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SOME PROPERTIES OF THE SCHURER TYPE OPERATORS

Lucia C˘abulea

Abstract. _{In this paper are presented first the operators of Schurer which} have been introduced and investigated by F. Schurer [11] in 1962 and theirs properties. Then, we study a generalization of Durrmeyer type and a general-ization of Kantorovich type the operators of Schurer and we estimate the values of this operators for the test functions. By means of the modulus of continuity of the function used one gives evaluations of the orders of approximation by the considered operators.

1. Introduction

Let p _∈N be fixed. In 1962, F. Schurer [11], introduced and investigated the linear positive operator Bm,p : C([0,1 +p]) → C([0,1]) , defined for any

m_∈N and any f _∈C([0,1 +p]) by

(Bm,p) (x) = m+p

X

k=0

m+p k

!

xk(1₋x)m+p−k

f k

m

!

,

where Bm,p are the operators of Bernstein-Schurer. One observe that forp = 0, Bm,0 we obtain the operators of Bernstein Bm.

Theorem. 1.1. _{The operators of Bernstein-Schurer have the following}

properties:

i) (Bm,pe0) (x) = 1,(Bm,pe1) (x) =

1 + p m

x ,

(Bm,pe1) (x) = m+p_m2 [(m+p)x2+x(1−x)] ;

ii) lim

n→∞Bm,pf =f uniformly on [0,1] , for any f ∈C([0,1 +p]) ,

iii) _|(Bm,pf) (x)−f(x)| ≤ 2ω(f;δm,p,x) , for any f ∈ C([0,1 +p]) and

any x_∈[0,1] ;

iv) _|(Bm,pf) (x)−f(x)| ≤ _mpx|f′(x)|+ 2δm,p,xω(f′;δm,p,x) , for any f ∈

C1_([0_,_{1 +}_p_]) _{and any} _x_∈_[0_,_1] _{, if we noticed} _δ

m,p,x=

√

p2_x2_+(m+p)x(1

−x)

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2. A generalization of Durrmeyer type for the operators of Schurer

We consider the operators of Schurer modified into integral form [2] B∗∗

m,p:

C([0,1 +p]) _→ C([0,1]) , defined for any m _∈ N and any f _∈ C([0,1 +p]) by

B∗∗

m,pf

(x) = (m+p+ 1) m+p

X

k=0

qk m,p(x)

1

∫

0

qk

m,p(t)f(t)dt, (2.1) where

qk_m,p(x) = m+p

k

!

xk(1₋x)m+p−k are the fundamental Schurer polynomials.

Theorem. 2.1. _{The operators defined by (2.1) have the properties:} i) B∗∗

m,pe0

(x) = 1 ; ii) B∗∗

m,pe1

(x) = (m+p)x+1_m+p+2 ; iii) B∗∗

m,pe2

(x) = (m+p)(m+p−1)x2_+4(m+p)x+2

(m+p+2)(m+p+3) .

Proof.

i)

B∗∗

m,pe0

(x) = (m+p+ 1) m+p

X

k=0

m+p k

!

xk(1₋x)m+p−k 1

m+p+ 1 = 1 if on used the relations:

m+p X

k=0

m+p k

!

xk(1₋x)m+p−k = 1

and 1

∫

0

m+p k

!

tk(1₋t)m+p−k

dt= m+p

k

!

β(k+ 1, m+p₋k+ 1)

= 1

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ii) We have

B∗∗

m,pe1

(x) = (m+p+ 1) m+p

X

k=0

m+p k

!

xk(1₋x)m+p−k

β(k+ 2, m+p₋k+ 1) = (m+p+ 1) m+p

X

k=0

m+p k

!

xk(1₋k)m+p−k

·

· k+ 1

(m+p+ 2) (m+p+ 1)

= 1

m+p+ 2 +

m+p m+p+ 2x

m+p X

k=1

m+p₋1

k₋1 !

xk−1₍₁₋_x₎m+p−k

= (m+p)x+ 1

m+p+ 2 . iii) We find

B∗∗

m,pe1

(x) = (m+p+ 1) m+p

X

k=0

m+p k

!

xk(1₋x)m+p−k

·β(k+3, m+p₋k+1)

= (m+p+ 1) m+p

X

k=0

m+p k

!

xk₍₁

−x)m+p−k

·₍_m₊_p_{+ 3)(}(k_m+ 2)(₊_pk_{+ 2)(}+ 1)_m₊_p_{+ 1)}

= (m+p) (m+p−1)x

2_{+ 4 (}_m₊_p₎_x_{+ 2} (m+p+ 2) (m+p+ 3) .

Theorem. 2.2. _{The operators defined (2.1) have the properties:} i) lim

n→∞B ∗∗

m,pf =f uniformly on [0,1], (∀)f ∈C([0,1 +p]) , ii)

B∗∗

m,pf

(x)₋f(x) ≤2ω

f;√ 1

2(m+p+3)

,(_∀)f _∈C([0,1 +p]),(_∀)x_∈

[0,1], m_≥3 , p_∈N is fixed. Proof.

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If L is a linear positive operator L:C(I)_→C(I) , such thatLe0 =e0 then

|(Lf) (x)₋f(x)_{| ≤}1 +δ−1q₍_Lϕ2 x) (x)

ω(f;δ) , (_∀)f _∈ CB(I) , (∀)x∈I ,

δ_i0 and ϕx =|t−x| . We have

B∗∗

m,pf

(x)₋f(x) ≤

1 +δ−1q

B∗∗

m,pϕ2x

ω(f;δ),

B∗∗

m,pϕ2x

(x) = B∗∗

m,pe2

(x)₋2xB∗∗

m,pe1

(x) +x2B∗∗

m,pe0

(x) = 2 (m+p−3)x(1−x) + 2

(m+p+ 2) (m+p+ 3) . If m+p_≥3 it is maximal for x = 1

2 and we find

B∗∗

m,pϕ 2 x

(x)_≤ m+p+ 1

2(m+p+ 2)(m+p+ 3). We get

B∗∗

m,pf

(x)₋f(x)

≤ 1 +δ

−1

s

m+p+ 1

2(m+p+ 2(m+p+ 3) !

ω(f;δ)

≤ 1 +δ−1 s

1 2(m+p+ 3)

!

ω(f;δ).

For δ= _√ 1

2(m+p+3) we obtain the inequalities

B∗∗

m,pf

(x)₋f(x) ≤2ω



f;

1 q

2(m+p+ 3) 

.

3. A generalizations of Kantorovich type for the operators of Schurer

We consider the operators of Schurer modified into integral form [3]

B∗

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B∗

m,pf

(x) = (m+p+ 1) m+p

X

k=0

m+p k

!

xk(1₋x)m+p−k

k+1 m+p+1

Z

k m+p+1

f(t)dt

(3.1) Theorem. 3.1. _{The operators defined by (3.1) have the properties:} i) B∗

m,pe0

(x) = 1 ; ii) B∗

m,pe1

(x) = _m+p+m+p x+ 1 2(m+p+1) ; iii) B∗

m,pe2

(x) = (m+p)(m+p−1) (m+p+1)2 x2+

2(m+p) (m+p+1)2x+

1 3(m+p+1)2 .

Proof.

i) B∗

m,pe0

(x) = (m+p+ 1)m+pP k=0

m+p k

!

xk₍₁

−x)m+p−k_t k+1 m+p+1 k m+p+1 = = (m +p+ 1)m+pP

k=0

m+p k

!

xk₍₁

−x)m+p−k 1

m+p+1 = 1 , if on used the relation:

m+p X

k=0

m+p k

!

xk(1₋x)m+p−k_{= 1}_. ii) We have

B∗

m,pe1

(x) = (m+p+ 1)m+pP k=0

m+p k

!

xk₍₁

−x)m+p−k t2

2 k+1 m+p+1 k m+p+1 = = _2(m+p+1)1 m+pP

k=0

m+p k

!

xk₍₁

−x)m+p−k₍₂_k_{+ 1) =} m+p m+p+1x+

1 2(m+p+1) . iii) We find

B∗

m,pe2

(x) = (m+p+ 1)m+pP k=0

m+p k

!

xk₍₁

−x)m+p−k t3

3 k+1 m+p+1 k m+p+1 = = _3(m+p+1)1 m+pP

k=0

m+p k

!

xk₍₁

−x)m+p−k₍₃_k2_{+ 3}_k_{+ 1) =} = (m+p)(m+p+1)_(m+p+1)2 x

2₊ 2(m+p) (m+p+1)2x+

1 3(m+p+1)2 .

Theorem. 3.2. _{The operators defined by (3.1) have the properties:} i) lim m→∞ B∗ m,pf

(x) =f(x) , uniformly on [0,1] , (_∀)f _∈C([0,1 +p]) ;

ii) B∗ m,pf

(x)₋f(x) ≤2ω

f; 1 2√m+p+1

, (_∀)f _∈C([0,1 +p]), (_∀)x_∈

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i) It results from Bohman-Korovkin theorem ii) We used the properties:

If L is a linear positive operator L:C(I)_→C(I) , such thatLe0 =e0 then

|(Lf) (x)₋f(x)_{| ≤}1 +δ−1q₍_Lϕ2 x) (x)

ω(f;δ) , (_∀)f _∈ CB(I) , (∀)x ∈I,

δ_i0 and ϕx =|t−x| . We have

B∗

m,pf

(x)₋f(x) ≤

1 +δ−1q

B∗

m,pϕ2x

ω(f;δ),

B∗

m,pϕ 2 x

(x) = m+p−1

(m+p+ 1)2x(1−x) +

1

3(m+p+ 1)2. For δ < ₂√ 1

m+p+1 , we find the inequality:

B∗

m,pf

(x)₋f(x) ≤2ω



f;

1 q

2(m+p+ 1) 

.

References

[1] Agratini, O., Aproximare prin operatori liniari, Presa Universitar˜a Clu-jean˘a, 2000.

[2] C˘abulea, L., Todea, M., Generalizations of Durrmeyer type, Acta Uni-versitatis Apulensis, no. 4, 2002, 37-44.

[3] C˘abulea, L., Generalizations of Kantorovich type, Analele Universit˘at¸ii Aurel Vlaicu, Arad, 2002.

[4] Derriennic, M. M.,Sur l’approximation des functions int´egrales sur [0,1] pardes polynomes de Bernstein modifi´es, J. Approx. Theory, 31 (1981), 325-343.

[5] Durrmeyer, J. L., Une formule d’inversion de la transformée de Laplace : Application à la théorie des moments, Thède de 3e cycle, Faculté des Sciences de l’Université de Paris, 1967.

[6] Gavrea, I., The approximation of the continuous functions by means of some linear positive operators, Results in Mathematics, 30(1996), 55-66.

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[8] P˘alt˘anea, R.,Sur un operateur polynomial defin sur l’ensemble des func-tions integrables, Itinerant Seminar on Functional Equations, Approximation andConvexity, Cluj-Napoca, 1983, 100-106.

[9] Razi, Q.,Approximation of function by Kantorovich type operators, Mat. Vesnic. 41 (1989), 183-192.

[10] Sendov, B., Popov, V. A., The Averaged Moduli of Smoothness, Pure and Applied Mathematics, John Wiley & Soons, 1988.

[11] Schurer, F., Linear positive operators in approximation theory, Math. Inst. Techn. Univ. Delft Report, 1962.

[12] Stancu, D. D., Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl. 8(1969), 1173-1194. [13] Stancu, D. D., Coma, Gh., Agratini, O., Trˆambit¸a¸s , R., Analiz˘a nu-meric˘a ¸si teoria aproxim˘arii, vol. I, Presa Universitar˘a Clujean˘a, 2001.

[14] Zeng, X-M., Chen, W., On the rate of convergence of the generaliyed Durrmeyer type operators for functions of bounded variation, J. Approx. The-ory, 102(2000), 1-12.

Author:

Lucia C˘abulea