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

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Linear Positive Operators Ovidiu T. Pop and Mircea D. F ˘arca¸s

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ABOUT A CLASS OF LINEAR POSITIVE OPERATORS OBTAINED BY CHOOSING THE

NODES

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

National College "Mihai Eminescu"

5 Mihai Eminescu Street Satu Mare 440014, Romania

EMail:{ovidiutiberiu,mirceafarcas2005}@yahoo.com

Received: 15 June, 2007

Accepted: 18 March, 2009

Communicated by: S.S. Dragomir

2000 AMS Sub. Class.: 41A10, 41A25, 41A35, 41A36.

Key words: Linear positive operators, convergence theorem, the first order modulus of smoothness, approximation theorem.

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

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1. Introduction

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

Let Nbe the set of positive integers and N0 = N∪ {0}. For m ∈ N, let B_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) pm,k(x) =

m k

x^k(1−x)^m−k,

for any x ∈ [0,1]and any k ∈ {0,1, . . . , m}(see [5] or [24]). For the following construction, see [15]. Define the natural numberm₀ by

(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 number x.

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.

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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 ≥m₀and for anyk ∈ {0,1, . . . , m}.

For the real numbers α and β, α ≥ 0, m₀ and µ^(α,β) defined by (1.3) – (1.6), let the operatorsPm^(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 number m, 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 is C([0,1]) and the numbersαandβverify 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) (Lmf)(x) = 1 (1 +x)^m

m

X

k=0

m k

x^kf

k m+ 1−k

,

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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 functionf ∈C₂([0,∞))by

(1.10) (S_mf) (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 functionf ∈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(V_m)_m≥1are 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 operators. After a slight adjustment, given by E.W. Cheney and A. Sharma

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in [6], these operators take the form Z_m : B([0,1)) → C([0,1)), defined for any functionf ∈B([0,1))by

(1.12) (Zmf) (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.

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

(1.13) (Rmf)(x) = e⁻^1+x^mx

∞

X

k=0

m(m+k)^k−1 k!

x 1 +x

k

e⁻^1+x^kx 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 on I, B(I) = f|f :I →R, f bounded onI , C(I) =

f|f : I →R,f continuous onI and C_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}.

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2. Preliminaries

For the following construction and result see [16] and [18], wherep_m = mfor any m ∈Norp_m = ∞for any m ∈N. LetI, J ⊂ [0,∞)be intervals withI ∩J 6=∅.

For any m∈N and k ∈ {0,1, ..., p_m} ∩ N0 consider the nodes x_m,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 functions ψ_x : I →R, ψ_x(t) = t−xande_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→∞(Lmf)(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≥1 we have that the following con- vergences

(2.3) lim

m→∞(Lmei)(x) = ei(x),

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i∈ {0,1,2}and

(2.4) lim

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

Remark 2. From Remark1it results that for any compactK ⊂I∩J the sequences (u_m(K))m≥1,(v_m(K))m≥1,(w_m(K))m≥1depending onK exist, so that the conver- gences

(2.5) lim

m→∞um(K) = lim

m→∞vm(K) = lim

m→∞wm(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_mψ²_x)(x)≤w_m(K),

for anyx∈K and anym ∈N.

In the following, form ∈Nandk ∈ {0,1, . . . , p_m} ∩N⁰ we consider the nodes y_m,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, . . . , pm} ∩N⁰ we note thatαm,k =xm,k−ym,k.

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Definition 2.1. Form∈N, define the operatorK_m :E(I)→F(J)by

(2.11) (K_mf)(x) =

pm

X

k=0

ϕ_m,k(x)f(y_m,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].

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3. Main Results

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

ϕ_m,k(x)(x_m,k −α_m,k) +x²

pm

X

k=0

ϕ_m,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)

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≤(Lmψ²_x)(x) + 2αm(Lme1)(x) + (α²_m+ 2xαm)(Lme0)(x).

Taking Remark1and Remark2into account, it results that (3.1) holds.

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

≤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_m and taking the inequality from the proof of the Theorem3.1into account verified by (K_mψ²_x)(x)and Remark2, the inequality (3.2) follows.

Corollary 3.3. If (3.3)

pm

X

k=0

ϕm,k(x) = 1

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

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

wm(K) + 2αmvm(K) +α²_m+ 4bαm.

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Proof. It results from Theorem 3.2, because (L_me₀)(x) = 1, for any m ∈ N and x∈J, sou_m(K) = 0, for anym ∈N.

Remark 4. From the conditions of Theorem3.2we 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 onK.

In the following, by particularisation of the sequence y_m,k, m ∈ N, k ∈ {0,1, . . . , p_m} ∩N0 and applying Theorem3.1and Corollary3.3, we can obtain a convergence and approximation theorem for the new operators. In Applications1–2, let p_m =m,ϕ_m,k(x) = p_m,k(x), wherem∈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 thatu_m([0,1]) = 0, v_m([0,1]) = 0andw_m([0,1]) = _4m¹ ,m∈N. We consider the nodesy_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→∞αm = 0. In this case, the operators(K_m)m≥1 have 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.

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Application 2. We study a particular case of the Stancu operators. Letα = 10and β =−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. IfI = J = [0,∞), E(I) = C₂([0,∞)), F(J) = C([0,∞)), K = [0, b], p_m = ∞, x_m,k = _m^k, ϕ_m,k(x) = e^−mx^(mx)_k!^k, m ∈ N, k ∈ N0, we obtain the Mirakjan-Favard-Szász operators and we have that u_m(K) = 0, v_m(K) = 0 and w_m(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 ∈

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N⁰. In this case, we obtain the Baskakov operators and we have that u_m(K) = 0, vm(K) = 0andwm(K) = ^b(1+b)_2m ,m∈N. We consider the nodesym,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. IfI = J = [0,∞), E(I) = F(J) = C([0,∞)),K = [0, b], p_m =

∞,x_m,k = _m^k,

ϕm,k(x) = m(m+k)^k−1 k!

x 1 +x

k

e

−(k+m)x

1+x , m∈N, k ∈N⁰,

we obtain the Ismail-May operators and we have thatu_m(K) = 0,v_m(K) = 0and w_m(K) = ^b(1+b)_m ²,m∈N. We consider the nodesy_m,k =

√3

k²(k+1)

m , m∈N,k ∈N0

and we have thatα_m = _3m¹ . In this case, the operators(K_m)_m≥1 have the form

(Kmf)(x) =e^−mx^1+x

∞

X

k=0

m(m+k)^k−1 k!

x 1 +x

k

e⁻^1+x^kx f p3

k²(k+ 1) m

! ,

wheref ∈C([0,∞)),m∈Nandδ⁰_m =

qb(7+6b+3b²)

3m +_9m¹2,m∈N.

Application 6. We consider I = J = [0,∞), E(I) = F(J) = CB([0,∞)), K = [0, b], pm = m, xm,k = _m+1−k^k , ϕm,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

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have that u_m(K) = 0, v_m(K) = b _1+b^b m

and w_m(K) = ^4b(1+b)_m+2², m ∈ N. We consider the nodesy_m,k = _m+1−k^β^m^k , 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) = 0 and we have αm =m|1−βm|,m ∈N. The operators(Km)m≥1 have 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. If I = J = [0,1], E(I) = B([0,1]), F(J) = C([0,1]), K = [0,1], pm = ∞, xm,k = _m+k^k , ϕm,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 thatum([0,1]) = 0, v_m([0,1]) = 0and w_m([0,1]) = _4(m+1)¹ , m ∈ N. We consider the nodes y_m,k =

k+βm

m+k+βm,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 thatαm = _m+β^β^m

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

(Kmf)(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) +^β^m_(m+β^(4m+5β^m⁾

m)² ,m∈N.

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References

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JJ II

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