Markov type operator interpolating the functions at the extremities of the domain [0,1]. Moreover, Bn|Xα,β is a contraction for all α,β, where Lipschitz constant is λn=121n.

Consequently, applying (11) we deduce

limk𝐁k(𝐟)=𝐟0,

where all components of the function 𝐟0 are given by pα,β=f(0)e0+(f(1)f(0))e1.

For p=1 we reobtain the identity (2) which is the classical result of Kelisky and Rivlin [11, Eq. (2.4)].

Remark. At first we recall the notion of weakly Picard operator, see e.g., [21]. Let (X,d) be a metric space. An operator U:XX is a weakly Picard operator (abbreviated WPO) if the sequence (Uk(y))k1 converges for all yX and the limit (which may depend on y) is a fixed point of U. Set

U:XX,U(y)=limkUk(y),yX.

By using this concept, Kelisky-Rivlin result can be reformulated as follows: for each n, Bn is WPO and (Bnf)=f(0)e0+(f(1)f(0))e1.

Recently, following the same route of WPO, in [4], the limit of iterates of modified Bernstein operators in Durrmeyer sense has been approached.

3.2. Application 2

In our attention is a generalization of Bernstein operators, the basis of its construction being a combinatorial identity of Jensen [10]

(15) (x+y+nγ)n=ν=0n(nν)x(x+νγ)ν1[y+(nν)γ]nν.

The inception of its motivation is Lagrange’s formula

u1(z)1zu2(z)u2(z)=ν=01ν!dνdzν((u2(z))νu1(z))(zu2(z))ν

and proceeds by setting u1(z)=exz, u2(z)=eγz. Choosing y=1x in (15), Cheney and Sharma [5] have investigated the operators

(16) (Qnf)(x)=k=0nqn,k(x;γ)f(kn),fC([0,1]),x[0,1],n,

where γ is a non-negative parameter and

qn,k(x;γ)=(1+nγ)1n(nk)x(x+kγ)k1(1x)(1x+(nk)γ)n1k.

Obviously, the Bernstein polynomials represent a particular case of (16) obtained by setting γ=0.

The authors proved that each Qn preserves the constant functions. In [19] was shown that Qn also reproduces the monomial e1. It is easy to see that for all (α,β)×, Xα,β defined by (3) with a=0, b=1 is an invariant set of Qn. Moreover, Qn|Xα,β is a contraction. Indeed, if f1 and f2 belong to Xα,β, then we get

|(Qnf1)(x)(Qnf2)(x)| (1qn,0(x;γ)qn,n(x;γ))supx[0,1]|f1(x)f2(x)|
(121n(1+nγ)1n)f1f2,x[0,1].

Examining (4), we can choose λn=121n(1+nγ)1n<1 and all requirements for one-dimensional operators LnQn are met. Thus, we can form the multidimensional operators 𝐋𝐐 as in (9) and the relation (11) takes place (a=0, b=1).

3.3. Application 3

Here we have in mind operators constructed with the help of binomial polynomials. This involves a foray into umbral calculus, consequently at the beginning we will point out the basics. The first rigorous version of this calculus belongs to Gian-Carlo Rota and his collaborators, see, e.g., [16].

For any n0={0}, we denote by Πn the linear space of polynomials of degree no greater than n and by Πn the set of all polynomials of degree n.

A sequence p=(pn)n0 such that pnΠn for every n0 is called a polynomial sequence.

A polynomial sequence b=(bn)n0 is called of binomial type if for any (x,y)× the following equalities

(17) bn(x+y)=k=0n(nk)bk(x)bnk(y),n0,

hold. We get b0(x)=1 and by induction we obtain bn(0)=0 for any n. Set

Π:=n0Πn.

The space of all linear operators T:ΠΠ will be denoted by . Among these operators an important role will be played by the shift operators, named Ea, defined by

(Eap)(x)=p(x+a),pΠ.

An operator T which switches with all shift operators, that is TEa=EaT for every a, is called a shift-invariant operator and the set of these operators are denoted by s.

An operator Q is called a delta operator if Qs and Qe1 is a nonzero constant. Let δ denote the set of all delta operators. More generally, according to [16, Proposition 2] for every Qδ we have

Q(Πn)Πn1,n.

A polynomial sequence p=(pn)n0 is called the sequence of basic polynomials associated to the delta operator Q if, for any n and x, we get

(i) p0(x)=1,

(ii) pn(0)=0,

(iii) (Qpn)(x)=npn1(x).

It was proved [16, Proposition 3] that every delta operator has a unique sequence of basic polynomials.

For a good understanding, we collect below some results established in [16] on this topic.

{proposition}

(a) If p=(pn)n0 is a basic sequence for some delta operator Q, then it is a sequence of binomial type. Reciprocally, if p is a sequence of binomial type, then it is a basic sequence for some delta operator.

(b) Let Ts and Qδ with the basic sequence p=(pn)n0. One has

T=k0(Tpk)(0)k!Qk.

(c) An isomorphism Ψ exists from the ring (,+,) of the formal power series in the variable t over field, onto (s,+,) such that

(18) Ψ(ϕ(t))=T, where ϕ(t)=k0akk!tk and T=k0akk!Qk.

(d) An operator Ps is a delta operator if and only if it corresponds under the isomorphism defined by (18), to a formal series ϕ(t) such that ϕ(0)=0 and ϕ(0)0.(e) Let Qδ with p=(pn)n0 its sequence of basic polynomials. Led ϕ(D)=Q and τ(t) be the inverse formal series of ϕ(t), where D indicates the derivative operator. Then one has

(19) exp(xτ(t))=n0pn(x)n!tn,

where τ(t) has the form c1t+c2t2+ (c10).

At this moment we are ready to present a new class of operators. Let Q be a delta operator and p=(pn)n0 be its sequence of basic polynomials under the additional assumption pn(1)0 for every n. We define LnQ:C([0,1])C([0,1]) as follows

(20) (LnQf)(x)=1pn(1)k=0n(nk)pk(x)pnk(1x)f(kn),n.

P. Sablonniere [18] called them Bernstein-Sheffer operators, but as D.D. Stancu and M.R. Occorsio motivated in [20] these operators can be namely Popoviciu operators. These operators check all the requirements in order to be able to create the multidimensional operators 𝐋Q, see (9).

The operators LnQ, n, are linear and reproduce the constants. Indeed, choosing in (17) y:=1x, from (20) we obtain LnQe0=e0. The positivity of these operators are given by the sign of the coefficients of the series τ(t) defined at (19). Tiberiu Popoviciu [14] and later P. Sablonniere [18, Theorem 1] have established the following result.

{lemma}

LnQ defined by (20) is a positive operator on C([0,1]) for every n if and only if

(21) c1>0 and cn0 for all n2.

In [18, Theorem 2(i)] it was shown that if (21) takes place, then LnQe1=e1.

Further, from the definition of basic polynomials we get pk(0)=δk,0, consequently (LnQf)(x0)=f(x0) for x0=0 and x0=1. It remains to check if LnQ|Xα,β is a contraction. Let f1 and f2 belong to Xα,β, where α, β are arbitrarily fixed. Using (20) we can write

|(LnQf1)(x)(LnQf2)(x)| (1pn(1x)+pn(x)pn(1))supx[0,1]|(f1f2)(x)|
(11pn(1)minx[0,1](pn(1x)+pn(x)))f1f2
:=λnf1f2.

Due to positivity of LnQ operators, n, see relation (21) corroborated with (19), we have 0λn<1. Thus, we deduce that LnQ|Xα,β is a contraction and

limn(LnQ)mf=pf(0),f(1).

Considering that the additional condition (21) occurs, we can apply our result for multidimensional operators. Consequently, we state that (11) is valid for 𝐋𝐋Q.

Remark. We can show that a particular case of LnQ operators defined by (20) leads us to Cheney-Sharma operators discussed in Application 2. Let γ be a fixed non-negative parameter. We define γ~=(anγ)n0 the following sequence of polynomials

a0γ=1,anγ(x)=x(x+nγ)n1,n.

It represents Abel sequence and verifies relation (17). Further, we consider Abel operator Aγ:=DEγ. For every pΠ,

(Aγp)(x)=dpdx(x+γ).

In this case γ~ forms the sequence of basic polynomials associated to delta operator Aγ. Choosing in (20) Q:=Aγ one obtains Cheney-Sharma operator and 𝐋𝐋Aγ.

We end this application by indicating Crăciun’s paper [6]. Here a more complex class of linear operators is built that mixes two sequences p=(pn)n0, s=(sn)n0, the first containing basic polynomials associated to a delta operator Q and the second is defined by the identity sn=S1pn, where S is an invertible shift invariant operator.

References

  • [1] Agratini, O., Rus, I.A., Iterates of a class of discrete linear operators via contraction principle, Comment. Math. Univ. Carolinae, 44(2003), 555-563.
  • [2] Agratini, O., Rus, I.A., Iterates of some bivariate approximation process via weakly Picard operators, Nonlinear Anal. Forum, 8(2003), No. 2, 159-168.
  • [3] Berinde, V., Approximating fixed points of weak contractions using Picard iteration, Nonlinear Anal. Forum, 9(2004), No. 1, 43-53.
  • [4] Cătinaş, T., Iterates of a modified Bernstein type operator, J. Numer. Anal. Approx. Theory, 48(2019), No. 2, 144-147.
  • [5] Cheney, E.W., Sharma, A., On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma (2), 5(1964), 77-84.
  • [6] Crăciun, M., Approximation operators constructed by means of Sheffer sequences, Rev. Anal. Numér. Théor. Approx., 30(2001), No. 2, 135-150.
  • [7] Filip, A.-D., Petruşel, A., Existence and uniqueness of the solution for a general system of Fredholm integral equations, Math. Methods Appl. Sci., (2020), doi.org/10.1002/mma.6737.
  • [8] Cvetković, M., Rakoc̆ević, V., Extensions of Perov theorem, Carpathian J. Math., 31(2015), No. 2, 181-188.
  • [9] Gavrea, I., Ivan, M., On the iterates of positive linear operators preserving the affine functions, J. Math. Anal. Appl., 372(2010), 366-368.
  • [10] Jensen, J.L.W.V., Sur une identité d’Abel et sur d’autres formules analogues, Acta Math., 26(1902), 307-318.
  • [11] Kelisky, R.P., Rivlin, T.J., Iterates of Bernstein operators, Pacific J. Math., 21(1967), 511-520.
  • [12] Novac, A., Precup, R., Perov type results in gauge spaces and their applications to integral systems on semi-axis, Math. Slovaca, 64(2014), 961-972.
  • [13] Perov, A.I., On Cauchy problem for a system of ordinary differential equation, Priblizhen. Metody Reshen. Differ. Uravn., 2(1964), 115-134.
  • [14] Popoviciu, T., Remarques sur le polynômes binomiaux, Bul. Soc. Sci. Cluj (Roumanie), 6(1931), 146-148 (also reproduced in Mathematica (Cluj), 6(1932), 8-10).
  • [15] Precup, R., The role of matrices that are convergent to zero in the study of semilinear operator systems, Math. Comput. Modelling, 49(2009), 703-708.
  • [16] Rota, G.-C., Kahaner, D., Odlyzko, A., On the Foundations of Combinatorial Theory. VIII. Finite operator calculus, J. Math. Anal. Appl., 42(1973), 685-760.
  • [17] Rus, I.A., Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl., 292(2004), 259-261.
  • [18] Sablonniere, P., Positive Bernstein-Sheffer operators, J. Approx. Theory, 83(1995), 330-341.
  • [19] Stancu, D.D., Cismaşiu, C., On an approximating linear positive operators of Cheney-Sharma, Rev. Anal. Numér. Théor. Approx., 26(1997), Nos. 1-2, 221-227.
  • [20] Stancu, D.D., Occorsio, M.R., On approximation by binomial operators of Tiberiu Popoviciu type, Rev. Anal. Numér. Théor. Approx., 27(1998), No. 1, 167-181.
  • [21] Zeidler, E., Nonlinear Functional Analysis and Its Applications: Fixed-point Theorems, Transl. from the German by Peter R. Wadsack, Springer New York, 1993.

[1] Agratini, O., Rus, I. A. Iterates of a class of discrete linear operators via contraction principle. Comment. Math. Univ. Carolinae 44 (2003), 555–563.

[2] Agratini, O., Rus, I.A. Iterates of some bivariate approximation process via weakly Picard operators. Nonlinear Anal. Forum 8 (2003), no. 2, 159–168.

[3] Berinde, V., Approximating fixed points of weak contractions using Picard iteration. Nonlinear Anal. Forum 9 (2004), no. 1, 43–53.

[4] Catinas, T., Iterates of a modified Bernstein type operator. J. Numer. Anal. Approx. Theory 48 (2019), no. 2, 144–147.

[5] Cheney, E. W., Sharma, A., On a generalization of Bernstein polynomials. Riv. Mat. Univ. Parma (2) 5 (1964), 77–84.

[6] Craciun, M., Approximation operators constructed by means of Sheffer sequences. Rev. Anal. Numer. Theor. Approx. 30 (2001), no. 2, 135–150.

[7] Cvetkovic, M., Rakocevic, V., Extensions of Perov theorem. Carpathian J. Math. 31 (2015), no. 2, 181–188.

[8] Filip, A.-D., Petrusel, A,. Existence and uniqueness of the solution for a general system of Fredholm integral equations. Math. Methods Appl. Sci. (2020), doi.org/10.1002/mma.6737.

[9] Gavrea, I., Ivan, M., On the iterates of positive linear operators preserving the affine functions. J. Math. Anal. Appl. 372 (2010), 366–368.

[10] Jensen, J. L., W. V. Sur une identite d’Abel et sur d’autres formules analogues. Acta Math. 26 (1902), 307–318.

[11] Kelisky, R. P., Rivlin, T. J. Iterates of Bernstein operators. Pacific J. Math. 21 (1967), 511–520.

[12] Novac, A., Precup, R. Perov type results in gauge spaces and their applications to integral systems on semi-axis. Math. Slovaca 64 (2014), 961–972.

[13] Perov, A. I., On Cauchy problem for a system of ordinary differential equation. Priblizhen. Metody Reshen. Differ. Uravn. 2 (1964), 115–134.

[14] Popoviciu, T., Remarques sur le polynomes binomiaux. Bul. Soc. Sci. Cluj (Roumanie) 6 (1931), 146–148 (also reproduced in Mathematica (Cluj) 6 (1932), 8–10).

[15] Precup, R., The role of matrices that are convergent to zero in the study of semilinear operator systems. Math. Comput. Modelling 49 (2009), 703–708.

[16] Rota, G.-C., Kahaner, D.; Odlyzko, A. On the Foundations of Combinatorial Theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42 (1973), 685–760.

[17] Rus, I. A., Iterates of Bernstein operators, via contraction principle. J. Math. Anal. Appl. 292 (2004), 259–261.

[18] Sablonniere, P. Positive Bernstein-Sheffer operators. J. Approx. Theory 83 (1995), 330–341.

[19] Stancu, D. D., Cismasiu, C. On an approximating linear positive operators of Cheney-Sharma. Rev. Anal. Numer. Theor. Approx. 26 (1997), nos. 1–2, 221–227.

[20] Stancu, D. D., Occorsio, M. R. On approximation by binomial operators of Tiberiu Popoviciu type. Rev. Anal. Numer. Theor. Approx. 27 (1998), no. 1, 167–181.

[21] Zeidler, E., Nonlinear Functional Analysis and Its Applications: Fixed-point Theorems. Transl. from the German by Peter R. Wadsack, Springer New York, 1993.

2022

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