The iterates of positive linear operators preserving constants

Applied Mathematics Letters 24 (2011) 2068–2071

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Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml

The iterates of positive linear operators preserving constants Ioan Gavrea, Mircea Ivan ∗ Department of Mathematics, Technical University of Cluj Napoca, Str. Memorandumului nr. 28, 400114 Cluj-Napoca, Romania

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Article history: Received 15 December 2010 Received in revised form 28 May 2011 Accepted 31 May 2011

In this note we introduce a simple and efficient technique for studying the asymptotic behavior of the iterates of a large class of positive linear operators preserving constant functions. © 2011 Elsevier Ltd. All rights reserved.

Keywords: Iterates Linear positive operator Cesàro mean operator

1. Introduction In 1967, Kelisky and Rivlin [1] studied the limit behavior of the iterates of Bernstein’s operators. Three years later, Karlin and Ziegler [2] provided new insights into the study of the limit behavior of the iterates of linear operators defined on C [0, 1]. Their results have attracted much attention lately and several new proofs and generalizations have been given; see [1–19], and the references therein. Recently, Galaz Fontes and Solís [15] studied the asymptotic behavior of the classical Cesàro operator C : C [0, 1] → C [0, 1] defined by

C f ( x) =

 f (0∫), 

x

x = 0,

x

1

f (t ) dt ,

x ∈ (0, 1],

(1)

0

and they proved (cf., [15, Th. 1]) that limk→∞ C k f = f (0). Throughout the work, we use the following notation/symbols: ei : [0, 1] → R, the monomial functions ei (x) = xi , i = 0, 1, . . .;

δx : C [0, 1] → C [0, 1], the Dirac operator, also known as the evaluation at x ∈ [0, 1], which is defined by δx (f ) = f (x); I: C [0, 1] → C [0, 1], the identity operator; u

−→, the uniform convergence of a functional sequence; s

−→, the strong convergence of an operator sequence. When there is no danger of confusion we shall denote the constant function e0 by 1. To avoid excessive formalism, we shall use the same symbol ‘‘0’’ to denote both the zero function and the real number zero. It should be understood from the context which one is meant.

∗

Corresponding author. E-mail addresses: [email protected] (I. Gavrea), [email protected] (M. Ivan).

0893-9659/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2011.05.044

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The goal of this work is to introduce a Korovkin type technique which entirely solves the problem of the asymptotic behavior of the iterates of positive linear operators preserving constants, and which enlarges the class of operators for which the limit of the iterates can be computed. The main result is the following theorem. Theorem 1. Let Uk : C [0, 1] → C [0, 1], k = 0, 1, . . ., be a sequence of positive linear operators preserving constant functions. If u

s

there exists a function ϕ ∈ C [0, 1] strictly positive on (0, 1], such that Uk ϕ −→ 0, then Uk −→ δ0 . We note that Theorem 1 was first communicated at ‘‘The First Jaen Conference on Approximation Theory’’, Ubeda, Jaén, Spain, July 4th–9th, 2010. We apply our method to calculate the limit of certain classical operators for which all previous attempts to compute the limit have failed: the Cheney–Sharma operator, the Schurer operators, the generalized Cesàro operators, etc. It is worth noting that all these operators were refractory to other methods. We would like to underline the simplifying outcome of our result: Whereas previously elaborate calculations were needed to find the asymptotic behavior of the iterates of the Stancu–Bernstein operators (see, e.g., [12]), the new technique being proposed in the work provides the means to characterize this behavior almost instantly. In fact, this holds true not only for the Stancu operators, but for all operators satisfying the simple restrictions of Theorem 1. To further illustrate the relevance of our result, we should like to refer to another argument: In order to calculate the limit of the iterates of the Cesàro operators, the authors of [15] use, among other things, the Weierstrass approximation theorem. For the sake of simplicity we restrict ourself to the space C [0, 1] endowed with the sup-norm, and mention that our results apply to operators defined on C [a, b]. The organization of this note is as follows. In Section 2 we collect some examples of positive linear operators preserving the constants, and in Section 3 we give the proof of Theorem 1 and some of its applications. 2. Examples of classical positive linear operators preserving constants In this section we give some examples of positive linear operators U: C [0, 1] → C [0, 1] preserving constants and satisfying the condition u

Uk e1 −→ 0.

(2)

We note that, for the quasi-totality of the classical positive linear operators preserving constants, e1 is an eigenvector, i.e., Ue1 = λ e1 , for some λ ∈ R. Since U is positive, we obtain that λ ≥ 0. The inequality e1 ≤ e0 yields Ue1 ≤ e0 ; hence λ e1 ≤ e0 , and consequently, 0 ≤ λ ≤ 1. We have studied the case λ = 1, i.e., the case of operators preserving linear functions, in [17]. The cases 0 ≤ λ < 1 fit condition (2). The evaluation at 0. The operator δ0 is linear, positive, and satisfies the conditions δ0 e0 = e0 and δ0k e1 = 0, for all k = 0, 1, . . . . The Cesàro operator. The Cesàro operator (1) satisfies

C e0 = e0 and C k e1 =

 k 1

u

e1 −→ 0

2

as k → ∞.

The Stancu operator. The Bernstein–Stancu operator Sn,β : C [0, 1] → C [0, 1] is given by Sn,β f (x) =

n   − n k =0

k

(1 − x)n−k xk f





k n+β

,

n = 0, 1, . . . ; β ≥ 0.

(3)

For β > 0, it satisfies (see, e.g., [20]) the following identities: Sn,β e0 = e0

and Snk,β e1 =



k

n n+β

e1 .

(4)

The Cheney–Sharma operator. Suppose that tn ≥ 0, n ∈ N, and let CSn : C [0, 1] → C [0, 1] be the nth Bernstein–Cheney–Sharma operator, defined by CSn f (x) = (1 + n tn )−n

n   − n k=0

k

x (x + k tn )k−1 (1 − x + (n − k)tn )n−k f

  k

n

.

(5)

It is known that 0 ≤ CSn e1 ≤ 1+1t (see, e.g., [21] and [3, (5.3.7)]. For tn > 0 one has that n e

CSn e0 = e0

and

0 ≤ CSnk e1 ≤



1 1 + tn

k

u

e1 −→ 0.

(6)

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The Schurer operator. For p, n ∈ N, n ≥ p, the Bernstein–Schurer type operator [3, (5.3.1)], ßn : C [0, 1] → C [0, 1] is defined by ßn f (x) =

 n −p    − k n−p f

n

k=0

k

xk (1 − x)n−p−k .

(7)

For p > 0, one can prove that this operator satisfies ßn e0 = e0

and

ßkn e1 =



n−p

k

n

u

e1 −→ 0.

(8)

A Bernstein type operator. For n, j ∈ N, 0 < j < n, 0 < a < 1, consider the Bernstein type operator [22, Proposition 11] Bn : C [0, 1] → C [0, 1],

Bn f (x) =

n   − n

k

k=0

 f

a

k(k − 1) · · · (k − j + 1)

1/j 

n(n − 1) · · · (n − j + 1)

xk (1 − x)n−k .

(9)

The operator Bn satisfies u

Bn e0 = e0 and Bnk ej = ak ej −→ 0.

(10)

3. Proof of the main result and its applications In this section we begin with the proof of the main result. Proof of Theorem 1. Suppose that f ∈ C [0, 1] and that ε > 0. Since f is continuous and ϕ is strictly positive on (0, 1], for sufficiently large a > 0, the following inequality is satisfied:

|f − f (0)| < ε + a ϕ. Using the linearity and the positivity of Uk we obtain that

|Uk f − f (0)| ≤ ε + a Uk ϕ,

k = 0, 1, . . . . u

Taking the limit k → ∞ in the preceding inequality, since Uk ϕ −→ 0, we obtain that

     lim Uk f − f (0) ≤ ε, k→∞  and the theorem is proved.



When Uk = U and ϕ = e1 one has, as a consequence of Theorem 1, the following result. k

u

Corollary 2. Let U: C [0, 1] → C [0, 1] be a positive linear operator preserving constants and satisfying the condition U k e1 −→ s

0. Then, U k −→ δ0 . Moreover, the following equivalence is satisfied: s

u

U k −→ δ0 ⇐⇒ U k e1 −→ 0. The first application of the main result is related to the generalized weighted mean Cesàro operator. Let µ: [a, b] → R be a nondecreasing function and recall that the generalized weighted mean Cesàro operator C: C [0, 1] → C [0, 1] is defined in terms of the Stieltjes integral by Cf (x) =

1

∫

f (x t ) dµ(t ),

x ∈ [0, 1],

(11)

0

with

1 0

dµ(t ) = 1 (see, e.g., [15,23]). We have 1

∫ Ce0 = e0

t dµ(t ).

and Ce1 = e1 0

1

1

Suppose first that 0 t dµ(t ) = 1. We deduce that 0 (1 − t ) dµ(t ) = 0. Since 1 − t is positive, and vanishes at t = 1, we get that Cf (x) = f (x), for f ∈ C [0, 1] and x ∈ [0, 1], and it follows that C is the identity operator I. Thus, if C ̸= I, we have 0≤

1 0

t dµ(t ) < 1, and

Ck e1 = e1

1

∫

k

t dµ(t )

u

−→ 0.

0

As a particular case of Corollary 2, when U Galaz Fontes and Solís [15, Theorem 1].

=

C, we obtain the following generalization of a result of

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Corollary 3. The sequence (Ck )∞ k=0 of the iterates of the generalized weighted mean Cesàro operator (11) satisfies s

Ck −→



δ0 ,

I,

if C ̸= I , if C = I .

Using the properties (4), (6) and (8) we obtain, on the basis of Corollary 2, the following. Corollary 4. For all f ∈ C [0, 1], the sequences of the iterates of the Stancu operators (Snk,β f )∞ k=0 (3), the Cheney–Sharma operators k ∞ (CSnk f )∞ k=0 (5), and the Schurer operators (ßn f )k=0 (7) converge uniformly to f (0).

By using (10), from Theorem 1, we obtain: Corollary 5. For all f ∈ C [0, 1], the sequences of the iterates of the Bernstein type operators (Bnk f )∞ k=0 (9) converge uniformly to f (0). Acknowledgment The authors are grateful to the reviewers for their helpful comments and suggestions used to improve the work. References [1] R.P. Kelisky, T.J. Rivlin, Iterates of Bernstein polynomials, Pacific J. Math. 21 (1967) 511–520. [2] S. Karlin, Z. Ziegler, Iteration of positive approximation operators, J. Approximation Theory 3 (1970) 310–339. [3] F. Altomare, M. Campiti, Korovkin-type approximation theory and its applications, in: de Gruyter Studies in Mathematics, vol. 17, Walter de Gruyter & Co., Berlin, 1994. [4] H.H. Gonska, X.L. Zhou, Approximation theorems for the iterated Boolean sums of Bernstein operators, J. Comput. Appl. Math. 53 (1994) 21–31. [5] H. Oruç, N. Tuncer, On the convergence and iterates of q-Bernstein polynomials, J. Approx. Theory 117 (2002) 301–313. [6] S. Ostrovska, q-Bernstein polynomials and their iterates, J. Approx. Theory 123 (2003) 232–255. [7] J.P. King, Positive linear operators which preserve x2 , Acta Math. Hungar. 99 (2003) 203–208. [8] I.A. Rus, Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl. 292 (2004) 259–261. [9] U. Itai, On the eigenstructure of the Bernstein kernel, Electron. Trans. Numer. Anal. 25 (2006) 431–438. [10] H. Gonska, D. Kacsó, P. Piţul, The degree of convergence of over-iterated positive linear operators, J. Appl. Funct. Anal. 1 (2006) 403–423. [11] H. Gonska, I. Raşa, The limiting semigroup of the Bernstein iterates: degree of convergence, Acta Math. Hungar. 111 (2006) 119–130. [12] H. Gonska, P. Piţul, I. Raşa, Over-iterates of Bernstein–Stancu operators, Calcolo 44 (2007) 117–125. [13] H.J. Wenz, On the limits of (linear combinations of) iterates of linear operators, J. Approx. Theory 89 (1997) 219–237. [14] O. Agratini, On the iterates of a class of summation-type linear positive operators, Comput. Math. Appl. 55 (2008) 1178–1180. [15] F. Galaz Fontes, F.J. Solís, Iterating the Cesàro operators, Proc. Am. Math. Soc. 136 (2008) 2147–2153. [16] U. Abel, M. Ivan, Over-iterates of Bernstein’s operators: a short and elementary proof, Amer. Math. Monthly 116 (2009) 535–538. [17] I. Gavrea, M. Ivan, On the iterates of positive linear operators preserving the affine functions, J. Math. Anal. Appl. 372 (2010) 366–368. [18] I. Rasa, Asymptotic behaviour of certain semigroups generated by differential operators, Jaen J. Approx. 1 (2009) 27–36. [19] I. Rasa, C0 semigroups and iterates of positive linear operators: asymptotic behaviour, Rend. Circ. Mat. Palermo (2) Suppl. 82 (2010) 123–142. [20] D.D. Stancu, Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl. 13 (1968) 1173–1194. [21] E.W. Cheney, A. Sharma, On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma (2) 5 (1964) 77–84. [22] J.M. Aldaz, O. Kounchev, H. Render, Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces, Numer. Math. 114 (2009) 1–25. [23] K. Jichang, The norm inequalities for the weighted Cesaro mean operators, Comput. Math. Appl. 56 (2008) 2588–2595.