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volume 5, issue 3, article 53, 2004.

Received 10 November, 2003;

accepted 29 April, 2004.

Communicated by:A.M. Fink

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Journal of Inequalities in Pure and Applied Mathematics

PARTIAL SUMS OF CERTAIN MEROMORPHIC FUNCTIONS

DAN B ˘ARBOSU

North University of Baia Mare Faculty of Sciences

Department of Mathematics and Computer Science

Victoriei 76, 4800 Baia Mare, ROMANIA.

EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 160-03

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Kantorovich-Stancu Type Operators Dan B ˘arbosu

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J. Ineq. Pure and Appl. Math. 5(3) Art. 53, 2004

http://jipam.vu.edu.au

Abstract

Considering two given real parameters α, β which satisfy the condition 0 ≤ α≤β, D.D. Stancu ([11]) constructed and studied the linear positive operators P_m^(α,β):C([0,1])→C([0,1]), defined for anyf∈C([0,1])and anym∈Nby

P_m^(α,β)f

(x) =

m

X

k=0

pmk(x)f k+α

m+β

.

In this paper, we are dealing with the Kantorovich form of the above operators.

We construct the linear positive operatorsKm^(α,β):L1([0,1])→C([0,1]), defined for anyf ∈L1([0,1])and anym∈Nby

K_m^(α,β)f

(x) = (m+β+ 1)

m

X

k=0

pm,k(x) Z ^k+α+1

m+β+1

k+α m+β+1

f(s)ds

and we study some approximation properties of the sequencen Km^(α,β)

o

m∈N

.

2000 Mathematics Subject Classification:41A36, 41A25

Key words: Linear positive operators, Bernstein operator, Kantorovich operator, Stancu operator, First order modulus of smoothness, Shisha-Mond theorem.

1. Preliminaries

Starting with two given real parametersα, β satisfying the conditions0≤α≤ β in 1968, D.D. Stancu (see [11]) constructed and studied the linear positive operatorsPm^(α,β) : C([0,1]) → C([0,1]) defined for anyf ∈ C([0,1]) and any m ∈Nby

(1.1) P_m^(α,β)f

=

m

X

k=0

p_m,k(x)f

k+α m+p

,

wherep_m,k(x) = ^m_k

x^k(1−x)^m−kare the fundamental Bernstein polynomials ([5]).

The operators (1.1) are known in mathematical literature as "the operators of D.D. Stancu" (see ([2])).

Note that for α =β = 0, the operatorPm^(0,0) is the classical Bernstein operator B_m([5]).

In 1930, L.V. Kantorovich constructed and studied the linear positive operators K_m : L₁([0,1]) → C([0,1]) defined for any f ∈ L₁([0,1]) and any non-negative integermby

(1.2) (Kmf) (x) = (m+ 1)

m

X

k=0

pm,k(x) Z _m+1^k+1

k m+1

f(s)ds.

The operators (1.2) are known as the Kantorovich operators. These operators are obtained from the classical Bernstein operators (1.1), replacing there the value f(k/m)of the approximated function by the integral off in a neighbor- hood ofk/m.

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Following the ideas of L.V. Kantorovich ([7]), let us consider the operators Km^(α,β) : L₁([0,1]) → C([0,1]), defined for anyf ∈ C([0,1]) and anym ∈ N by

(1.3) K_m^(α,β)f

(x) = (m+β+ 1)

m

X

k=0

p_m,k(x)

Z _m+β+1^k+α+1

k+α m+β+1

f(s)ds

obtained from the Stancu type operators (1.1).

Section 2 provides some interesting approximation properties of operators (1.3), called "Kantorovich-Stancu type operators" because they are obtained starting from the Stancu type operators (1.1) following Kantorovich’s ideas (see also G.G. Lorentz [9]).

A convergence theorem for the sequencen

Km^(α,β)fo

m∈N

is proved and the rate of convergence under some assumptions on the approximated functionfis evaluated.

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2. Main Results

Lemma 2.1. The Kantorovich-Stancu type operators (1.3) are linear and posi- tive.

Proof. The assertion follows from definition (1.3).

In what follows we will denote bye_k(s) =s^k, k ∈N, the test functions.

Lemma 2.2. The operators (1.3) verify K_m^(α,β)e₀

(x) = 1, (2.1)

K_m^(α,β)e₁

(x) = m

m+β+ 1x+ α

m+β+ 1 + m+β 2(m+β+ 1)², (2.2)

(2.3) K_m^(α,β)e₂ (x)

= 1

(m+β+ 1)²

m²x²+mx(1−x) + 2αm²

m+β +α²(3m+β) m+β

+ 1

(m+β+ 1)²{mk +α}+ 1 3(m+β+ 1)² for anyx∈[0,1].

Proof. It is well known (see [11]) that the Stancu type operators (1.1) satisfy P_m^(α,β)e₀

(x) = 1 P_m^(α,β)e₁

(x) = m

m+β x+ α m+β

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P_m^(α,β)e₂

(x) = 1

(m+β)²

m²x² +mx(1−x) + 2 αm²

m+β x+ 3α²m m+β

Next we apply the definition (2.1).

Lemma 2.3. The operators (1.3) satisfy

(2.4) K_m^(α,β)((e₁−x)²;x)

= (β+ 1)²

(m+β+ 1)² x²+ m

(m+β+ 1)² x(1−x)

+ m

(m+β+ 1)²(m+β){m+ 2α(m−β−1)}x + 3α²(3m+β) + (m+β)(1−3m−3β)

3(m+β)(m+β+ 1)² for anyx∈[0,1].

Proof. From the linearity ofKm^(α,β), we get K_m^(α,β)((e1−x)²;x)

= K_m^(α,β)e₂

(x)−2x K_m^(α,β)(e₁;x) +x² K_m^(α,β)e₀ (x)

Next, we apply Lemma2.2.

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Theorem 2.4. The sequence n

Km^(α,β)f o

m∈N converges tof, uniformly on[0,1], for anyf ∈L₁([0,1]).

Proof. Using Lemma2.3, we get

m→∞lim K_m^(α,β)((e₁−x)²;x) = 0

uniformly on[0,1]. We can then apply the well known Bohman-Korovkin The- orem (see [6] and [8]) to obtain the desired result.

Next, we deal with the rate of convergence for the sequence n

Km^(α,β)fo

m∈N

, under some assumptions on the approximated functionf. In this sense, the first order modulus of smoothness will be used.

Let us recall that ifI ⊆Ris an interval of the real axis andf is a real valued function defined on I and bounded on this interval, the first order modulus of smoothness forf is the functionω1 : [0,+∞)→R, defined for anyδ≥0by (2.5) ω₁(f;δ) = sup{|f(x⁰)−f(x⁰⁰)|:x⁰, x⁰⁰∈I, |x⁰−x⁰⁰| ≤δ}. For more details, see for example [1].

Theorem 2.5. For any f ∈ L1([0,1]), any α, β ≥ 0 satisfying the condition α ≤βand eachx∈[0,1]the Kantorovich-Stancu type operators (1.3) satisfy (2.6)

K_m^(α,β)f

(x)−f(x) ≤2ω1

f;

q

δm^(α,β)(x)

, where

(2.7) δ_m^(α,β)(x) = K_m^(α,β)((e₁−x)²;x)

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Proof. From Lemma2.2follows (2.8)

K_m^(α,β)f

(x)−f(x)

≤(m+β+ 1)

m+p

X

k=0

Z _m+β+1^k+α+1

k+α m+β+1

|f(s)−f(x)|ds

On the other hand

|f(s)−f(x)| ≤ω₁(f;|s−x|)≤(1 +δ⁻²(s−x)²)ω₁(f;δ).

For|s−x|< δ, the lost increase is clear. For|s−x| ≥δ, we use the following properties

ω₁(f;λδ)≤(1 +λ)ω₁(f;δ)≤(1 +λ²)ω₁(f;δ), where we chooseλ =δ⁻¹· |s−x|.

This way, after some elementary transformation, (2.8) implies (2.9)

K_m^(α,β)f

(x)−f(x)

≤

K_m^(α,β)e0

(x) +δ⁻²K_m^(α,β)((e1−x)²;x ω1(f;δ) for anyδ >0and eachx∈[0,1].

Using next Lemma2.2and Lemma2.3, from (2.9) one obtains

(2.10)

K_m^(α,β)f

(x)−f(x)

≤ 1 +δ⁻²δ_m^(α,β)(x)

ω₁(f;δ) for anyδ≥0and eachx∈[0,1].

Taking into account Lemma 2.1, it follows that δm^(α,β)(x) ≥ 0 for each x ∈ [0,1]. Consequently, we can takeδ :=δ^(α,β)m (x)in (2.9), arriving at the desired result.

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Theorem 2.6. For anyf ∈L₁([0,1])and anyx∈[0,1]the following

(2.11)

K_m^(α,β)f

(x)−f(x) ≤2ω₁

f;

q

δ^(α,β)m ,1

holds, where

(2.12) δ_m,1^(α,β) = (β+ 1)²

(m+β+ 1)² + m²(2α+ 1)

(m+β)(m+β+ 1)² + m 4(m+β+ 1)² +3α²(3m+β) + (m+β)(1−3m−3β)

3(m+β)(m+β+ 1)² . Proof. For anyx∈[0,1], the inequality

K_m^(α,β)((e₁−x)²;x)≤δ_m,1^(α,β) holds. Consequently, applying Theorem2.5we get (2.11).

Remark 2.1. Theorem 2.5 gives us the order of local approximation (in each pointx∈[0,1]), while Theorem2.6contains an evaluation for the global order of approximation (in any pointx∈[0,1]).

Because the maximum of δm^(α,β)(x)from (2.6) depends on the relations between αandβ, it follows that it can be refined further.

Taking into account the inclusion C([0,1]) ⊂ L₁([0,1]), as consequences of Theorem2.5and Theorem2.6, follows the following two results.

Corollary 2.7. For any f ∈ C([0,1]), any α, β ≥ 0 satisfying the condition α ≤βand eachx∈[0,1], the inequality (2.6) holds.

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Corollary 2.8. For any f ∈ C([0,1]), any α, β ≥ 0 satisfying the condition α ≤βand anyx∈[0,1], the inequality (2.11) holds.

Further, we estimate the rate of convergence for smooth functions.

Theorem 2.9. For any f ∈ C¹([0,1]) and each x ∈ [0,1]the operators (1.3) verify

(2.13)

K_m^(α,β)f

(x)−f(x)

≤ |f⁰(x)| ·

m+β

2(m+β+ 1)² − β+ 1 (m+β+ 1)² x

+ 2 q

2δm^(α,β)(x)ω1

f⁰;

q

δm^(α,β)(x)

,

whereδ^(α,β)m (x)is given in (2.7).

Proof. Applying a well known result due to O. Shisha and B. Mond (see [10]), it follows that

(2.14)

K_m^(α,β)f

(x)−f(x)

≤ |f(x)| ·

K_m^(α,β)e₀

(x)−1

+|f⁰(x)| ·

K_m^(α,β)e₁

(x)−x K_m^(α,β)e₀ (x)

+ q

Km^(α,β)((e₁−x)²;x)

× (r

Km^(α,β)e₀

(x) +δ⁻¹ q

Km^(α,β)((e₁−x)²;x) )

ω₁(f⁰;δ).

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From (2.14), using Lemma2.2and Lemma2.3, we get (2.15)

K_m^(α,β)f

(x)−f(x)

≤ |f⁰(x)| ·

m+β

(m+β+ 1)² − β+ 1 (m+β+ 1)²x

+ q

δ^(α,β)m (x)

1 +δ⁻¹ q

δm^(α,β)(x)

ω₁(f⁰;δ).

Choosingδ= q

δm^(α,β)(x)in (2.15), we arrive at the desired result.

Theorem 2.10. For any f ∈ C¹([0,1]) and anyx ∈ [0,1]the operators (1.3) verify

(2.16)

K_m^(α,β)f

(x)−f(x)

≤ m+β

(m+β+ 1)²M₁+ 2√ δω₁

f⁰;√ δ

,

where

M₁ = max

x∈[0,1]|f⁰(x)|, δ= max

x∈[0,1]δ^(α,β)_m (x). Proof. The assertion follows from Theorem2.9.

Remark 2.2. Because δdepends on the relation between αand β, (2.16) can be further refined, following the ideas of D.D. Stancu [11,12].

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References

[1] O. AGRATINI, Aproximare prin operatori liniari (Romanian), Presa Uni- versitar˘a Clujean˘a, Cluj-Napoca, 2000.

[2] F. ALTOMAREANDM. CAMPITI, Korovkin-type Approximation Theory and its Applications, de Gruyter Series Studies in Mathematics, vol. 17, Walter de Gruyter & Co. Berlin, New-York, 1994.

[3] D. B ˘ARBOSU, A Voronovskaja type theorem for the operator of D.D.

Stancu, Bulletins for Applied & Computer Mathematics, BAM 1998- C/2000, T.U. Budapest (2002), 175–182.

[4] D. B ˘ARBOSU, Kantorovich-Schurer operators (to appear in Novi-Sad Journal of Mathematics)

[5] S.N. BERNSTEIN, Démonstration du théorème de Weierstrass fondée sur le calcul de probabilités, Commun. Soc. Math. Kharkow, 13(2) (1912- 1913), 1–2.

[6] H. BOHMAN, On approximation of continuous and analytic functions, Ark. Mat., 2 (1952), 43–56.

[7] L.V. KANTOROVICH, Sur certain développements suivant les polynômes de la forme de S. Bernstein, I, II, C.R. Acad. URSS (1930), 563–568, 595–

600.

[8] P.P. KOROVKIN, On convergence of linear operators in the space of con- tinuous functions (Russian), Dokl. Akad. Nauk. SSSR (N.S.), 90 (1953), 961–964.

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[9] G.G. LORENTZ, Bernstein Polynomials, Toronto: Univ. of Toronto Press, 1953.

[10] O. SHISHAANDB. MOND, The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. U.S.A, 60 (1968), 1196–2000.

[11] D.D. STANCU, Approximation of function by a new class of polynomial operators, Rev. Roum. Math. Pures et Appl., 13(8) (1968), 1173–1194.

[12] D.D. STANCU, Asupra unei generaliz˘ari a polinoamelor lui Bernstein (Romanian), Studia Universitatis Babe¸s-Bolyai, 14(2) (1969), 31–45.

[13] D.D. STANCU, GH. COMAN, O. AGRATINI AND R. TRˆIMBI ¸TA ¸S, Analiz˘a Numeric˘a ¸si Teoria Aproxim˘arii, Vol.I (Romanian), Presa Uni- versitar˘a Clujean˘a, Cluj-Napoca, 2001.