accepted 18 March, 2009 Communicated by S.S Dragomir ABSTRACT

ABOUT A CLASS OF LINEAR POSITIVE OPERATORS OBTAINED BY CHOOSING THE NODES

OVIDIU T. POP AND MIRCEA D. F ˘ARCA ¸S NATIONALCOLLEGE"MIHAIEMINESCU"

5 MIHAIEMINESCUSTREET

SATUMARE440014, ROMANIA

[email protected] [email protected] Received 15 June, 2007; accepted 18 March, 2009

Communicated by S.S Dragomir

ABSTRACT. In this paper we consider the given linear positive operators(Lm)m≥1 and with their help, we construct linear positive operators(Km)m≥1. We study the convergence, the eval- uation for the rate of convergence in terms of the first modulus of smoothness for the operators (Km)_m≥1.

Key words and phrases: Linear positive operators, convergence theorem, the first order modulus of smoothness, approxima- tion theorem.

2000 Mathematics Subject Classification. 41A10, 41A25, 41A35, 41A36.

1. INTRODUCTION

In this section, we recall some notions and operators which we will use in this article.

LetNbe the set of positive integers andN0 = N∪ {0}. Form ∈ N, letB_m : C([0,1]) → C([0,1])be Bernstein operators, defined for any functionf ∈C([0,1])by

(1.1) (B_mf)(x) =

m

X

k=0

p_m,k(x)f k

m

,

wherep_m,k(x)are the fundamental polynomials of Bernstein, defined as follows

(1.2) p_m,k(x) =

m k

x^k(1−x)^m−k,

for anyx ∈[0,1]and anyk ∈ {0,1, . . . , m}(see [5] or [24]). For the following construction, see [15]. Define the natural numberm₀by

(1.3) m₀ =

( max(1,−[β]), if β ∈R−Z; max(1,1−β), if β ∈Z,

where[x],{x}denote the integer and fractional parts respectively of a real numberx.

201-07

For the real numberβ, we have that

(1.4) m+β ≥γ_β

for any natural numberm,m ≥m₀, where

(1.5) γ_β =m₀ +β =

( max (1 +β,{β}), if β ∈R−Z; max(1 +β,1), if β ∈Z. For the real numbersα, β,α≥0, we note

(1.6) µ^(α,β) =

( 1, if α≤β;

1 + ^α−β_γ

β , if α > β.

For the real numbersαandβ,α≥0, we have that1≤µ^(α,β)and

(1.7) 0≤ k+α

m+β ≤µ^(α,β)

for any natural numberm,m ≥m0 and for anyk ∈ {0,1, . . . , m}.

For the real numbersαandβ,α≥0,m₀andµ^(α,β)defined by (1.3) – (1.6), let the operators Pm^(α,β) :C [0, µ^(α,β)]

→C [0,1]

, defined for any functionf ∈C [0, µ^(α,β)] by

(1.8) P_m^(α,β)f

(x) =

m

X

k=0

p_m,k(x)f

k+α m+β

,

for any natural numberm, m ≥ m₀ and for any x ∈ [0,1]. These operators are called Stancu operators, and were introduced and studied in 1969 by D.D. Stancu in the paper [23]. In [23], the domain of definition of Stancu’s operators isC([0,1])and the numbersα andβ verify the condition0≤α≤β.

In 1980, G. Bleimann, P. L. Butzer and L. Hahn introduced in [4] a sequence of linear positive operators(L_m)_m≥1,L_m :C_B([0,∞))→C_B([0,∞)), defined for any functionf ∈C_B([0,∞)) by

(1.9) (L_mf)(x) = 1

(1 +x)^m

m

X

k=0

m k

x^kf

k m+ 1−k

,

for anyx ∈ [0,∞)and anym ∈ N, whereC_B([0,∞)) = {f|f : [0,∞) → R, f is bounded and continuous on[0,∞)}.

Form ∈N, consider the operatorsS_m :C₂([0,∞)) → C([0,∞))defined for any function f ∈C₂([0,∞))by

(1.10) (Smf) (x) = e^−mx

∞

X

k=0

(mx)^k k! f

k m

, for anyx∈[0,∞), where

C₂([0,∞)) =

f ∈C([0,∞)) : lim

x→∞

f(x)

1 +x² exists and is finite

.

The operators(S_m)_m≥1 are called Mirakjan-Favard-Szász operators and were introduced in 1941 by G. M. Mirakjan in [12].

They were intensively studied by J. Favard in 1944 in [8] and O. Szász in 1950 in [25].

For m ∈ N, the operator V_m : C₂([0,∞)) → C([0,∞))is defined for any function f ∈ C₂([0,∞))by

(1.11) (V_mf) (x) = (1 +x)^−m

∞

X

k=0

m+k−1 k

x 1 +x

k

f k

m

, for anyx∈[0,∞).

The operators(Vm)_m≥1 are named Baskakov operators and they were introduced in 1957 by V. A. Baskakov in [2].

W. Meyer-König and K. Zeller have introduced in [11] a sequence of linear and positive op- erators. After a slight adjustment, given by E.W. Cheney and A. Sharma in [6], these operators take the formZ_m :B([0,1)) →C([0,1)), defined for any functionf ∈B([0,1))by

(1.12) (Z_mf) (x) =

∞

X

k=0

m+k k

(1−x)^m+1x^kf k

m+k

, for anym∈Nand for anyx∈[0,1).

These operators are called the Meyer-König and Zeller operators.

Observe thatZ_m :C([0,1]) →C([0,1]),m ∈N.

In [10], M. Ismail and C.P. May consider the operators(R_m)_m≥1.

Form∈N,R_m :C([0,∞))→C([0,∞))is defined for any functionf ∈C([0,∞))by (1.13) (R_mf)(x) =e^−1+xmx

∞

X

k=0

m(m+k)^k−1 k!

x 1 +x

k

e^−1+xkx f k

m

for anyx∈[0,∞).

We considerI ⊂ R, I an interval and we shall use the following function sets: E(I), F(I) which are subsets of the set of real functions defined onI, B(I) =

f|f :I →R,f bounded onI ,C(I) =

f|f :I →R,f continuous onI andC_B(I) = B(I)∩C(I).

If f ∈ B(I), then the first order modulus of smoothness of f is the function ω(f; ·) : [0,∞)→Rdefined for anyδ ≥0by

(1.14) ω(f;δ) = sup{|f(x⁰)−f(x⁰⁰)|:x⁰, x⁰⁰ ∈I,|x⁰−x⁰⁰| ≤δ}. 2. PRELIMINARIES

For the following construction and result see [16] and [18], where p_m = mfor anym ∈ N orp_m = ∞for anym ∈ N. Let I, J ⊂ [0,∞)be intervals with I ∩J 6= ∅. For anym∈N andk∈ {0,1, ..., pm} ∩N⁰ consider the nodesxm,k ∈ I and the functionsϕm,k : J →Rwith the property thatϕm,k(x) ≥ 0for anyx ∈ J. LetE(I)andF(J)be subsets of the set of real functions defined onI, respectivelyJ so that the sum

pm

X

k=0

ϕ_m,k(x)f(x_m,k)

exists for anyf ∈E(I),x ∈J andm ∈N. For anyx∈I consider the functionsψ_x :I →R, ψ_x(t) = t−xand e_i : I → R, e_i(t) = tⁱ for anyt ∈ I, i ∈ {0,1,2}. In the following, we suppose that for anyx∈I we haveψ_x ∈E(I)ande_i ∈E(I),i∈ {0,1,2}.

Form∈N, let the given operatorL_m :E(I)→F(J)defined by

(2.1) (L_mf)(x) =

pm

X

k=0

ϕ_m,k(x)f(x_m,k)

with the property that the convergence

(2.2) lim

m→∞(L_mf)(x) =f(x) is uniform on any compactK ⊂I∩J, for anyf ∈E(I)∩C(I).

Remark 1. From (2.2), for the operators(L_m)m≥1we have that the following convergences

(2.3) lim

m→∞(L_me_i)(x) = e_i(x), i∈ {0,1,2}and

(2.4) lim

m→∞(L_mψ_x²)(x) = 0 are uniform on any compactK ⊂I∩J.

Remark 2. From Remark 1 it results that for any compactK ⊂I∩Jthe sequences(um(K))m≥1, (vm(K))m≥1,(wm(K))m≥1 depending onKexist, so that the convergences

(2.5) lim

m→∞u_m(K) = lim

m→∞v_m(K) = lim

m→∞w_m(K) = 0 are uniform onK and

(2.6) |(L_me₀)(x)−1| ≤u_m(K),

(2.7) |(L_me₁)(x)−x| ≤v_m(K),

(2.8) (L_mψ_x²)(x)≤w_m(K),

for anyx∈K and anym∈N.

In the following, form∈Nandk ∈ {0,1, . . . , pm} ∩N⁰ we consider the nodesym,k ∈Iso that

(2.9) α_m = sup

k∈{0,1,...,pm}∩N0

|x_m,k−y_m,k|<∞ for anym∈Nand

(2.10) lim

m→∞α_m = 0.

Form∈Nandk ∈ {0,1, . . . , p_m} ∩N0 we note thatα_m,k =x_m,k −y_m,k. Definition 2.1. Form∈N, define the operatorK_m:E(I)→F(J)by

(2.11) (Kmf)(x) =

pm

X

k=0

ϕm,k(x)f(ym,k), for anyx∈I and anyf ∈E(I).

Remark 3. Similar ideas to the construction above can be found in the recent papers [9] and [13].

3. MAINRESULTS

In this section, we study the operators defined by (2.11).

Theorem 3.1. For anyf ∈E(I)∩C(I)we have that the convergence

(3.1) lim

m→∞(Kmf)(x) = f(x) is uniform on any compactK ⊂I∩J.

Proof. Forx∈Kandm∈Nwe have that

(K_mψ_x²)(x) = (K_me₂)(x)−2x(K_me₁)(x) +x²(K_me₀)(x)

=

pm

X

k=0

ϕ_m,k(x)y_m,k² −2x

pm

X

k=0

ϕ_m,k(x)y_m,k +x²

pm

X

k=0

ϕ_m,k(x)

=

pm

X

k=0

ϕ_m,k(x)(x_m,k −α_m,k)²

−2x

pm

X

k=0

ϕm,k(x)(xm,k −αm,k) +x²

pm

X

k=0

ϕm,k(x)

=

pm

X

k=0

ϕ_m,k(x)x²_m,k −2

pm

X

k=0

ϕ_m,k(x)x_m,kα_m,k

+

pm

X

k=0

ϕ_m,k(x)α_m,k² −2x

pm

X

k=0

ϕ_m,k(x)x_m,k

+ 2x

pm

X

k=0

ϕ_m,k(x)α_m,k +x²

pm

X

k=0

ϕ_m,k(x)

≤(Lmψ_x²)(x) + 2αm(Lme1)(x) + (α_m² + 2xαm)(Lme0)(x).

Taking Remark 1 and Remark 2 into account, it results that (3.1) holds.

Theorem 3.2. If f ∈ E(I ∩J)∩C(I ∩J), then for anyx ∈ K = [a, b] ⊂ I ∩J and any m∈N, we have that

|(K_mf)(x)−f(x)| ≤ |f(x)| |(L_me₀(x))−1|+ ((L_me₀)(x) + 1)ω(f;δ_m,x) (3.2)

≤M u_m(K) + (2 +u_m(K))ω(f;δ_m), where

δ_m,x =p

(L_me₀)(x)[(L_mψ_x²)(x) + 2α_m(L_me₁)(x) + (α²_m+ 2xα_m)(L_me₀)(x)], δ_m =p

(1 +u_m(K))[w_m(K) + 2α_m(b+v_m(K) + (α²_m+ 2bα_m)(1 +u_m(K))]

and

M = sup{|f(x)|:x∈K}.

Proof. We apply the Shisha-Mond Theorem (see [22] or [24]) for the operatorK_mand taking the inequality from the proof of the Theorem 3.1 into account verified by(Kmψ²_x)(x)and Remark

2, the inequality (3.2) follows.

Corollary 3.3. If (3.3)

pm

X

k=0

ϕ_m,k(x) = 1

for anyx ∈ J, then for anyf ∈ E(I∩J)∩C(I ∩J), anyx ∈ K = [a, b] ⊂ I ∩J and any m∈Nwe have that

(3.4) |(K_mf)(x)−f(x)| ≤2ω(f;δ_m,x)≤2ω(f;δ⁰_m) whereδ_m⁰ =p

w_m(K) + 2α_mv_m(K) +α²_m+ 4bα_m.

Proof. It results from Theorem 3.2, because (L_me₀)(x) = 1, for any m ∈ N and x ∈ J, so

u_m(K) = 0, for anym∈N.

Remark 4. From the conditions of Theorem 3.2 we have that

|(K_mf)(x)−f(x)| ≤M u_m(K) + (2 +u_m(K))ω(f;δ_m) and because lim

m→∞δ_m = 0, it results that the convergence lim

m→∞(K_mf)(x) =f(x)is uniform on K.

In the following, by particularisation of the sequencey_m,k, m∈ N,k ∈ {0,1, . . . , p_m} ∩N0

and applying Theorem 3.1 and Corollary 3.3, we can obtain a convergence and approximation theorem for the new operators. In Applications 1 – 2, letp_m =m,ϕ_m,k(x) = p_m,k(x), where m∈N,k ∈ {0,1, . . . , m}andK = [0,1].

Application 1. If I = J = [0,1], E(I) = F(J) = C([0,1]), x_m,k = _m^k, m ∈ N, k ∈ {0,1, . . . , m}, we obtain the Bernstein operators. We have that um([0,1]) = 0, vm([0,1]) = 0 and w_m([0,1]) = _4m¹ , m ∈ N. We consider the nodes y_m,k =

√

k(k+1)

m , m ∈ N, k ∈ {0,1, . . . , m}. Then it is verified immediately thatα_m = ¹

m+√

m(m+1),m ∈ Nand lim

m→∞α_m = 0. In this case, the operators(K_m)_m≥1have the form

(K_mf)(x) =

m

X

k=0

p_m,k(x)f

pk(k+ 1) m

! , f ∈C([0,1]),x∈[0,1],m ∈Nandδ_m⁰ <q ₅

4m + ²

m+

√

m(m+1) < ₂^√3_m,m∈N.

Application 2. We study a particular case of the Stancu operators. Letα = 10andβ = −¹₂. We obtainI = [0,22]and for anyf ∈C([0,22]),x∈[0,1]andm ∈N

P_m^(10,−1/2)f (x) =

m

X

k=0

p_m,k(x)f

2k+ 20 2m−1

.

We consider the nodesy_m,k = ^(4k+40)m_(2m−1)2 . In this case, the operators(K_m)m≥1 have the form (Kmf)(x) =

m

X

k=0

pm,k(x)f

m(4k+ 40) (2m−1)²

, wheref ∈C([0,22]),x∈[0,1],m∈Nandδ_m⁰ <

√

36m³+2220m²−399m+81

(2m−1)² < ^√_2m−1⁴⁵ ,m ∈N. Application 3. If I = J = [0,∞), E(I) = C2([0,∞)), F(J) = C([0,∞)), K = [0, b], pm =∞, xm,k = _m^k, ϕm,k(x) = e^−mx(mx)_k!^k,m ∈ N, k ∈ N⁰, we obtain the Mirakjan-Favard- Szász operators and we have that um(K) = 0, vm(K) = 0 and wm(K) = _m^b, m ∈ N. We consider the nodesy_m,k = _m(2k+1)^2k(k+1),m ∈N,k∈N0 and we have thatα_m = _2m¹ ,m∈N. In this case, the operators(K_m)m≥1 have the form

(K_mf)(x) =e^−mx

∞

X

k=0

(mx)^k k! f

2k(k+ 1) m(2k+ 1)

,

wheref ∈C2([0,∞)),x∈[0,∞),m∈Nandδ⁰_m = q3b

m +_4m¹2,m∈N.

Application 4. Let I = J = [0,∞), E(I) = C₂([0,∞)), F(J) = C([0,∞)), K = [0, b], p_m =∞, x_m,k = _m^k, ϕ_m,k(x) = (1 +x)^{−m m+k−1}_{k x}

1+x

k

, m ∈ N, k ∈ N0. In this case, we obtain the Baskakov operators and we have thatu_m(K) = 0,v_m(K) = 0andw_m(K) = ^b(1+b)_2m , m∈N. We consider the nodesy_m,k =

√4k²+4k+2

2m ,m∈N,k ∈N0and we have thatα_m = ¹

m√ 2. The operators(K_m)m≥1have the form

(K_mf)(x) = (1 +x)^−m

∞

X

k=0

m+k−1 k

x 1 +x

k

f

√4k²+ 4k+ 2 2m

! , wheref ∈C₂([0,∞)),x∈[0,∞),m∈Nandδ⁰_m =

qb(b+1+2√ 2)

m +_2m¹2,m∈N.

Application 5. If I = J = [0,∞), E(I) = F(J) = C([0,∞)), K = [0, b], p_m = ∞, x_m,k = _m^k,

ϕ_m,k(x) = m(m+k)^k−1 k!

x 1 +x

k

e^−(k+m)x1+x , m∈N, k ∈N0,

we obtain the Ismail-May operators and we have thatu_m(K) = 0,v_m(K) = 0andw_m(K) =

b(1+b)²

m , m ∈ N. We consider the nodes y_m,k =

√3

k²(k+1)

m , m ∈ N, k ∈ N0 and we have that α_m = _3m¹ . In this case, the operators(K_m)m≥1 have the form

(K_mf)(x) =e^−mx1+x

∞

X

k=0

m(m+k)^k−1 k!

x 1 +x

k

e^−1+xkx f p3

k²(k+ 1) m

! , wheref ∈C([0,∞)),m∈Nandδ⁰_m =

qb(7+6b+3b²)

3m + _9m¹2,m∈N.

Application 6. We considerI = J = [0,∞),E(I) = F(J) = C_B([0,∞)), K = [0, b], p_m = m, x_m,k = _m+1−k^k ,ϕ_m,k(x) = _(1+x)¹ m

m k

x^k,m ∈N,k ∈ {0,1, . . . , m}. In this case we obtain the Bleimann-Butzer-Hahn operators and we have thatum(K) = 0, vm(K) = b _1+b^b m

and w_m(K) = ^4b(1+b)_m+2², m ∈N. We consider the nodes y_m,k = _m+1−k^βmk , m ∈N, k ∈ {0,1, . . . , m}, where(β_m)_m≥1 is a sequence of positive real numbers such that lim

m→∞m(1−β_m) = 0and we haveα_m=m|1−β_m|,m∈N. The operators(K_m)m≥1have the form

(K_mf)(x) = (1 +x)^−m

m

X

k=0

m k

x^kf

β_mk m+ 1−k

, wherex∈[0,∞),m∈N,f ∈C_B([0,∞)).

Application 7. IfI = J = [0,1], E(I) = B([0,1]), F(J) = C([0,1]),K = [0,1], p_m = ∞, x_m,k = _m+k^k , ϕ_m,k(x) = ^m+k_k

(1−x)^m+1x^k, m ∈ N, k ∈ N0, we obtain the Meyer-König and Zeller operators and we have thatu_m([0,1]) = 0,v_m([0,1]) = 0andw_m([0,1]) = _4(m+1)¹ , m ∈ N. We consider the nodes y_m,k = _m+k+β^k+βm

m, m ∈ N, k ∈ N0, where (β_m)m≥1 is a sequence of positive real numbers so that lim

m→∞

βm

m+βm = 0. Then it is verified immediately that α_m = _m+β^βm

m,m∈Nand the operators(K_m)m≥1have the form (K_mf)(x) =

∞

X

k=0

m+k k

(1−x)^m+1x^kf

k+β_m m+k+β_m

,

wheref ∈B([0,1]),x∈[0,1],m ∈Nandδ_m⁰ = q 1

4(m+1) +^βm_(m+β^(4m+5βm)

m)² ,m∈N. REFERENCES

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

[2] V.A. BASKAKOV, An example of a sequence of linear positive operators in the space of continuous functions, Dokl. Acad. Nauk, USSR, 113 (1957), 249–251.

[3] M. BECKER AND R.J. NESSEL, A global approximation theorem for Meyer-König and Zeller operators, Math. Zeitschr., 160 (1978), 195–206.

[4] G. BLEIMANN, P.L. BUTZERANDL.A. HAHN, Bernstein-type operator approximating contin- uous functions on the semi-axis, Indag. Math., 42 (1980), 255–262.

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

[6] E.W. CHENEYANDA. SHARMA, Bernstein power series, Canadian J. Math., 16(2) (1964), 241–

252.

[7] Z. DITZIANANDV. TOTIK, Moduli of Smoothness, Springer Verlag, Berlin, 1987.

[8] J. FAVARD, Sur les multiplicateurs d’interpolation, J. Math. Pures Appl., 23(9) (1944), 219–247.

[9] M. F ˘ARCA ¸S, An extension for the Bernstein-Stancu operators, An. Univ. Oradea Fasc. Mat., Tom XV (2008), 23–27.

[10] M. ISMAIL AND C.P. MAY, On a family of approximation operators, J. Math. Anal. Appl., 63 (1978), 446–462.

[11] W. MEYER-KÖNIGANDK. Zeller, Bernsteinsche Potenzreihen, Studia Math., 19 (1960), 89–94.

[12] G.M. MIRAKJAN, Approximation of continuous functions with the aid of polynomials, Dokl.

Acad. Nauk SSSR, 31 (1941), 201–205 (Russian).

[13] C. MORTICIANDI. OANCEA, A nonsmooth extension for the Bernstein-Stancu operators and an application, Studia Univ. ”Babe¸s-Bolyai”, Mathematica, LI(2) (2006), 69–81.

[14] M.W. MÜLLER, Die Folge der Gammaoperatoren, Dissertation, Stuttgart, 1967.

[15] O.T. POP, New properties of the Bernstein-Stancu operators, An. Univ. Oradea Fasc. Mat., Tom XI (2004), 51–60.

[16] O.T. POP, The generalization of Voronovskaja’s theorem for a class of linear and positive operators, Rev. Anal. Num. Théor. Approx., 34(1) (2005), 79–91.

[17] O.T. POP, About a class of linear and positive operators, Carpathian J. Math., 21(1-2) (2005), 99–108.

[18] O.T. POP, About some linear and positive operators defined by infinite sum, Dem. Math., XXXIX(2) (2006), 377–388.

[19] O.T. POP, On operators of the type Bleimann, Butzer and Hahn, Anal. Univ. Timi¸soara, XLIII(1) (2005), 115–124.

[20] O.T. POP, The generalization of Voronovskaja’s theorem for exponential operators, Creative Math

& Inf., 16 (2007), 54–62.

[21] O.T. POP, About a general property for a class of linear positive operators and applications, Rev.

Anal. Num. Théor. Approx., 34(2) (2005), 175–180.

[22] O. SHISHAAND B. MOND, The degree of convergence of linear positive operators, Proc. Nat.

Acad. Sci. USA, 60 (1968), 1196–1200.

[23] D.D. STANCU, Asupra unei generaliz˘ari a polinoamelor lui Bernstein, Studia Univ. Babe¸s-Bolyai, Ser. Math.-Phys., 14 (1969), 31–45 (Romanian).

[24] D.D. STANCU, GH. COMAN, O. AGRATINI AND R. TRÎMBI ¸TA ¸S, Analiz˘a numeric˘a ¸si teoria aproxim˘arii, I, Presa Universitar˘a Clujean˘a, Cluj-Napoca, 2001 (Romanian).

[25] O. SZÁSZ, Generalization of. S.N. Bernstein’s polynomials to the infinite interval, J. Research, National Bureau of Standards, 45 (1950), 239–245.

[26] A.F. TIMAN, Theory of Approximation of Functions of Real Variable, New York: Macmillan Co., 1963, MR22#8257.