Density by Moduli and Korovkin Type Approximation Theorem of Boyanov and Veselinov

doi:10.5666/KMJ.2018.58.4.733

Kyungpook Mathematical Journal 2018; 58(4): 733-746

Department of Mathematics, Kurukshetra University, Kurukshetra-136119, India, e-mail : vinodk_bhj@rediffmail.com, Department of Mathematics, KVA DAV College for Women Karnal-132001, India, e-mail : shwetadhawan_dav@rediffmail.com

^*Corresponding Author.

Received: April 4, 2018; Revised: August 9, 2018; Accepted: August 13, 2018; Published online: December 23, 2018.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract: The concept of f-statistical convergence which is, in fact, a generalization of statistical convergence, has been introduced recently by Aizpuru et al. (Quaest. Math. 37: 525–530, 2014). The main object of this paper is to prove an f-statistical analog of the classical Korovkin type approximation theorem of Boyanov and Veselinov. It is shown that the f-statistical analog is intermediate between the classical theorem and its statistical analog. As an application, we estimate the rate of f-statistical convergence of the sequence of positive linear operators defined from C^*[0,∞) into itself.; Keywords: modulus function, statistical convergence, positive linear operator, rate of convergence, Korovkin type approximation theorem.

1. Introduction

The idea of statistical convergence was given by Zygmund [36] in the first edition of his monograph published in Warsaw in 1935. The concept of statistical convergence was formally introduced by Steinhaus [35] and Fast [16] and later reintroduced by Schoenberg [34]. Various generalizations and applications of statistical convergence have been studied by Šalát [33], Fridy [17], Aizpuru et al. [1], Gadjiev and Orhan [18], Mursaleen and Alotaibi [28] and many others.

Statistical convergence depends on the natural density of subsets of the set ℕ = {1, 2, 3, …}. The natural density d(K) of set K ⊆ ℕ (see [29, Chapter 11]) is defined by

(1.1)

d (K) = lim_{n \to \infty} \frac{1}{n} ∣ {k \leq n : k \in K} ∣,

where |{ k ≤ n : k ∈ K}| denotes the number of elements of K not exceeding n. Obviously we have d(K) = 0 provided that K is finite.

The concept of statistical convergence was studied by Kolk [22] in the more general setting of normed spaces.

Definition 1.1

Let X be a normed space. A sequence x = (x_k) in X is said to be statistically convergent to some x ∈ X if, for each ε > 0 the set { k ∈ ℕ : ||x_k −x|| ≥ ε } has natural density zero, i.e.,

lim_{n \to \infty} \frac{1}{n} ∣ {k \leq n : ‖ x_{k} - x ‖ \geq ɛ} ∣ = 0,

and we write it as st – lim x_k = x.

Recall [25, 32] that a modulus f is a function from ℝ⁺ to ℝ⁺ such that

f(x) = 0 if and only if x = 0,

f(x + y) ≤ f(x) + f(y) for x ≥ 0, y ≥ 0,

f is increasing,

f is continuous from the right at 0.

If f, g are moduli and a, b are positive real numbers, then f ∘ g, af+bg, and f∨ g are moduli. A modulus may be unbounded or bounded. For example, the modulus f(x) = x^p where 0 < p ≤ 1, is unbounded, but $g (x) = \frac{x}{(1 + x)}$ is bounded. For the work related to sequence spaces defined by a modulus one may refer to [1, 5, 6, 7, 9, 10, 25] and many others.

Aizpuru et al. [1] have recently introduced a new concept of density by moduli and consequently obtained a new concept of non-matrix convergence, namely, f-statistical convergence which is, in fact, a generalization of the concept of statistical convergence and intermediate between the ordinary convergence and the statistical convergence. This idea of replacing natural density with density by moduli has also been recently used to study the concepts of f-statistical convergence of order α [5], f-lacunary statistical convergence [6] and f-statistical boundedness [10].

Definition 1.2.([1])

For any unbounded modulus f, the f-density of a set K ⊂ ℕ is denoted by d^f (K) and is defined by

d^{f} (K) = lim_{n \to \infty} \frac{f (∣ {k \leq n : k \in K} ∣)}{f (n)}

in case this limit exists.

Definition 1.3.([1])

Let f be an unbounded modulus and X be a normed space. A sequence x = (x_k) in X is said to be f-statistically convergent to x, if, for each ɛ > 0,

\begin{array}{r} d^{f} ({k \in ℕ : ‖ x_{k} - x ‖ \geq ɛ}) = 0, \\ i . e ., lim_{n \to \infty} \frac{1}{f (n)} f (∣ {k \leq n : ‖ x_{k} - x ‖ \geq ɛ} ∣) = 0, \end{array}

and we write it as f – st lim x_k = x or S^f – lim x_k = x.

Remark 1.4.([1])

For any unbounded modulus f, every convergent sequence is fstatistically convergent which in turn is statistically convergent, but not conversely.

The theory of approximation is an area of mathematical analysis, which, at its core, is concerned with the approximation of functions by simpler and more easily calculated functions. In the fifties, the theory of approximation of functions by positive linear operators developed a lot, when Popoviciu [31], Bohman [11] and Korvokin [23, 24], discovered, independently, a simple and easily applicable criterion to check if a sequence of positive linear operators converges uniformly to the function to be approximated. This criterion says that the necessary and sufficient condition for the uniform convergence of the sequence (L_n) of positive linear operators to the continuous function g on the compact interval [a, b], is the uniform convergence of the sequence (L_ng) to g for the only three functions e_n(x) = xⁿ, n = 0, 1, 2. This classical result of approximation theory is mostly known under the name of Bohman-Korovkin theorem, because Popoviciu’s contribution in [31] remained unknown for a long time.

Due to this classical result, the monomials e_n, n = 0, 1, 2, play an important role in the approximation theory of linear and positive operators on spaces of continuous functions. These monomials are often called Korovkin test-functions. This elegant and simple result has inspired many mathematicians to extend this result in different directions, generalizing the notion of sequence and considering different spaces. In this way a special branch of approximation theory arose, called Korovkin-type approximation theory. A complete and comprehensive exposure on this topic can be found in [2].

Statistical convergence had not been examined in approximation theory until 2002. The Korovkin first and second approximation theorems were first proved via statistical convergence by Gadjiev and Orhan [18] and Duman [14], in years 2002 and 2003, respectively. After this, Korovkin-type approximation theorems have been studied via various summability methods by many mathematicians [4, 13, 21, 26, 27, 28]. Quite recently Bhardwaj and Dhawan [8] have obtained f-statistical analogs of the Korovkin first and second approximation theorem.

In 1970, Boyanov and Veselinov [12] have proved the Korovkin theorem on C^*[0,∞), by using the test functions 1, e⁻^x, e⁻²^x, where C^*[0,∞) denotes the Banach space of all real-valued continuous functions defined on [0,∞), with the property that lim_x_→∞f(x) exists, with the uniform norm ||.||_∞. The statistical analog of this well-known result was given by Duman et al. [15]. In this paper, we generalize the classical result of Boyanov and Veselinov by using the notion of f-statistical convergence. A relationship between the classical theorem of Boyanov and Veselinov and their statistical and f-statistical analogs has also been studied. We also estimate the rate of f-statistical convergence of the sequence of positive linear operators defined from C^*[0,∞) into itself.

Before proceeding to establish the proposed results, we recall [20, 30] that for any linear spaces X, Y of real functions

the mapping L : X → Y is called linear operator if

$L (α g_{1} + β g_{2}) = α L (g_{1}) + β L (g_{2}) for g_{1}, g_{2} \in X and α, β \in ℝ,$

if g ≥ 0, g ∈ X ⇒ Lg ≥ 0, then L is a positive linear operator,

in order to highlight the argument of the function Lg ∈ Y we use the notation L(g, x),

if L is a positive linear operator, then for every g ∈ X we have |Lg| ≤ L(|g|).

2. Korovkin Type Approximation Theorem of Boyanov and Veselinov via f-statistical Convergence

We begin this section by recalling the classical Korovkin type approximation theorem due to of Boyanov and Veselinov [12].

Theorem 2.1

If the sequence (L_n) of positive linear operators L_n : C^*[0,∞) → C^*[0,∞) satisfies the conditions

\begin{array}{r} lim {‖ L_{n} (1, x) - 1 ‖}_{\infty} = 0, \\ lim {‖ L_{n} (e^{- t}, x) - e^{- x} ‖}_{\infty} = 0, \\ lim {‖ L_{n} (e^{- 2 t}, x) - e^{- 2 x} ‖}_{\infty} = 0, \end{array}

then for any function g ∈ C^*[0,∞), we have

lim {‖ L_{n} (g, x) - g (x) ‖}_{\infty} = 0.

The statistical analog of this theorem, given by Duman et al. [15], is as follows.

Theorem 2.2

Let (L_n) be a sequence of positive linear operators L_n : C^*[0,∞) → C^*[0,∞). Then, for all g ∈ C^*[0,∞),

s t - lim {‖ L_{n} (g, x) - g (x) ‖}_{\infty} = 0

if and only if

\begin{array}{r} s t - lim {‖ L_{n} (1, x) - 1 ‖}_{\infty} = 0, \\ s t - lim {‖ L_{n} (e^{- t}, x) - e^{- x} ‖}_{\infty} = 0, \\ s t - lim {‖ L_{n} (e^{- 2 t}, x) - e^{- 2 x} ‖}_{\infty} = 0. \end{array}

We now state and prove an f-statistical analog of the classical theorem of Boyanov and Veselinov.

Theorem 2.3

Let f be an unbounded modulus and (L_n) be a sequence of positive linear operators L_n : C^*[0,∞) → C^*[0,∞). Then, for all g ∈ C^*[0,∞),

(2.1)

f - s t lim {‖ L_{n} (g, x) - g (x) ‖}_{\infty} = 0

if and only if

(2.2)

f - s t lim {‖ L_{n} (1, x) - 1 ‖}_{\infty} = 0,

(2.3)

f - s t lim {‖ L_{n} (e^{- t}, x) - e^{- x} ‖}_{\infty} = 0,

(2.4)

f - s t lim {‖ L_{n} (e^{- 2 t}, x) - e^{- 2 x} ‖}_{\infty} = 0.

Proof

Since each of 1, e⁻^t, e⁻²^t belongs to C^*[0,∞), conditions (2.2) – (2.4) follow immediately from (2.1). Now, let the conditions (2.2) – (2.4) hold. Let g ∈ C^*[0,∞). We follow the proof of Theorem 2 of Boyanov and Veselinov [12] up to a certain stage. Since the function g is bounded on [0,∞), we can write

(2.5)

∣ g (t) - g (x) ∣ < 2 M, 0 \leq t, x < \infty .

It is easy to prove that for a given ε > 0 there is a δ > 0 such that

(2.6)

∣ g (t) - g (x) ∣ < ɛ whenever ∣ e^{- t} - e^{- x} ∣ < δ for all t, x \in [0, \infty) .

Putting ψ(t) = (e⁻^t–e⁻^x)² (x an arbitrary number but fixed in the interval [0,∞)) and using inequalities (2.5) and (2.6), we have

∣ g (t) - g (x) ∣ < ɛ + \frac{2 M}{δ^{2}} ψ (t) for all t \in [0, \infty) .

This means,

(2.7)

- ɛ - \frac{2 M}{δ^{2}} ψ (t) < g (t) - g (x) < ɛ + \frac{2 M}{δ^{2}} ψ (t) for all t \in [0, \infty) .

In fact, if |e⁻^t – e⁻^x| < δ, then (2.6) implies (2.7) since ψ(t) = (e⁻^t – e⁻^x)² ≥ 0 and if |e⁻^t – e⁻^x| ≥ δ, then

\frac{2 M}{δ^{2}} ψ (t) \geq \frac{2 M}{δ^{2}} δ^{2} = 2 M,

and (2.7) follows from (2.5) since ε > 0.

In view of monotonicity and linearity of the operators L_n(g, x), the inequality (2.7) implies

- ɛ L_{n} (1, x) - \frac{2 M}{δ^{2}} L_{n} (ψ, x) \leq L_{n} (g, x) - L_{n} (g (x), x) \leq ɛ L_{n} (1, x) + \frac{2 M}{δ^{2}} L_{n} (ψ, x) .

Note that x is fixed and so g(x) is a constant number. Therefore,

(2.8)

- ɛ L_{n} (1, x) - \frac{2 M}{δ^{2}} L_{n} (ψ, x) \leq L_{n} (g, x) - g (x) L_{n} (1, x) \leq ɛ L_{n} (1, x) + \frac{2 M}{δ^{2}} L_{n} (ψ, x) .

But

(2.9)

L_{n} (g, x) - g (x) = [L_{n} (g, x) - g (x) L_{n} (1, x)] + g (x) [L_{n} (1, x) - 1] .

Using (2.8) and (2.9), we have

(2.10)

L_{n} (g, x) - g (x) \leq ɛ L_{n} (1, x) + \frac{2 M}{δ^{2}} L_{n} (ψ, x) + g (x) [L_{n} (1, x) - 1] .

Now, let us estimate L_n(ψ, x).

(2.11)

\begin{array}{l} L_{n} (ψ, x) = L_{n} (e^{- 2 t} - 2 e^{- t} e^{- x} + e^{- 2 x}, x) \\ = L_{n} (e^{- 2 t}, x) - 2 e^{- x} L_{n} (e^{- t}, x) + e^{- 2 x} L_{n} (1, x) \\ = [L_{n} (e^{- 2 t}, x) - e^{- 2 x}] - 2 e^{- x} [L_{n} (e^{- t}, x) - e^{- x}] + e^{- 2 x} [L_{n} (1, x) - 1] . \end{array}

Using (2.11) in (2.10), we have

\begin{array}{l} L_{n} (g, x) - g (x) \leq ɛ L_{n} (1, x) \\ + \frac{2 M}{δ^{2}} {[L_{n} (e^{- 2 t}, x) - e^{- 2 x}] - 2 e^{- x} [L_{n} (e^{- t}, x) - e^{- x}] + e^{- 2 x} [L_{n} (1, x) - 1]} \\ + g (x) [L_{n} (1, x) - 1] \\ = (ɛ + \frac{2 M}{δ^{2}} e^{- 2 x} + g (x)) [L_{n} (1, x) - 1 + \frac{2 M}{δ^{2}} [L_{n} (e^{- 2 t}, x) - e^{- 2 x}] \\ - \frac{4 M}{δ^{2}} e^{- x} [L_{n} (e^{- t}, x) - e^{- x}] + ɛ . \end{array}

Therefore, using the fact that |e⁻^t| ≤ 1 and |e⁻²^t| ≤ 1, for all t ∈ [0,∞), we have

(2.12)

\begin{array}{l} {‖ L_{n} (g, x) - g (x) ‖}_{\infty} \leq (ɛ + \frac{2 M}{δ^{2}} + M) {‖ L_{n} (1, x) - 1 ‖}_{\infty} \\ + \frac{2 M}{δ^{2}} {‖ L_{n} (e^{- 2 t}, x) - e^{- 2 x} ‖}_{\infty} + \frac{4 M}{δ^{2}} {‖ L_{n} (e^{- t}, x) - e^{- x} ‖}_{\infty} + ɛ \\ \leq K ({‖ L_{n} (1, x) - 1 ‖}_{\infty} + {‖ L_{n} (e^{- 2 t}, x) - e^{- 2 x} ‖}_{\infty} \\ + {‖ L_{n} (e^{- t}, x) - e^{- x} ‖}_{\infty}) + ɛ, \end{array}

where $K = max {ɛ + \frac{2 M}{δ^{2}} + M, \frac{4 M}{δ^{2}}, \frac{2 M}{δ^{2}}}$ . For any ε′ > 0, choose ε > 0 such that ε < ε′. Now, from inequality (2.12), we have

(2.13)

∣ {n \leq N : {‖ L_{n} (g, x) - g (x) ‖}_{\infty} \geq ɛ^{'}} ∣ \leq | {n \leq N : {‖ L_{n} (1, x) - 1 ‖}_{\infty} + {‖ L_{n} (e^{- t}, x) - e^{- x} ‖}_{\infty} + {‖ L_{n} (e^{- 2 t}, x) - e^{- 2 x} ‖}_{\infty} \geq \frac{ɛ^{'} - ɛ}{K}} | .

Now, write

\begin{array}{l} D & : = {n : {‖ L_{n} (1, x) - 1 ‖}_{\infty} + {‖ L_{n} (e^{- t}, x) - e^{- x} ‖}_{\infty} + {‖ L_{n} (e^{- 2 t}, x) - e^{- 2 x} ‖}_{\infty} \geq \frac{ɛ^{'} - ɛ}{K}}, \\ D_{1} & : = {n : {‖ L_{n} (1, x) - 1 ‖}_{\infty} \geq \frac{ɛ^{'} - ɛ}{3 K}}, \\ D_{2} & : = {n : {‖ L_{n} (e^{- t}, x) - e^{- x} ‖}_{\infty} \geq \frac{ɛ^{'} - ɛ}{3 K}}, \\ D_{3} & : = {n : {‖ L_{n} (e^{- 2 t}, x) - e^{- 2 x} ‖}_{\infty} \geq \frac{ɛ^{'} - ɛ}{3 K}} . \end{array}

It is easy to see that D ⊂ D₁ ∪ D₂ ∪ D₃. Now, from (2.13), we have

\begin{array}{l} ∣ {n \leq N : {‖ L_{n} (g, x) - g (x) ‖}_{\infty} \geq ɛ^{'}} ∣ \leq | {n \leq N : {‖ L_{n} (1, x) - 1 ‖}_{\infty} \geq \frac{ɛ^{'} - ɛ}{3 K}} | \\ + | {n \leq N : {‖ L_{n} (e^{- t}, x) - e^{- x} ‖}_{\infty} \geq \frac{ɛ^{'} - ɛ}{3 K}} | \\ + | {n \leq N : {‖ L_{n} (e^{- 2 t}, x) - e^{- 2 x} ‖}_{\infty} \geq \frac{ɛ^{'} - ɛ}{3 K}} |, \end{array}

which yields that

\begin{array}{l} \frac{1}{f (N)} f (∣ {n \leq N : {‖ L_{n} (g, x) - g (x) ‖}_{\infty} \geq ɛ^{'}} ∣) \\ \leq \frac{1}{f (N)} f (| {n \leq N : {‖ L_{n} (1, x) - 1 ‖}_{\infty} \geq \frac{ɛ^{'} - ɛ}{3 K}} |) \\ + \frac{1}{f (N)} f (| {n \leq N : {‖ L_{n} (e^{- t}, x) - e^{- x} ‖}_{\infty} \geq \frac{ɛ^{'} - ɛ}{3 K}} |) \\ + \frac{1}{f (N)} f (| {n \leq N : {‖ L_{n} (e^{- 2 t}, x) - e^{- 2 x} ‖}_{\infty} \geq \frac{ɛ^{'} - ɛ}{3 K}} |) \end{array}

and using (2.2) – (2.4), we get

(2.14)

f - s t lim {‖ L_{n} (g, x) - g (x) ‖}_{\infty} = 0, for all g \in C^{*} [0, \infty) .

Remark 2.4

Since every convergent sequence is f-statistically convergent, it immediately follows that any sequence satisfying the conditions of the classical theorem of Boyanov and Veselinov (Theorem 2.1 ) automatically satisfies the conditions of its f-statistical analog (Theorem 2.3).

Our next example shows that there may exist a sequence of positive linear operators which satisfies the conditions of Theorem 2.3 but does not satisfy the conditions of Theorem 2.1, thereby showing that our result is stronger than the classical one.

Example 2.5

Consider the sequence Q_n : C^*[0,∞) → C^*[0,∞) of positive linear operators defined by

Q_{n} (g, x) = (1 + α_{n}) V_{n} (g, x),

where (V_n) is the sequence of classical Baskakov operators [3] defined by

V_{n} (g, x) = \sum_{k = 0}^{\infty} (\begin{matrix} n + k - 1 \\ k \end{matrix}) x^{k} {(1 + x)}^{- n - k} g (\frac{k}{n}), x \geq 0, n \geq 1.

and (α_n) is the sequence of scalars which is f-statistically convergent to zero for some unbounded modulus f but not convergent to zero. From Corollary 3.3 of [19] it follows that

(0.15)

lim {‖ V_{n} (1, x) - 1 ‖}_{\infty} = 0,

(0.16)

lim {‖ V_{n} (e^{- t}, x) - e^{- x} ‖}_{\infty} = 0,

(0.17)

lim {‖ V_{n} (e^{- 2 t}, x) - e^{- 2 x} ‖}_{\infty} = 0.

Now making use of the fact (Remark 1.4 ) that every convergent sequence is f-statistically convergent, we see that the sequence (Q_n) satisfies conditions (2.2) — (2.4) of Theorem 2.3 and hence we have

f - s t lim {‖ Q_{n} (g, x) - g (x) ‖}_{\infty} = 0 for all g \in C^{*} [0, \infty) .

On the other hand,

Q_{n} (1, x) = (1 + α_{n}) V_{n} (1, x) = (1 + α_{n})

and so,

lim {‖ Q_{n} (1, x) - 1 ‖}_{\infty} = lim ‖ 1 + α_{n} - 1 ‖ = lim ‖ α_{n} ‖ = lim ∣ α_{n} ∣ \neq 0,

from where it follows that (Q_n) does not satisfy the conditions of Theorem 2.1.

Remark 2.6

Since every f-statistically convergent sequence is statistically convergent, it immediately follows that any sequence satisfying the conditions of the f-statistical analog of the classical theorem of Boyanov and Veselinov (Theorem 2.3) automatically satisfies the conditions of the statistical analog of the classical theorem (Theorem 2.2).

Our next example shows that there may exist a sequence of positive linear operators which satisfies the conditions of Theorem 2.2 but not of Theorem 2.3, thereby showing that the statistical analog is stronger than the f-statistical analog.

Example 2.7

Consider the sequence Q_n : C^*[0,∞) → C^*[0,∞) of positive linear operators defined by

Q_{n} (g, x) = (1 + α_{n}) V_{n} (g, x),

where (V_n) is the sequence of Baskakov operators and (α_n) is the sequence of scalars which is statistically convergent to zero but not f-statistically convergent to zero for some unbounded modulus f. Proceeding as in Example 2.5, it is easy to see that the sequence (Q_n) satisfies the conditions of the statistical analog of the classical theorem of Boyanov and Veselinov but does not satisfy the conditions of the f-statistical analog of the classical theorem of Boyanov and Veselinov.

Remark 2.8

From Remarks 2.4, 2.6 and Examples 2.5, 2.7 it follows that the classical Korovkin type approximation theorem of Boyanov and Veselinov implies its f-statistical analog which in turn implies its statistical analog but not conversely. Hence our result is intermediate between the classical theorem and its statistical analog.

3. Rate of f-statistical Convergence

In this section, using f-statistical convergence we study the rate of convergence of a sequence of positive linear operators defined from C^*[0,∞) into itself with the help of modulus of continuity. We begin by presenting the following definition.

Definition 3.1

Let X be a normed space and (a_n) be a positive sequence of real numbers such that a_n → 0 as n → ∞. A sequence (x_k) in X is said to be f-statistically convergent to some x ∈ X with the rate o(a_n) if for every ε > 0,

lim_{n \to \infty} \frac{1}{a_{n} f (n)} f (∣ {k \leq n : ‖ x_{k} - x ‖ \geq ɛ} ∣) = 0.

In this case, we write x_k – x = S^f – o(a_n).

Lemmma 3.2

Let f be an unbounded modulus, (a_n) and (b_n) be two sequences of positive real numbers. Let (x_k) and (y_k) be two sequences of real numbers such that x_k –x = S^f –o(a_n) and y_k –y = S^f –o(b_n). Let c_n = max{a_n, b_n}. Then we have

(x_k – x) ± (y_k – y) = S^f – o(c_n),

(x_k – x)(y_k – y) = S^f – o(c_n),

α(x_k – x) = S^f – o(a_n), for any scalar α.

Proof

(i) Assume that x_k – x = S^f – o(a_n) and y_k – y = S^f – o(b_n). Then, for every ε > 0, we have

(3.1)

\begin{array}{l} \frac{1}{c_{n} f (n)} f (∣ {k \leq n : ∣ (x_{k} - x) \pm (y_{k} - y) ∣ \geq ɛ} ∣) \\ \leq \frac{1}{c_{n} f (n)} f (∣ {k \leq n : ∣ x_{k} - x ∣ \geq \frac{ɛ}{2}} ∣) + \frac{1}{c_{n} f (n)} f (∣ {k \leq n : ∣ y_{k} - y ∣ \geq \frac{ɛ}{2}} ∣) \\ \leq \frac{1}{a_{n} f (n)} f (∣ {k \leq n : ∣ x_{k} - x ∣ \geq \frac{ɛ}{2}} ∣) + \frac{1}{b_{n} f (n)} f (∣ {k \leq n : ∣ y_{k} - y ∣ \geq \frac{ɛ}{2}} ∣) \end{array}

Now by taking the limit as n→∞ in (3.1) and using the hypothesis, we conclude that

lim_{n \to \infty} \frac{1}{c_{n} f (n)} f (∣ {k \leq n : ∣ (x_{k} - x) \pm (y_{k} - y) ∣ \geq ɛ} ∣) = 0,

where c_n = max{a_n, b_n}. Since the other assertions can be proved similarly, we omit the details.

Recall [20] that for a continuous function g ∈ C(I) defined on an interval I ⊂ ℝ, the function ω : C(I) × [0,∞) → ℝ ∪ {∞}, defined by:

ω (g, δ) = sup {∣ g (t) - g (x) ∣ : t, x \in I, ∣ t - x ∣ \leq δ}

is called its usual modulus of continuity.

To estimate the rate of convergence of a sequence of positive linear operators defined from C^*[0,∞) into itself, we make use of the following modulus of continuity:

ω^{*} (g, δ) = sup_{\begin{matrix} t, x \geq 0 \\ ∣ e^{- t} - e^{- x} ∣ \leq δ \end{matrix}} ∣ g (t) - g (x) ∣,

defined for every δ ≥ 0 and every function g ∈ C^*[0,∞). In fact, this modulus (known as weighted modulus of continuity) can be expressed in terms of the usual modulus of continuity by the relation [19]:

ω^{*} (g, δ) = ω (g^{*}, δ),

where g^* is the continuous function defined on [0, 1] by

g^{*} (x) = {\begin{array}{l} g (- ln x), & x \in (0, 1], \\ {lim}_{t \to \infty} f (t), & x = 0 \end{array}

It is well known ( e.g. see [20]) that for any δ > 0

∣ g (t) - g (x) ∣ \leq (1 + \frac{{(e^{- t} - e^{- x})}^{2}}{δ^{2}}) ω^{*} (g, δ) for all g \in C^{*} [0, \infty) .

We are now in a position to give the promised estimation.

Theorem 3.3

Let (L_n) be a sequence of positive linear operators from C^*[0,∞) into C^*[0,∞). Suppose that

L_n(1, x) –1 = S^f − o(a_n),

ω^*(g, λ_n) = S^f − o(b_n), where $λ_{n} = \sqrt{{‖ L_{n} (Ψ_{x}, x) ‖}_{\infty}}$ and Ψ_x(t) = (e⁻^t − e⁻^x)².

Then for all g ∈ C^*[0,∞), we have

L_{n} (g, x) - g (x) = S^{f} - o (c_{n}),

where c_n = max{a_n, b_n}.

Proof

Let g ∈ C^*[0,∞) and x ∈ [0,∞). Then using the monotonicity of (L_n), we see (for any δ > 0 and n ∈ ℕ) that

\begin{array}{l} ∣ L_{n} (g, x) - g (x) ∣ = ∣ L_{n} (g (t), x) - g (x) ∣ \\ = ∣ L_{n} (g (t) - g (x), x) + L_{n} (g (x), x) - g (x) ∣ \\ \leq ∣ L_{n} (g (t) - g (x), x) ∣ + ∣ L_{n} (g (x), x) - g (x) ∣ \\ \leq L_{n} (∣ g (t) - g (x) ∣, x) + ∣ g (x) L_{n} (1, x) - g (x) ∣ \\ \leq L_{n} (1 + \frac{{(e^{- t} - e^{- x})}^{2}}{δ^{2}}, x) ω^{*} (g, δ) + ∣ g (x) ∣ ∣ L_{n} (1, x) - 1 ∣ \\ = [L_{n} (1, x) + \frac{1}{δ^{2}} L_{n} ({(e^{- t} - e^{- x})}^{2}, x)] ω^{*} (g, δ) + ∣ g (x) ∣ ∣ L_{n} (1, x) - 1 ∣ \\ = [L_{n} (1, x) - 1] ω^{*} (g, δ) + ω^{*} (g, δ) + \frac{1}{δ^{2}} L_{n} ({(e^{- t} - e^{- x})}^{2}, x) ω^{*} (g, δ) + ∣ g (x) ∣ ∣ L_{n} (1, x) - 1 ∣ \end{array}

Putting $δ = λ_{n} = \sqrt{{‖ L_{n} ({(e^{- t} - e^{- x})}^{2}, x) ‖}_{\infty}}$ , we get

(3.2)

\begin{array}{l} {‖ L_{n} (g, x) - g (x) ‖}_{\infty} \leq {‖ (L_{n} (1, x) - 1) ‖}_{\infty} ω^{*} (g, λ_{n}) + 2 ω^{*} (g, λ_{n}) + {‖ g ‖}_{\infty} {‖ L_{n} (1, x) - 1 ‖}_{\infty} \\ \leq K {{‖ (L_{n} (1, x) - 1) ‖}_{\infty} ω^{*} (g, λ_{n}) + ω^{*} (g, λ_{n}) + {‖ L_{n} (1, x) - 1 ‖}_{\infty}}, \end{array}

where K = max{||g||_∞, 2}.

Now, for a given ε > 0, we consider the following sets

\begin{array}{l} D & : = {n \leq N : {‖ L_{n} (g, x) - g (x) ‖}_{\infty} \geq ɛ}, \\ D_{1} & : = {n \leq N : {‖ L_{n} (1, x) - 1 ‖}_{\infty} ω^{*} (g, λ_{n}) \geq \frac{ɛ}{3 K}}, \\ D_{2} & : = {n \leq N : ∣ ω^{*} (g, λ_{n}) ∣ \geq \frac{ɛ}{3 K}}, \\ D_{3} & : = {n \leq N : {‖ L_{n} (1, x) - 1 ‖}_{\infty} \geq \frac{ɛ}{3 K}} . \end{array}

Then it follows from (3.2) that D ⊂ D₁ ∪ D₂ ∪ D₃.

Now, since c_n = max{a_n, b_n}, for every N ∈ ℕ, we have

\begin{array}{l} \frac{1}{c_{N} f (N)} f (∣ {n \leq N : {‖ L_{n} (g, x) - g (x) ‖}_{\infty} \geq ɛ} ∣) \\ \leq \frac{1}{c_{N} f (N)} f (| {n \leq N : {‖ L_{n} (1, x) - 1 ‖}_{\infty} ω^{*} (g, λ_{n}) \geq \frac{ɛ}{3 K}} |) \\ + \frac{1}{b_{N} f (N)} f (| {n \leq N : ∣ ω^{*} (g, λ_{n}) ∣ \geq \frac{ɛ}{3 K}} |) \\ + \frac{1}{a_{N} f (N)} f (| {n \leq N : {‖ L_{n} (1, x) - 1 ‖}_{\infty} \geq \frac{ɛ}{3 K}} |) \end{array}

from where it follows that L_n(g, x) – g(x) = S^f – o(c_n).

Acknowledgements: The authors wish to thank the referee for his valuable suggestions, which have improved the presentation of the paper.

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