About a class of linear positive operators

About a class of linear positive operators

Ovidiu T. Pop and Mircea D. F˘arca¸s

Abstract

In this paper we construct a class of linear positive operators (Lm)m≥1 with the help of some nodes. We study the convergence and we demonstrate the Voronovskaja-type theorem for them. By particularization, we obtain some known operators.

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

Key words: Linear positive operators, convergence theorem.

1 Introduction

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

LetN be the set of positive integers and N₀ =N∪ {0}. For m ∈N, let pm,k(x) the fundamental polynomials of Bernstein, defined as follows

(1.1) pm,k(x) =

µm k

¶

x^k(1−x)^m−k,

1Received 9 November 2007

Accepted for publication (in revised form) 4 December 2007

59

for anyx∈[0,1] and anyk ∈ {0,1, . . . , m}(see [5] or [21]). For the following construction see [15]. Define the natural number m₀ by

(1.2) m0 =

( max{1,−[β]}, if β ∈R\Z max{1,1−β}, if β ∈Z. For the real number β, we have that

(1.3) m+β ≥γ_β

for any natural number m, m≥m₀, where (1.4) γβ =m0+β =

( max©

1 +β,{β}ª

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

(1.5) µ^(α,β) =







1, if α≤β

1 + α−β

γ_β , if α > β.

For the real numbersα and β,α ≥0, we have that 1≤µ^(α,β) and

(1.6) 0≤ k+α

m+β ≤µ^(α,β)

for any natural number m, m≥m0 and for any k∈ {0,1, . . . , m}.

For the real numbers α and β, α ≥ 0, m0 and µ^(α,β) defined by (1.2)- (1.6), let the operators Pm^(α,β) : C¡

[0, µ^(α,β)]¢

→ C¡ [0,1]¢

, defined for any function f ∈C¡

[0, µ^(α,β)]¢ by

(1.7) ¡

P_m^(α,β)f¢ (x) =

m

X

k=0

pm,k(x)f

µk+α m+β

¶ ,

for any natural number m, m ≥ m0 and for any x ∈ [0,1]. These opera- tors are named Stancu operators, introduced and studied in 1969 by D. D.

Stancu in the paper [20]. In [20], the domain of definition of the Stancu op- erators isC([0,1]) and the numbersαandβverify the condition 0 ≤α ≤β.

Remark 1.1. For α =β = 0 we obtain the Bernstein operators.

Remark 1.2. For α= 0, p∈N₀ and choosing m bym+p andp bym−p, we obtain the Schurer operators.

In 1980, G. Bleimann, P. L. Butzer and L. Hahn introduced in [4] a se- quence of linear positive operators (Lm)m≥1,Lm :CB([0,∞))→CB([0,∞)), defined for any function f ∈CB([0,∞)) by

(1.8) (Lmf)(x) = 1 (1 +x)^m

m

X

k=0

µm k

¶ x^kf

µ k m+ 1−k

¶ ,

for anyx∈[0,∞) and any m∈N, whereCB([0,∞)) ={f|f : [0,∞)→R, f bounded and continuous on [0,∞)}.

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

(1.9) (Smf) (x) = e^−mx

∞

X

k=0

(mx)^k k! f

µk m

¶ ,

for anyx∈[0,∞), whereC2([0,∞)) = n

f ∈C([0,∞)) : lim

x→∞

f(x)

1 +x² exists and is finite o

.

The operators (Sm)_m≥1 are named Mirakjan-Favard-Sz´asz operators and were introduced in 1941 by G. M. Mirakjan in [11].

They were intensively studied by J. Favard in 1944 in [8] and O. Sz´asz in 1950 in [22].

Let for m ∈ N the operators V_m : C₂([0,∞)) → C([0,∞)) be defined for any function f ∈C₂([0,∞)) by

(1.10) (Vmf) (x) = (1 +x)^−m

∞

X

k=0

µm+k−1 k

¶ µ x 1+x

¶k

f µk

m

¶ , for any x∈[0,∞).

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

W. Meyer-K¨onig and K. Zeller have introduced in [10] a sequence of linear and positive operators. After a slight adjustment given by E. W. Ch- eney and A. Sharma in [6], these operators take the form Z_m :B([0,1)) → C([0,1)), defined for any functionf ∈B([0,1)) by

(1.11) (Zmf) (x) =

∞

X

k=0

µm+k k

¶

(1−x)^m+1x^kf µ k

m+k

¶ , for any m ∈N and for any x∈[0,1).

These operators are named the Meyer-K¨onig and Zeller operators.

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

In the paper [9], M. Ismail and C. P. May consider the operators (Rm)m≥1. For m ∈ N, R_m : C([0,∞)) → C([0,∞)) is defined for any function f ∈C([0,∞)) by

(1.12) (Rmf)(x) = e^−1+xmx

∞

X

k=0

m(m+k)^k−1 k!

µ x 1 +x

¶k

e^−1+xkx f µk

m

¶

for any x∈[0,∞).

We considerI ⊂R,I an interval and we shall use the following functions 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 on Iª

, C(I) = ©

f|f : I → R, f continuous on Iª

and C_B(I) = B(I)∩C(I). For any x ∈ I, consider the function ψ_x :I →R defined by ψ_x(t) =t−x, for any t∈I.

2 Preliminaries

The following construction is about the idea from [15]. Let I, J be real intervals with I∩J 6=∅andpm =mfor anym∈N(the finite case) orpm =

∞for anym ∈N(the infinite case). For anym∈Nandk∈ {0,1, ..., pm}∩N₀, consider the nodes xm,k ∈ I and the functions ϕm,k : J → R with the property thatϕ_m,k(x)≥0, for anyx∈J. We suppose that for any compact K ⊂I∩J there exists the sequence (um(K))m≥1, depending onK such that

(2.1) lim

m→∞um(K) = 0

uniformly on K and (2.2)

¯

¯

¯

¯

¯

pm

X

k=0

ϕm,k(x)−1

¯

¯

¯

¯

¯

≤um(K)

for any x∈K, any m ∈Nand we note u(K) = sup{u_m(K) :m∈K}.

Remark 2.1. From (2.1)it result that lim

m→∞

pm

X

k=0

ϕ_m,k(x) = 1, for anyx∈J. Let a fixed function w:I →(0,∞), called the weight function and the set functions

(2.3) E_w(I) ={f|f :I →Rsuch thatwf is bounded onI}.

Forf ∈E_w(I) there exists a positive constant M(f), depending onf, such that w(x)|f(x)| ≤ M(f) for any x ∈ I. Then, for m ∈ N and x ∈ J, and taking in the end (2.2) into account, we have

¯

¯

¯

¯

¯

pm

X

k=0

ϕ_m,k(x)f(xm,k)

¯

¯

¯

¯

¯

≤

pm

X

k=0

ϕ_m,k(x)|f(x_m,k)| ≤ M(f) w(x)

pm

X

k=0

ϕ_m,k(x)≤

≤ M(f)

w(x)(1 +um(K))≤ M(f)

w(x)(1 +u(K)), from where it results that the sum

pm

X

k=0

ϕm,k(x)f(xm,k) exists.

We consider the operators (Lm)m≥1 defined by

(2.4) (Lmf)(x) =

pm

X

k=0

ϕ_m,k(x)f(xm,k) for any f ∈E_w(I),x∈J and m∈N.

Proposition 2.1. The operators(Lm)m≥1 are linear and positive onE_w(I).

Proof. The proof follows immediately.

3 Main results

In the following, let s be fixed natural number, s even. For any x ∈I ∩J we suppose that ψ_xⁱ ∈ E_w(I), where i ∈ {0,1, . . . , s+ 2}. For m ∈ N and i∈ {0,1, . . . , s+ 2} define

(3.1) (TiLm)(x) =mⁱ(Lmψ_xⁱ)(x) = mⁱ

pm

X

k=0

(xm,k−x)ⁱϕm,k(x) for any x∈I∩J.

Theorem 3.1. Let x∈I∩J and we suppose that there exist αs+2 ≥0 and m(s) ∈ N such that ^(Ts+2_mαs+2^Lm)(x) is bounded for any m ∈ N, m ≥ m(s). If γ ∈R verify γ < s+ 2−α_s+2 and δ > 0, then

(3.2) lim

m→∞m^γ X

|xm,k−x|≥δ

(xm,k −x)^sϕm,k(x) = 0.

If for the compact interval K ⊂ I ∩J exist m(s) ∈ N and the constant ks+2(K) ∈ R, depending on K, such that for any m ∈ N, m ≥ m(s) and x∈K we have

(3.3) (T_s+2L_m)(x)

m^αs+2 ≤k_s+2(K), then the convergence given in (3.2) is uniform on K.

Proof. We have X

|xm,k−x|≥δ

(xm,k−x)^sϕm,k(x)≤ 1 δ²

X

|xm,k−x|≥δ

(xm,k −x)^s+2ϕm,k(x)≤

≤

pm

X

k=0

(xm,k−x)^s+2ϕ_m,k(x) = 1

δ²m^s+2(Ts+2L_m)(x), so

(3.4) m^γ X

|xm,k−x|≥δ

(xm,k −x)^sϕm,k(x)≤ 1

δ²m^s+2−γ(Ts+2Lm)(x).

But 1

δ²m^s+2−γ(Ts+2L_m)(x) = 1

δ²m^{s+2−αs+2−γ} · (Ts+2L_m)(x) m^αs+2

and becauseγ < s+2−α_s+2, we gets+2−α_s+2−γ >0. Because ^(Ts+2_mαs+2^Lm)(x)

is bounded for any m∈N, m≥m(s), it results that

m→∞lim

1

δ²m^{s+2−αs+2−γ} · (Ts+2L_m)(x) m^αs+2 = 0.

Considering the limit compute above, the fact that s is even and (3.4), we obtain (3.2).

Remark 3.1. In Theorem 3.1 we choose the smallest α_s+2 and the bigger γ, if they exists.

In the following, we suppose that exists M >0 such that the inequality

(3.5)

pm

X

k=0

ϕ_m,k(x)≤M holds for any x∈J and any m∈N.

Theorem 3.2. If f ∈E_w(I)is as times differentiable function at x∈I∩J (if s = 0 we consider that f is continuous on I ∩J) and we suppose that exists α_s+2 ≥ 0 and m(s) ∈ N such that ^(Ts+2_mαs+2^Lm)(x) is bounded for any m ∈N, m≥m(s), then for any γ which verify

(3.6) γ < s+ 2−αs+2

we have

(3.7) lim

m→∞m^γ

"

(Lmf)(x)−

s

X

i=0

1

mⁱi!(TiLm)(x)f⁽ⁱ⁾(x)

#

= 0.

If f ∈E_w(I) is a s times differentiable function on I and for the compact intervalK ⊂I∩J existm(s)∈Nand the constantk_s+2(K)∈R, depending on K, such that for any m∈N, m ≥m(s) and x∈K we have

(3.8) (T_s+2L_m)(x)

m^αs+2 ≤k_s+2(K), then the convergence given in (3.7) is uniform on K.

Proof. According to Taylor’s formula for the functionf arroundx, we have

(3.9) f(t) =

s

X

i=0

(t−x)ⁱ

i! f⁽ⁱ⁾(x) + (t−x)^sµ(t−x) where µ is a bounded function and lim

t→xµ(t−x) = 0. Then exists a neigh- borhood V = [−a, a] of the point 0 such that for any ǫ >0, exists δ_ǫ > 0, for any h ∈V with |h|< δ_ǫ, we have

(3.10) |µ(h)|< ǫ.

If we replacetwithx_m,k in (3.9), multiply byϕ_m,k(x) and sum afterk, when k ∈ {0,1, . . . , p_m}, we obtain

(Lmf)(x) =

pm

X

k=0 s

X

i=0

(x_m,k−x)ⁱ

i! ϕ_m,k(x)f⁽ⁱ⁾(x)+

+

pm

X

k=0

(xm,k −x)^sϕ_m,k(x)µ(xm,k−x) =

=

s

X

i=0

1 mⁱi!

"

mⁱ

pm

X

k=0

(xm,k−x)ⁱϕm,k(x)

#

f⁽ⁱ⁾(x)+

+

pm

X

k=0

(xm,k −x)^sϕm,k(x)µ(xm,k−x), or

(Lmf)(x)−

s

X

i=0

1

mⁱi!(TiLm)(x)f⁽ⁱ⁾(x) =

pm

X

k=0

(xm,k −x)^sϕm,k(x)µ(xm,k−x) and thus

(3.11) m^γ

"

(Lmf)(x)−

s

X

i=0

1

mⁱi!(TiLm)(x)f⁽ⁱ⁾(x)

#

= (Rmf)(x), where

(3.12) (Rmf)(x) =m^γ

pm

X

k=0

(xm,k−x)^sϕ_m,k(x)µ(xm,k−x).

Consider δ_ǫ from (3.10), I_m ={0,1, . . . , pm} ∩N, I_m,1 ={k ∈I_m : |x_m,k− x|< δ_ǫ} and I_m,2 ={k∈I_m :|x_m,k−x| ≥δ_ǫ}. Then

|(R_mf)(x)| ≤m^γ

pm

X

k=0

(xm,k−x)^sϕ_m,k(x)|µ(x_m,k−x)|=

=m^γ X

k∈Im,1

(xm,k−x)^sϕ_m,k(x)|µ(x_m,k−x)|+

+m^γ X

k∈Im,2

(xm,k−x)^sϕ_m,k(x)|µ(x_m,k−x)|

and taking (3.10) into account, and considering the fact thatµis bounded, so sup

t∈V

|µ(t)|=η, we have

|(R_mf)(x)| ≤m^γǫ X

k∈Im,1

(xm,k−x)^sϕ_m,k(x)+

(3.13)

+m^γη X

k∈Im,2

(xm,k −x)^sϕm,k(x).

But (xm,k−x)^s ≤(2a)^s, so (3.14) X

k∈Im,1

(xm,k −x)^sϕm,k(x)≤(2a)^s X

k∈Im,1

ϕm,k(x)≤(2a)^s

pm

X

k=0

ϕm,k(x).

Taking (3.5) and (3.14) into account, we have that

(3.15) m^γǫ X

k∈Im,1

(xm,k−x)^sϕ_m,k(x)≤m^γ(2a)^sM.

From (3.6), we have that γ < s+ 2−α_s+2 and then from Theorem 3.1 we obtain lim

m→∞m^γ X

k∈Im,2

(x_m,k −x)^sϕ_m,k(x) = 0, thus for ǫ from (3.10), there exists m(ǫ)∈N, for anym ∈N,m ≥m(ǫ), we have

(3.16) m^γη X

k∈Im,2

(xm,k −x)^sϕm,k(x)< ǫ.

Choose ǫ = _m[mγ(2a)^{1 s}M+1] and there exists m(ǫ) ∈ N, for any m ∈ N, m ≥m(ǫ), from (3.13)-(3.16) it results that |(Rmf)(x)|< _m¹, and so

(3.17) lim

m→∞(Rmf)(x) = 0.

From (3.11) and (3.13), (3.7) follows. For the second afirmation from The- orem 3.2, we apply in the proof above the Theorem 3.1.

For s = 0, respectively s = 2 in Theorem 3.2 we obtain the Corollary 3.1.

Corollary 3.1. Iff ∈Ew(I)is as times differentiable function atx∈I∩J and we suppose that exist α_s+2 ≥ 0 and m(s) ∈ N such that ^(Ts+2_mαs+2^Lm)(x) is bounded for any m∈N, m≥m(s), then for any γ which verify

(3.18) γ < s+ 2−α_s+2,

we have

(3.19) lim

m→∞m^γ[(Lmf)(x)−(T0Lm)(x)f(x)] = 0 if s = 0, and

m→∞lim m^γ

·

(L_mf)(x)−(T₀L_m)(x)f(x)− 1

m(T₁L_m)(x)f⁽¹⁾(x)−

(3.20)

− 1

2m²(T₂L_m)(x)f⁽²⁾(x)

¸

= 0, if s = 2.

If f ∈ Ew(I) is a s times differentiable function on I and for the compact K ⊂I∩J exist m(s)∈N and the constant ks+2(K)∈R, depending on K such that for any m∈N, m≥m(s) and x∈K we have

(3.21) (T_s+2L_m)(x)

m^αs+2 ≤k_s+2(K),

where s ∈ {0,2}, then the convergences given in (3.19) and (3.20) are uniform on K.

Remark 3.2. The relation (3.20)from Corollary 3.1is a Voronovskaja- type identity.

In the following, in every application we have

pm

X

k=0

ϕ_m,k(x) = 1, so (T₀L_m)(x) = 1 for any x ∈ J, m ∈ N, u_m(K) = 0 for any K ⊂ I ∩J and m ∈N,α2 = 1, α4 = 2, γ = 0 if s= 0 andγ = 1 if s= 2.

In the following, by particularization of the sequence xm,k, m ∈ N, k ∈ {0,1, . . . , pm} ∩N₀ and applying Corollary 3.1, we can obtain convergence theorems and Voronovskaja-type theorems for the operators from the first section of this paper. Because every application is a simple substitute in the Corollary 3.1, we won’t replace anything.

Application 3.1. We study a particular case of the Stancu operators. Let α = 10 and β = −¹₂. We obtain I = [0,22], K = J = [0,1] and for any f ∈C([0,22]), x∈[0,1] and m∈N

(P_m^(10,−1/2)f)(x) =

m

X

k=0

p_m,k(x)f

µ2k+ 20 2m−1

¶ ,

where ϕm,k(x) = pm,k(x) and xm,k = ^2k+20_2m−1, k ∈ {0,1, . . . , m}. We obtain (T1Pm^(10,−1/2))(x) = ^m(20+x)_2m−1 , (T2Pm^(10,−1/2))(x) = m^{2 4}mx(1−x)+(20+x)²

(2m−1)² for any m ∈N and x∈[0,1],k2(K) = ⁵₄, k4(K) = ¹⁹₁₆ (see [19]).

For the Bleimann-Butzer-Hahn operators and for the Meyer-K¨onig and Zeller operators we only give the convergence theorems.

Application 3.2. We consider I = J = [0,∞), Ew(I) = CB([0,∞)), w(x) = 1 for any x ∈ [0,∞), K = [0, b], b > 0, 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}, x ∈ [0,∞) and in this case we obtain the Bleimann-Butzer-Hahn operators. We have (T1L_m)(x) =

−mx¡ _x

1+x

¢m

, x∈K and k₂(K) = 4b(1 +b)² for m≥24(1 +b) (see [17]).

Application 3.3. If I = J = [0,1], w(x) = 1 for any x ∈ [0,1], Ew(I) = B([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, x∈[0,1], we obtain the Meyer-K¨onig and Zeller operators and we have (T1Z_m)(x) = 0, m∈N,x∈[0,1], andk₂(K) = 2 (see [16]).

Application 3.4. If I = J = [0,∞), w(x) = _1+x¹ 2 for any x ∈ [0,∞), E_w(I) = C₂([0,∞)), K = [0, b], b > 0, p_m = ∞, x_m,k = _m^k, ϕ_m,k(x) = e^−mx(mx)_k!^k,m∈N,k ∈N₀,x∈[0,∞), we obtain the Mirakjan-Favard-Sz´asz operators. We have (T1Sm)(x) = 0, (T2Sm)(x) = mx, m ∈ N, x ∈ [0,∞), k2(K) =b and k4(K) = 3b²+b (see [16]).

Application 3.5. Let I = J = [0,∞), w(x) = _1+x¹ 2 for any x ∈ [0,∞), E_w(I) =C₂([0,∞)), K = [0, b], b > 0,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 and x∈[0,∞). In this case we obtain the Baskakov operators and we have (T1Vm)(x) = 0, (T2Vm)(x) =mx(1+x), m ∈N,x∈[0,∞), k2(K) =b(1 +b) andk4(K) = 9b⁴+ 10b³+ 10b²+b (see [16]).

Application 3.6. If I = J = [0,∞), w(x) = 1 for any x ∈ [0,∞), E_w(I) = C([0,∞)), K = [0, b], b > 0, 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,x∈[0,∞), we obtain the Ismail- May operators. We have (T1R_m)(x) = 0, (T2R_m)(x) =mx(1 +x)²,m ∈N, x∈[0,∞),k₂(K) = 1 +b(1 +b)² and k₄(K) = 1 +b²(1 +b)⁴ (see [18]).

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[13] Pop, O. T., New properties of the Bernstein-Stancu operators, Anal.

Univ. Oradea, Fasc. Matematica, Tom XI (2004), 51-60

[14] Pop, O. T., The generalization of Voronovskaja’s theorem for a class of linear and positive operators, Rev. Anal. Num. Th´eor. Approx., 34 (2005), No. 1, 79-91

[15] Pop, O. T., About a class of linear and positive operators, Carpathian J. Math., 21 (2005), No. 1-2, 99-108

[16] Pop, O. T.,About some linear and positive operators defined by infinite sum, Dem. Math., XXXIX, No. 2 (2006), 377-388

[17] Pop, O. T.,On operators of the type Bleimann, Butzer and Hahn, Anal.

Univ. Timi¸soara, XLIII, Fasc. 1 (2005), 115-124

[18] Pop, O. T., The generalization of Voronovskaja’s theorem for exponen- tial operators, Creative Math & Inf., 16(2007), 54-62

[19] Pop, O.T., F˘arca¸s, M.D., The Voronovskaja-type theorem for a class of linear positive operators (to appear)

[20] Stancu, D. D., Asupra unei generaliz˘ari a polinoamelor lui Bernstein, Studia Univ. Babe¸s-Bolyai, Ser. Math.-Phys., 14 (1969), 31-45 (Roma- nian)

[21] Stancu, D. D., Coman, Gh., Agratini, O., Trˆımbit¸a¸s, R., Analiz˘a nu- meric˘a ¸si teoria aproxim˘arii, I, Presa Universitar˘a Clujean˘a, Cluj- Napoca, 2001 (Romanian)

[22] Sz´asz, O., Generalization of. S. N. Bernstein’s polynomials to the in- finite interval, J. Research, National Bureau of Standards, 45 (1950), 239-245

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

[24] Voronovskaja, E., D´etermination de la forme asymptotique d’approximation des fonctions par les polynˆomes de S. N. Bern- stein, C. R. Acad. Sci. URSS (1932), 79-85

Ovidiu T. Pop

Vest University ”Vasile Goldi¸s” of Arad Branch of Satu Mare,

26 Mihai Viteazul Street, Satu Mare,

440030, Romania

E-mail: [email protected]

M. D. F˘arca¸s National College ”Mihai Eminescu”

5 Mihai Eminescu Street, Satu Mare, 440014, Romania E-mail: [email protected]