Quantitative estimates for positive linear operators in weighted spaces

Quantitative estimates for positive linear operators in weighted spaces

Adrian Holho¸s

Abstract

We give some quantitative estimates for positive linear operators in weighted spaces by introducing a new modulus of continuity and then apply these results to the Bernstein-Chlodowsky polynomials.

2000 Mathematical Subject Classification: 41A36, 41A25.

1 Introduction

LetR+ = [0,∞) and let ϕ :R+→ R+ be an unbounded strictly increasing continuous function such there existM > 0 andα∈(0,1] with the property (1) |x−y| ≤M|ϕ(x)−ϕ(y)|^α, for every x, y ≥0.

Let ρ(x) = 1 +ϕ²(x) be a weight function and let B_ρ(R+) be the space defined by

B_ρ(R+) =

f :R+→R| f _ρ = sup

x≥0

f(x)

ρ(x) <+∞

. 99

We define also the spaces

C_ρ(R+) ={f ∈B_ρ(R+), f is continuous}, C_ρ^k(R₊) =

f ∈C_ρ(R₊), lim

x→+∞

f(x)

ρ(x) =K_f <+∞

, U_ρ(R₊) ={f ∈C_ρ(R₊), f /ρ is uniformly continuous}. We have the inclusions C_ρ^k(R+)⊂U_ρ(R+)⊂C_ρ(R+)⊂B_ρ(R+).

We consider (A_n)_n≥1 a sequence of positive linear operators acting from C_ρ(R+) to B_ρ(R+). In [1] is given the following

Theorem 1 If A_n : C_ρ(R+) → B_ρ(R+) is a sequence of linear operators such that

(2) lim

n→∞,,A_nϕⁱ−ϕⁱ,,

ρ = 0, i= 0,1,2, then for any function f ∈C_ρ^k(R+) we have

n→∞lim A_nf −f _ρ = 0.

Remark 1 The conditions (2) can be replaced with:

n→∞lim ,,A_nρ^i/2 −ρ^i/2,,

ρ = 0, i= 0,1,2 and the theorem remains valid. (see [2])

Remark 2 Taking f^∗(x) =ϕ²(x) cosπx, we notice that f^∗ ∈U_ρ(R₊). But it was proved in [1] that there is a sequence A_n of positive linear operators such that lim_n→∞ A_nf^∗−f^∗ _ρ ≥1. So, the space C_ρ^k(R₊), from Theorem 1 cannot be replaced by U_ρ(R+).

In [2] it was introduced a weighted modulus of continuity to estimate the rate of approximation in these spaces: for everyδ ≥0 and for every f ∈C_ρ(R+)

Ω_ρ(f, δ) = sup

x,y∈R+

|ρ(x)−ρ(y)|≤δ

|f(x)−f(y)| [|ρ(x)−ρ(y)|+ 1]ρ(x),

where ρ was defined as a continuously differentiable function on R+ with ρ(0) = 1 and inf_x≥0ρ(x)≥1. For this modulus, it was proved

Theorem 2 Let A_n be a sequence of positive linear operators such that ,,A_nρ⁰−ρ⁰,,

ρ =α_n, A_nρ−ρ _ρ =β_n, ,,A_nρ²−ρ²,,

ρ² =γ_n,

where α_n, β_n and γ_n tend to zero as n goes to the infinity. Then A_nf −f _ρ4 ≤16·Ω_ρ

f,

α_n+ 2β_n+γ_n

+ f _ρα_n for all f ∈C_ρ^k(R+) and n large enough.

We want to improve this result and give an application.

2 A new weighted modulus of continuity

For eachf ∈C_ρ(R+) and for every δ ≥0 we introduce ω_ϕ(f, δ) = sup

x,y≥0

|ϕ(x)−ϕ(y)|≤δ

|f(x)−f(y)| ρ(x) +ρ(y) .

Remark 3 Because of the symmetry we have

ω_ϕ(f, δ) = sup

y≥x≥0

|ϕ(y)−ϕ(x)|≤δ

|f(x)−f(y)| ρ(x) +ρ(y) .

We observe thatω_ϕ(f,0) = 0 andω_ϕ(f,·)is a nonnegative, increasing func- tion for all f ∈ C_ρ(R+). Moreover, ω_ϕ(f,·) is bounded, which follows from the inequality |f(y)−f(x)| ≤ f _ρ(ρ(y) +ρ(x)).

Remark 4 If ϕ(x) =x, then ω_ϕ is equivalent with Ωdefined by Ω(f, δ) = sup

x≥0,|h|≤δ

|f(x+h)−f(x)| (1 +h²)(1 +x²) , in the sense that

ω_ϕ(f, δ)≤Ω(f, δ)≤3·ω_ϕ(f, δ),

the first inequality being true for δ ≤ ^√1₂ and the second for all δ ≥ 0.

Indeed,ω_ϕ(f, δ)≤Ω(f, δ) is equivalent with the inequality

1 +x²+h² +x²h² ≤1 +x²+ 1 +x²+ 2xh+h², ∀x≥0 or x²(1−h²) + 2xh+ 1 ≥0,∀x≥0, which is true if 2h²−1≤0.

The inequality Ω(f, δ)≤3·ω_ϕ(f, δ) is equivalent with

1 +x² + 1 +x²+ 2xh+h² ≤3(1 +x²+h²+x²h²), ∀x≥0 or x²+ 2h²+ 2x²h² + (xh−1)² ≥0, which is true for all h, x∈R. Lemma 1 lim

δ0ω_ϕ(f, δ) = 0, for every f ∈U_ρ(R₊).

Proof. Lety ≥x≥0 such that 0≤ϕ(y)−ϕ(x)≤δ. Then

|f(x)−f(y)| ρ(x) +ρ(y) ≤

f(x)

ρ(x) − f(y) ρ(y)

· ρ(x)

ρ(x) +ρ(y) +|f(y)|

ρ(y) ·|ρ(x)−ρ(y)| ρ(x) +ρ(y)

≤ω f

ρ,|x−y|

·1

2 + f _ρ·|ϕ(x)−ϕ(y)|·[ϕ(x) +ϕ(y)]

2 +ϕ²(x) +ϕ²(y)

≤ 1 2 ·ω

f

ρ, M|ϕ(x)−ϕ(y)|^α

+ f _ρ· |ϕ(x)−ϕ(y)| 2

≤ M + 1 2 ·ω

f ρ, δ^α

+ f _ρ· δ 2,

whereω(f, δ) is the usual modulus of continuity. We obtain ω_ϕ(f, δ)≤ M + 1

2 ·ω f

ρ, δ^α

+ f _ρ· δ 2

The right-hand side tend to zero when δ tend to zero, because f /ρ is uni- formly continuous, so the lemma is proved.

Lemma 2 For every δ≥0 and λ ≥0 we have ω_ϕ(f, λδ)≤(2 +λ)·ω_ϕ(f, δ).

Proof. We prove thatω_ϕ(f, mδ)≤(m+ 1)·ω_ϕ(f, δ), for every nonnegative integer m. The property for λ ∈ R+ can be easily obtained by using the inequalities [λ]≤λ≤[λ] + 1, where [λ] denotes the greatest integer less or equal toλ.

Form = 0 and m = 1 the inequality is obvious. For m ≥2, let y > x≥ 0 such that ϕ(y)−ϕ(x) ≤ mδ. We construct, inductively, the sequence of pointsx=x₀ < x₁ <· · ·< x_m =ysuch that for each k ∈ {1, . . . , m},

ϕ(x_k)−ϕ(x_k−1) =c= ϕ(y)−ϕ(x)

m ≤δ.

For simplifying the computations we set a_k =ϕ(x_k)≥0. We have m

k=1

a²_k+a²_k−1−2 m

k=1

a_ka_k−1 = m

k=1

(a_k−a_k−1)² =mc², and

m

k=1

a²_k+a_ka_k−1+a²_k−1 = 1 c

m

k=1

a³_k−a³_k−1 = a³_m−a³₀

c =m(a²_m+a_ma₀+a²₀).

We deduce 3

m

k=1

a²_k+a²_k−1 = 2 m

k=1

a²_k+a_ka_k−1+a²_k−1+ m

k=1

a²_k−2a_ka_k−1+a²_k−1

= 2m(a²_m+a_ma₀+a²₀) +mc²

≤3m(a²_m+a²₀) + (a_m−a₀)²

≤3(m+ 1)(a²_m+a²₀).

Using this, we have

|f(y)−f(x)| ρ(y) +ρ(x) ≤

m

k=1

|f(x_k)−f(x_k−1)|

ρ(x_k) +ρ(x_k−1) · 2 +ϕ²(x_k) +ϕ²(x_k−1) ρ(y) +ρ(x)

≤ω_ϕ(f, δ) m

k=1

2 +a²_k+a²_k−1 2 +a²_m+a²₀

≤(m+ 1)ω_ϕ(f, δ).

The supremum being the least upper bound, we obtain ω_ϕ(f, mδ)≤(m+ 1)ω_ϕ(f, δ).

Lemma 3 For every f ∈C_ρ(R+), forδ >0 and for all x, y ≥0

|f(y)−f(x)| ≤(ρ(y) +ρ(x))

2 + |ϕ(y)−ϕ(x)| δ

ω_ϕ(f, δ).

Proof. From the definition of the modulus we deduce

|f(y)−f(x)| ≤[ρ(y) +ρ(x)]·ω_ϕ(f,|ϕ(y)−ϕ(x)|).

If|ϕ(y)−ϕ(x)| ≤δ then by the monotony of the modulus we have ω_ϕ(f,|ϕ(y)−ϕ(x)|)≤ω_ϕ(f, δ).

If|ϕ(y)−ϕ(x)| ≥δ then by the previous lemma ω_ϕ(f,|ϕ(y)−ϕ(x)|) =ω_ϕ

f,|ϕ(y)−ϕ(x)|

δ ·δ

≤

2 + |ϕ(y)−ϕ(x)| δ

ω_ϕ(f, δ).

Theorem 3 Let A_n : C_ρ(R+) → B_ρ(R+) be a sequence of positive linear operators with

,,A_nϕ⁰−ϕ⁰,,

ρ⁰ =a_n, A_nϕ−ϕ

ρ¹² =b_n, ,,A_nϕ²−ϕ²,,

ρ =c_n, ,,A_nϕ³−ϕ³,,

ρ³² =d_n,

where a_n, b_n, c_n and d_n tend to zero as n goes to the infinity. Then A_nf −f

ρ³² ≤(7 + 4a_n+ 2c_n)·ω_ϕ(f, δ_n) + f _ρa_n for all f ∈C_ρ(R+), where

δ_n= 2

(a_n+ 2b_n+c_n)(1 +a_n) +a_n+ 3b_n+ 3c_n+d_n. Proof. By the previous lemma and by the fact that

[ρ(x) +ρ(y)]|ϕ(y)−ϕ(x)| ≤

2ρ(x) +|ϕ²(y)−ϕ²(x)|

|ϕ(y)−ϕ(x)|

we obtain

|A_nf(x)−f(x)| ≤ |f(x)| · |A_nϕ⁰(x)−ϕ⁰(x)|+A_n(|f(y)−f(x)|, x)

≤ f _ρa_nρ(x) +ω_ϕ(f, δ_n)· 0

2A_nρ(x) + 2ρ(x)A_nϕ⁰(x)

+2ρ(x)A_n(|ϕ(y)−ϕ(x)|, x) +A_n([ϕ(y) +ϕ(x)][ϕ(y)−ϕ(x)]², x) δ_n

(3) 1

Applying Cauchy-Schwarz inequality we have A_n(|ϕ(y)−ϕ(x)|, x)≤ &

A_n([ϕ(y)−ϕ(x)]², x)'¹

2 ·&

A_nϕ⁰(x)'¹

2

and using

A_n([ϕ(y)−ϕ(x)]², x)

=A_nϕ²(x)−ϕ²(x)−2ϕ(x)[A_nϕ(x)−ϕ(x)] +ϕ²(x)[A_nϕ⁰(x)−ϕ⁰(x)]

≤ρ(x)c_n+ 2ρ¹²(x)ϕ(x)b_n+a_nϕ²(x), we obtain

A_n(|ϕ(y)−ϕ(x)|, x)≤ρ¹²(x)·

(a_n+ 2b_n+c_n)(1 +a_n).

Because

A_n(ϕ(y)[ϕ(y)−ϕ(x)]², x)

=A_nϕ³(x)−ϕ³(x)−2ϕ(x)[A_nϕ²(x)−ϕ²(x)] +ϕ²(x)[A_nϕ(x)−ϕ(x)]

≤ρ³²(x)d_n+ 2ρ(x)ϕ(x)c_n+b_nϕ²(x)ρ¹²(x), we obtain

A_n([ϕ(y) +ϕ(x)][ϕ(y)−ϕ(x)]², x)≤ρ³²(x)·(d_n+2c_n+b_n+a_n+2b_n+c_n).

Choosingδ_n = 2

(a_n+ 2b_n+c_n)(1 +a_n) +a_n+ 3b_n+ 3c_n+d_n

|A_nf(x)−f(x)|≤

2ρ(x)(2a_n+c_n+3) +ρ³²(x)

ω_ϕ(f, δ_n)+ f _ρa_nρ(x).

So,

A_nf −f

ρ³² ≤(7 + 4a_n+ 2c_n)·ω_ϕ(f, δ_n) + f _ρa_n.

Remark 5 In the conditions of the Theorem 3, using Lemma 1 we have

n→∞lim A_nf−f

ρ³² = 0, for all f ∈U^k

ρ³²(R+).

Corollary 1 Let A_n : C_ρ(R₊) → B_ρ(R₊) be a sequence of positive linear operators with

,,A_nϕ⁰−ϕ⁰,,

ρ⁰ =a_n, A_nϕ−ϕ

ρ¹² =b_n, ,,A_nϕ²−ϕ²,,

ρ =c_n, ,,A_nϕ³−ϕ³,,

ρ³² =d_n,

where a_n, b_n, c_n and d_n tend to zero as n goes to the infinity. Let η_n be a sequence of real numbers such that

n→∞lim η_n=∞ and lim

n→∞ρ¹²(η_n)δ_n = 0, whereδ_n= 2

(a_n+ 2b_n+c_n)(1 +a_n) +a_n+ 3b_n+ 3c_n+d_n. Then for every f ∈C_ρ(R+)

sup

0≤x≤ηn

|A_nf(x)−f(x)|

ρ(x) ≤(7 + 4a_n+ 2c_n)·ω_ϕ

f, ρ¹²(η_n)δ_n

+ f _ρa_n. Proof. Replacing δ_n from (3) with ρ¹2(η_n)δ_n we obtain the result.

3 Application

We want to apply the result obtained in the Corollary 1 for the weight ρ(x) = 1 +x² and the Bernstein-Chlodowsky operators defined by

B_nf(x) = n

k=0

f k

nb_n n k

x b_n

_k 1− x

b_{n n−k}

,

for 0≤ x ≤ b_n and B_nf(x) = f(x), for x > b_n, where b_n is a sequence of positive numbers such that

n→∞lim b_n =∞ and lim

n→∞

b_n n = 0.

The condition (1) overϕ(x) =x is verified for α= 1 and M = 1.

Theorem 4 If B_n : C_ρ(R+) → B_ρ(R+) is the sequence of Bernstein- Chlodowsky operators, then for all f ∈C_ρ(R₊)

(4) B_nf−f _ρ ≤

7 + b_n n

·ω_ϕ -

f, 1 +b²_n

-2b_n n + 3b_n

n + b²_n 2n²

..

. Proof. We have

B_ne₀(x) = 1, B_ne₁(x) =x,

B_ne₂(x) =x² +x(b_n−x)

n ,

B_ne₃(x) =x³ +x(b_n−x)[(3n−2)x+b_n] n²

We obtain

a_n = B_ne₀ −e_{0 ρ0} = 0, b_n = B_ne₁ −e₁

ρ¹² = 0, c_n = B_ne₂ −e_{2 ρ} = sup

0≤x≤bn

x(b_n−x)

n(1 +x²) = b²_n

2n

1 +b²_n+ 1

≤ b_n 2n, d_n = B_ne₃ −e₃

ρ³² ≤ sup

0≤x≤bn

x(b_n−x)

n(1 +x²)· sup

0≤x≤bn

(3n−2)x+b_n n√

1 +x²

≤ b_n 2n

3 + b_n

n

.

Settingη_n=b_n in the Corollary 1, and considering 2

(a_n+ 2b_n+c_n)(1 +a_n) +a_n+ 3b_n+ 3c_n+d_n ≤ 2b_n

n + 3b_n n + b²_n

2n², we obtain the estimation from the theorem.

Remark 6 In order to obtain

n→∞lim B_nf −f _ρ = 0,

in the relation (4) from Theorem 4, we must have f ∈U_ρ(R+) and

n→∞lim b³_n

n = 0.

References

[1] A.D.Gadzhiev, On P.P.Korovkin type theorems, Math.Zamet. 20, 5, 1976, 781-786. Transl. in Math. Notes 20, 5, 1976, 996-998.

[2] A.D.Gadjiev, A.Aral, The estimates of approximation by using a new type of weighted modulus of continuity, Comput. Math. Appl. 54, 2007, 127-135.

Adrian Holho¸s

Technical University of Cluj-Napoca, Department of Mathematics,

Str. C.Daicoviciu, nr.15, 400020, Cluj-Napoca, Romania e-mail: [email protected]