PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 102(116) (2017), 203–209 DOI: https://doi.org/10.2298/PIM1716203M

KOROVKIN TYPE THEOREM FOR FUNCTIONS OF TWO VARIABLES VIA LACUNARY EQUISTATISTICAL CONVERGENCE

M. Mursaleen

Abstract. Aktuğlu and Gezer [1] introduced the concepts of lacunary equistatistical convergence, lacunary statistical pointwise convergence and lacunary statistical uniform convergence for sequences of functions. Recently, Kaya and Gönül [11 ] proved some analogs of the Korovkin approximation theorem via lacunary equistatistical convergence by using test functions 1, \frac{x}{1+x} , \frac{y}{1+y} , ( \frac{x}{1+x} )^{2} + ( \frac{y}{1+y} )^{2}. We apply the notion of lacunary equistatistical convergence to prove a Korovkin type approximation theorem for functions of two variables by using test functions 1, \frac{x}{1−x} , \frac{y}{1−y} , ( \frac{x}{1−x} )^{2} + ( \frac{y}{1−y} )^{2}.

1. Introduction and preliminaries

The following concept of statistical convergence for sequences of real numbers was introduced by Fast [6]. Let K ⊆ ℕ and K_{n} = {j : j ⩽ n, j ∈ K}. Then the natural density of K is defined by δ(K) := lim_{n→∞} |K_{n}|/n if the limit exists, where |K_{n}| denotes the cardinality of the set K_{n}.

A sequence x = (x_{j}) of real numbers is said to be statistically convergent to the number L if, for every ϵ > 0, the set {j : j ∈ ℕ, |x_{j} − L| ⩾ ϵ} has natural density zero, that is, if, for each ϵ > 0, we have

lim_{n} \frac{1}{n} |\{j : j ⩽ n, |x_{j} − L| ⩾ ϵ\}| = 0.

By a lacunary sequence we mean an increasing integer sequence θ = {k_{r}} such that k_{0} = 0 and h_{r} = k_{r} − k_{r−1} → ∞ as r → ∞. Throughout this paper the intervals determined by θ will be denoted by I_{r} = (k_{r−1} , k_{r}], and the ratio k_{r}/k_{r−1} will be abbreviated by q_{r}.

2010 Mathematics Subject Classification: Primary 41A10, 41A25, 41A36; Secondary 40A30, 40G15.

Key words and phrases: statistical convergence, lacunary equistatistical convergence, positive linear operator, Korovkin type approximation theorem.

The present research was supported by the Department of Science and Technology, New Delhi, under grant No. SR/S4/MS:792/12.

Communicated by Gradimir Milovanović.

Fridy and Orhan [7] defined the notion of lacunary statistical convergence as follows. Let θ be a lacunary sequence; the number sequence x is S_{θ}-convergent to L provided that for every ϵ > 0),

lim_{r} \frac{1}{h_{r}} |\{k ∈ I_{r} : |x_{k} − L| ⩾ ϵ\}| = 0.

In this case we write S_{θ}-limit x = L or x_{k} → L(S_{θ}).

The concept of equistatistical convergence was introduced by Balcerzak et al. [2] and was subsequently applied for deriving approximation theorems in [1,8–10,19]. In [1], Aktuglu and Gezer [1] generalized the idea of statistical convergence to lacunary equistatistical convergence. Recently, Kaya and Gönül [11] established some analogs of the Korovkin approximation theorem via lacunary equistatistical convergence. Korovkin type approximation theorems for various kinds of statistical convergence are studied in [3–5, 14–18]. In this paper, we prove such type of theorem via lacunary equistatistical convergence by using the test functions 1, \frac{x}{1−x} and (\frac{x}{1−x} )^{2}.

Let C[a, b] be the linear space of all real-valued continuous functions f on [a, b]. We know that C[a, b] is a Banach space with the norm given by

∥f∥_{C[a,b]} := sup_{x∈[a,b]} |f(x)| (f ∈ C[a, b]).

Let f and f_{n} (n ∈ ℕ) be real-valued functions defined on a subset X of the set ℕ of positive integers.

Definition 1.1. A sequence (f_{k}) of real-valued functions is said to be lacunary equi-statistically convergent to f on X if, for every ϵ > 0, the sequence (S_{r}(ϵ, x))_{r∈ℕ} of real-valued functions converges uniformly to the zero function on X, that is, if, for every ϵ > 0, we have lim_{r→∞} ∥S_{r}(ϵ, x)∥_{C(X)} = 0, where

S_{r}(ϵ, x) := \frac{1}{h_{r}} |\{k : k ∈ I_{r}, |f_{k}(x) − f(x)| ⩾ ϵ\}|

and C(X) denotes the space of all continuous functions on X. In this case, we write

f_{k} ⇉ f (θ-equistat).

Definition 1.2. A sequence (f_{k}) is said to be lacunary statistically pointwise convergent to f on X if, for every ϵ > 0 and for each x ∈ X, we have

lim_{r} \frac{1}{h_{r}} |\{k : k ∈ I_{r}, |f_{k}(x) − f(x)| ⩾ ϵ\}| = 0.

In this case, we write f_{r} → f (θ-stat).

Definition 1.3. A sequence (f_{r}) is said to be lacunary statistically uniformly convergent to f on X if (for every ϵ > 0), we have

lim_{r} \frac{1}{h_{r}} |\{k : k ∈ I_{r}, ∥f_{k} − f∥_{C(X)} ⩾ ϵ\}| = 0.

In this case, we write f_{r} ⇉ f (θ-stat).

Definition 1.4. (see [10]). A sequence (f_{r}) of real-valued functions is said to be equistatistically convergent to f on X if, for every ϵ > 0, the sequence (P_{n,ϵ}(x))_{r∈ℕ} of real-valued functions converges uniformly to the zero function on X, that is, if (for every ϵ > 0) we have lim_{n→∞} ∥P_{n,ϵ}(x)∥_{C(X)} = 0, where

P_{n,ϵ}(x) = \frac{1}{n} |\{k : k ⩽ n. |f_{k}(x) − f(x)| ⩾ ϵ\}| = 0.

In this case, we write f_{k} ⇝ f (equistat).

The following implications of the above definitions and concepts are trivial.

f_{k} ⇉ f (θ-stat) ⇒ f_{k} ⇝ f (θ-equistat) ⇒ f_{k} → f (θ-stat).

Furthermore, in general, the reverse implications do not hold true.

2. Main Results

Let I = [0, A], J = [0, B], A, B ∈ (0, 1) and K = I ×J. We denote by C(K) the space of all continuous real valued functions on K. This space is equipped with the norm ∥f∥_{C(K)} := sup_{(x,y)∈K} |f(x, y)|, f ∈ C(K). Let H_{ω} (K) denote the space of all real valued functions f on K such that

|f(s, t) − f(x, y)| ⩽ ω( f; \sqrt{( \frac{s}{1 − s} − \frac{x}{1 − x} )^{2} + ( \frac{t}{1 − t} − \frac{y}{1 − y} )^{2}} ) ,

where ω is the modulus of continuity, i.e.

ω(f; δ) = sup_{(s,t),(x,y)∈K} \{|f(s, t) − f(x, y)| : \sqrt{(s − x)^{2} + (t − y)^{2}} ⩽ δ\}.

It is to be noted that any function f ∈ H_{ω} (K) is continuous and bounded on K.

In [1], Aktuğlu and Gezer proved the Korovkin theorem for lacunary eqistatistiacal convergence by using the test functions 1, x and x^{2}; while Kaya and Gönül [11] used the test functions 1, \frac{x}{1+x} , (\frac{y}{1+y} ), (\frac{x}{1+x} )^{2} + (\frac{y}{1+y} )^{2}. Recently, Srivastava et al [19] defined and studied the λ-eqistatistiacal convergence of positive linear operators by using the notion of λ-statistical convergence [15]. In this paper, we apply the notion of lacunary equistatistical convergence to prove a Korovkin type approximation theorem for functions of two variables by using test functions 1, \frac{x}{1−x} , (\frac{y}{1−y} ), (\frac{x}{1−x} )^{2} + (\frac{y}{1−y} )^{2}.

Let T be a linear operator which maps C[a, b] into itself. We say that T is positive if, for every non-negative f ∈ C[a, b], we have T (f, x) ⩾ 0 (x ∈ [a, b]).

We prove the following result:

Theorem 2.1. Let θ = (k_{r}) be a lacunary sequence, and let (L_{r}) be a sequence of positive linear operators from H_{ω} (K) into C_{B}(K). Then for all f ∈ H_{ω} (K)

(2.1)  L_{r}(f; x, y) ⇝ f(x, y) (θ-equistat)

if and only if

(2.2)  L_{r}(f; x, y) ⇝ g_{i}(x, y) (θ-equistat) (i = 0, 1, 2, 3),

with g_{0}(x) = 1, g_{1}(x) = \frac{x}{1−x} , g_{2}(x) = \frac{y}{1−y} and g_{3}(x) = ( \frac{x}{1−x} )^{2} + ( \frac{y}{1−y} )^{2}.

Proof. Since each of the functions f_{i} belongs to H_{ω} (K), conditions (2.2) follow immediately. Let g ∈ H_{ω} (K) and (x, y) ∈ K be fixed. Then for ε > 0, there exist δ_{1}, δ_{2} > 0 such that |f(s, t) − f(x, y)| < ε holds for all (s, t) ∈ K satisfying | \frac{s}{1−s} − \frac{x}{1−x} | < δ_{1}, | \frac{t}{1−t} − \frac{y}{1−y} | < δ_{2}. Let

K(δ_{1}, δ_{2}) := \{(s, t) ∈ K : | \frac{s}{1 − s} − \frac{x}{1 − x} | < δ_{1}, | \frac{t}{1 − t} − \frac{y}{1 − y} | < δ_{2}\}.

Hence

(2.3)  |f(s, t) − f(x, y)| = |f(s, t) − f(x, y)|_{χ_{K(δ_{1} ,δ_{2} )} (s,t)}
+ |f(s, t) − f(x, y)|_{χ_{K∖K(δ_{1} ,δ_{2} )} (s,t)}
⩽ ε + 2N_{χ_{K∖K(δ_{1} ,δ_{2} )} (s,t)} ,

where χ_{D} denotes the characteristic function of the set D and N = ∥f∥_{C_{B} (K)} Further we get

(2.4)  χ_{K∖K(δ_{1} ,δ_{2} )} (s, t) ⩽ \frac{1}{δ_{1}^{2}}( \frac{s}{1 − s} − \frac{x}{1 − x} )^{2} + \frac{1}{δ_{2}^{2}}( \frac{t}{1 − t} − \frac{y}{1 − y} )^{2}.

Combining (2.3) and (2.4) and choosing δ := min{δ_{1}, δ_{2}}, we get

|f(s, t) − f(x, y)| ⩽ ε + \frac{2N}{δ^{2}} \{( \frac{s}{1 − s} − \frac{x}{1 − x} )^{2} + ( \frac{t}{1 − t} − \frac{y}{1 − y} )^{2}\}.

After using the linearity and positivity of operators {L_{r}}, we get

|L_{r}(f; x, y) − f(x, y)| ⩽ ε + M{|L_{r}(g_{0}; x, y) − g_{0}(x, y)| + |L_{r}(g_{1}; x, y) − g_{1}(x, y)|
+ |L_{r}(g_{2}; x, y) − g_{2}(x, y)| + |L_{r}(g_{3}; x, y) − g_{3}(x, y)|},

which implies that

(2.5)  |L_{r}(f; x, y) − f(x, y)| ⩽ ϵ + B ∑_{i=0}^{3} |L_{r}(g_{i}; x, y) − g_{i}(x, y)|,

where M := ε + N + \frac{4N}{δ^{2}} . Now for a given ρ > 0, choose ϵ > 0 such that ϵ < ρ. Then, for each i = 0, 1, 2, 3, set ψ_{ρ}(x, y) := |{k ∈ ℕ : |L_{k}(f; x, y) − f(x, y)| ⩾ ρ}| and ψ_{i,ρ} (x, y) := |\{k ∈ ℕ : |L_{k}(g_{i}; x, y) − g_{i}(x, y)| ⩾ \frac{ρ−ϵ}{4K} \}| for (i = 0, 1, 2, 3), it follows from (2.5) that ψ_{ρ}(x, y) ⊆ ⋃ _{i=0}^{3} ψ_{i,ρ} (x, y). Hence

(2.6)  \frac{∥ψ_{ρ}(x, y)∥_{C_{B} (K)}}{h_{r}} ⩽ ∑_{i=o}^{3} ( \frac{∥ψ_{i,ρ} (x, y)∥_{C_{B} (K)}}{h_{r}} ) .

Now using hypothesis (2.2) and Definition 1.1, the right-hand side of (2.6) tends to zero as r → ∞. Therefore, we have lim_{r→∞} \frac{1}{h_{r}} ∥ψ_{ρ}(x, y)∥_{C_{B} (K)} = 0 for every ρ > 0, i.e.. (2.1) holds. □

Example 2.1. Consider the following Meyer-König and Zeller [13] (of two variables) operators:

B_{m,n} (f; x, y) := (1 − x)^{m+1} (1 − y)^{n+1}

KOROVKIN TYPE THEOREM FOR FUNCTIONS OF TWO VARIABLES… 207

× ∑_{j=0}^{∞} ∑_{k=0}^{∞} f( \frac{j}{j + m + 1} , \frac{k}{k + n + 1} ) \binom{m + j}{j} \binom{n + k}{k} x^{j} y^{k},

where f ∈ H_{ω} (K), and K = [0, A] × [0, B], A, B ∈ (0, 1).

Since, for x ∈ [0, A], A ∈ (0, 1), we have 1/(1 − x)^{n+1} = ∑ _{k=0}^{∞} \binom{n+k}{k}x^{k}, it is easy to see that B_{m,n} (g_{0}; x, y) = f_{0}(x, y). Also, we obtain

B_{m,n} (g_{1}; x, y) = (1 − x)^{m+1} (1 − y)^{n+1} ∑_{j=0}^{∞} ∑_{k=0}^{∞} \frac{j}{m + 1} \binom{m + j}{j} \binom{n + k}{k} x^{j} y^{k}

= (1 − x)^{m+1} (1 − y)^{n+1} x ∑_{j=0}^{∞} ∑_{k=0}^{∞} \frac{1}{m + 1} \frac{(m + j)!}{m!(j − 1)!} \binom{n + k}{k} x^{j−1} y^{k}

= (1 − x)^{m+1} (1 − y)^{n+1} x\frac{1}{(1 − x)^{m+2}} \frac{1}{(1 − y)^{n+1}} = \frac{x}{(1 − x)} ,

and similarly B_{m,n} (g_{2}; x, y) = \frac{y}{(1−y)} .

Finally, we get

B_{m,n} (g_{3}; x, y)

= (1−x)^{m+1} (1−y)^{n+1} ∑_{j=0}^{∞} ∑_{k=0}^{∞} \{( \frac{j}{m+1} )^{2}+ ( \frac{k}{n+1} )^{2}\} \binom{m+j}{j} \binom{n+k}{k} x^{j} y^{k}

= (1 − x)^{m+1} (1 − y)^{n+1} \frac{x}{m + 1} ∑_{j=0}^{∞} ∑_{k=0}^{∞} \frac{j}{m + 1} \frac{(m + j)!}{m!(j − 1)!} \binom{n + k}{k} x^{j−1} y^{k}

+ (1 − x)^{m+1} (1 − y)^{n+1} \frac{y}{n + 1} ∑_{j=0}^{∞} ∑_{k=0}^{∞} \frac{k}{n + 1} \binom{m + j}{j} \frac{(n + k)!}{n!(k − 1)!} x^{j} y^{k−1}

= (1 − x)^{m+1} (1 − y)^{n+1} \frac{x}{m + 1} \{ x ∑_{j=0}^{∞} ∑_{k=0}^{∞} \frac{(m + j + 1)!}{(m + 1)!(j − 1)!} \binom{n + k}{k} x^{j−1} y^{k}

+ ∑_{j=0}^{∞} ∑_{k=0}^{∞} \binom{m + j + 1}{j} \binom{n + k}{k} x^{j} y^{k}\}

+ (1 − x)^{m+1} (1 − y)^{n+1} \frac{y}{n + 1} \{ y ∑_{j=0}^{∞} ∑_{k=0}^{∞} \frac{(n + k + 1)!}{(n + 1)!(k − 1)!} \binom{m + j}{j} x^{j} y^{k−1}

+ ∑_{j=0}^{∞} ∑_{k=0}^{∞} \binom{n + k + 1}{k} \binom{m + j}{j} x^{j} y^{k}\}

= \frac{m + 2}{m + 1} ( \frac{x}{1 − x} )^{2} + \frac{1}{m + 1} \frac{x}{1 − x} + \frac{n + 2}{n + 1} ( \frac{y}{1 − y} )^{2} + \frac{1}{n + 1} \frac{y}{1 − y}

→ ( \frac{x}{1 − x} )^{2} + ( \frac{y}{1 − y} )^{2}.

Therefore, we have B_{n}(f_{i}; x, y) ⇝ g_{i}(x, y) (θ-equistat) (i = 0, 1, 2, 3). Hence by Theorem 2.1, we have B_{n}(f; x, y) ⇝ g(x, y) (θ-equistat).

3. Rate of Lacunary Equistatistical Convergence

In this section we study the rate of lacunary equistatistical convergence of a sequence of positive linear operators as given in [11].

Definition 3.1. Let (a_{n}) be a positive non-increasing sequence. A sequence (f_{r}) is said to be lacunary equistatistically convergent to a function f with the rate β (0 < β < 1) if for every ϵ > 0,

lim_{r→∞} \frac{Λ_{r,ϵ}(x, y)}{r^{−β}} = 0

uniformly with respect to (x, y) ∈ K or equivalently, for every ϵ > 0,

lim_{r→∞} \frac{∥Λ_{r,ϵ}(x, y)∥_{C_{B} (X)}}{r^{−β}} = 0,

where

Λ_{r}(x, ϵ) := \frac{1}{h_{r}} |\{k ∈ I_{r} : |f_{k}(x, y) − f(x, y)| ⩾ ε\}| = 0.

In this case, we write f_{r} − f = o(r^{−β} ) (θ-equistat) on K.

We have the following basic lemma.

Lemma 3.1. Let (f_{r}) and (g_{r}) be sequences of functions belonging to H_{ω} (K). Assume that f_{r} − f = o(r^{−β_{1}} ) (θ-equistat) on X and g_{r} − g = o(r^{−β_{2}} ) (θ-equistat). Let β = min{β_{1}, β_{2}}. Then the following statement holds:

(i) (f_{r} + g_{r}) − (f + g) = o(r^{−β} ) (θ-equistat), (ii) (f_{r} − f)(g_{r} − g) = o(r^{−β_{1}} ) (θ-equistat), (iii) μ(f_{r} − f) = o(r^{−β_{1}} ) (θ-equistat) for any real number μ, (iv) \sqrt{|f_{r} − f} | = o(r^{−β_{1}} ) (θ-equistat).

We recall that the modulus of continuity of a function f ∈ H_{ω} (K) is defined by

ω(f; δ) = sup_{s,x∈K} {|f(s) − f(x)| : |s − x| ⩽ δ} (δ > 0).

Now we prove the following result.

Theorem 3.1. Let {L_{r}} be a sequence of positive linear operators from H_{ω} (K) into C_{B}(K). Assume that the following conditions hold:

(a) L_{r}(g_{0}; x, y) − g_{0} = o(r^{−β_{1}} ) (θ-equistat) on K,

(b) ω(f; δ_{r,x} , δ_{r,y} ) = o(r^{−β_{2}} ) (θ-equistat) on K,

where δ_{r,x} = \sqrt{L_{r}((( \frac{s}{1−s} − \frac{x}{1−x} )^{2}, x))} and δ_{r,y} = \sqrt{L_{r}((( \frac{t}{1−t} − \frac{y}{1−y} )^{2}, y))} . Then for all f ∈ H_{ω} (K), we have L_{r}(f; x, y) − f(x, y) = o(r^{−β} ) (θ-equistat) on K, where β = min{β_{1}, β_{2}}.

Proof. Let f ∈ H_{ω} (K) and (x, y) ∈ K. Then it is well known that,

|L_{r}(f; x, y) − f(x, y)| ⩽ M|L_{r}(g_{0}; x, y) − g_{0}(x, y)|
+ (L_{r}(g_{0}; x, y) + \sqrt{L_{r}(g_{0}; x, y)} )ω(f; δ_{r,x} , δ_{r,y} ),

where M = ∥f∥_{H_{ω} (K)} . This yields that

|L_{r}(f; x, y) − f(x, y)| ⩽ M|(L_{r}(g_{0}; x, y) − g_{0}(x, y)| + 2ω(f; δ_{r,x} , δ_{r,y} )

+ ω(f; δ_{r,x} , δ_{r,y} )|(L_{r}(g_{0}; x, y) − g_{0}(x, y)| .

Now using the conditions (a), (b) and Lemma 3.1 in the above inequality, we get L_{r}(f) − f = o(r^{−β} ) (θ-equistat) on K. □

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Department of Mathematics (Received 15 11 2015) Aligarh Muslim University Aligarh, India

 mursaleenm@gmail.com