Statistical approximation to Bögel-type continuous and periodic functions

Cent. Eur. J. Math. • 7(3) • 2009 • 539-549DOI: 10.2478/s11533-009-0025-4

Central European Journal of Mathematics

Statistical approximation to Bögel-typecontinuous and periodic functions

Research Article

Fadime Dirik1∗, Oktay Duman2† , Kamil Demirci1‡

1 Department of Mathematics, Faculty of Arts and Sciences, Sinop University, Sinop, Turkey

2 Department of Mathematics, Faculty of Arts and Sciences, TOBB Economics and Technology University, Ankara, Turkey

Received 13 October 2008; accepted 1 May 2009

Abstract: In this paper, considering A-statistical convergence instead of Pringsheim’s sense for double sequences, weprove a Korovkin-type approximation theorem for sequences of positive linear operators defined on the spaceof all real valued Bögel-type continuous and periodic functions on the whole real two-dimensional space. Astrong application is also presented. Furthermore, we obtain some rates of A-statistical convergence in ourapproximation.

MSC: 41A25, 41A36

Keywords: The Korovkin theorem • B-continuous functions • B-2π-periodic functions • A-statistical convergence for doublesequences • Regularity for double sequences

© Versita Warsaw and Springer-Verlag Berlin Heidelberg.

1. Introduction

In order to improve the classical Korovkin theory, the space of Bögel-type continuous (or, simply, B-continuous) functionsinstead of the classical one has been used in [3–6]. Recall that the concept of B-continuity was first introduced in 1934by Bögel [8] (see also [9, 10]). On the other hand, this Korovkin theory has also been generalized via the concept ofstatistical convergence (see, for instance, [1, 12–14, 17]). It is well-known that every convergent sequence (in the usualsense) is statistically convergent but its converse is not always true. Also, statistical convergent sequences do not needto be bounded. With these properties, the usage of the statistical convergence in the approximation theory leads us tomore powerful results than the classical aspects.In this paper, we focus on the statistical approximation by means of positive linear operators defined on the space of allreal valued Bögel-type continuous and periodic functions on R2, the real two-dimensional space. In the first section, we

∗ E-mail: [email protected]† E-mail: [email protected]‡ E-mail: [email protected]

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Statistical approximation to Bögel-type continuous and periodic functions

recall the convergence methods for double sequences in Pringsheim’s sense and the statistical sense, while, in the secondsection, we obtain a strong Korovkin-type approximation theorem with the help of statistical convergence, B-continuityand B-periodicity. The third section is devoted to compute the statistical rates of our approximation.As usual, a double sequence

x = {xm,n}, m, n ∈ N,

is convergent in Pringsheim’s sense if, for every ε > 0, there exists N = N(ε) ∈ N such that |xm,n − L| < ε wheneverm, n > N. Then, L is called the Pringsheim limit of x and is denoted by P − lim x = L (see [20]). In this case, wesay that x = {xm,n} is “P-convergent to L”. Also, if there exists a positive number M such that |xm,n| ≤ M for all(m, n) ∈ N2 = N × N, then x = {xm,n} is said to be bounded. Note that in contrast to the case for single sequences, aconvergent double sequence need not to be bounded.Now let

A = [aj,k,m,n], j, k, m, n ∈ N,

be a four-dimensional summability matrix. For a given double sequence x = {xm,n}, the A-transform of x, denoted byAx := {(Ax)j,k}, is given by

(Ax)j,k =∑

(m,n)∈N2

aj,k,m,nxm,n, j, k ∈ N,

provided the double series converges in Pringsheim’s sense for every (m, n) ∈ N2. In summability theory, a two-dimensional matrix transformation is said to be regular if it maps every convergent sequence in to a convergent sequencewith the same limit. The well-known characterization for two dimensional matrix transformations is known as Silverman-Toeplitz conditions (see, for instance, [16]). In 1926, Robison [21] presented a four dimensional analog of the regularityby considering an additional assumption of boundedness. This assumption was made because a double P-convergentsequence is not necessarily bounded. The definition and the characterization of regularity for four dimensional matricesis known as Robison-Hamilton conditions, or briefly, RH-regularity (see, [15, 21]).Recall that a four dimensional matrix A = [aj,k,m,n] is said to be RH-regular if it maps every bounded P-convergentsequence into a P-convergent sequence with the same P-limit. The Robison-Hamilton conditions state that a fourdimensional matrix A = [aj,k,m,n] is RH-regular if and only if

(i) P − limj,k

aj,k,m,n = 0 for each (m, n) ∈ N2,

(ii) P − limj,k

∑(m,n)∈N2

aj,k,m,n = 1,

(iii) P − limj,k

∑m∈N

∣∣aj,k,m,n∣∣ = 0 for each n ∈ N,

(iv) P − limj,k

∑n∈N

∣∣aj,k,m,n∣∣ = 0 for each m ∈ N,

(v)∑

(m,n)∈N2

∣∣aj,k,m,n∣∣ is P−convergent,

(vi) there exist finite positive integers A and B such that∑

m,n>B

∣∣aj,k,m,n∣∣ < A holds for every (j, k) ∈ N2.

Now let A = (aj,k,m,n) be a non-negative RH−regular summability matrix, and let K ⊂ N2. Then A−density of K isgiven by

δ(2)A {K} := P − lim

j,k

∑(m,n)∈K

aj,k,m,n

provided that the limit on the right-hand side exists in Pringsheim’s sense. A real double sequence x = (xm,n) is said tobe A−statistically convergent to a number L if, for every ε > 0,

δ(2)A {(m, n) ∈ N2 : |xm,n − L| ≥ ε} = 0.

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F. Dirik, O. Duman, K. Demirci

In this case, we write st(2)A − limm,n xm,n = L. Observe that, a P-convergent double sequence is A-statistically convergent

to the same value but the converse does not hold true. For example, consider the double sequence x = {xm,n} given by

xm,n ={

mn, if m and n are squares,1, otherwise.

We should note that if we take A = C (1, 1), which is the double Cesáro matrix, then C (1, 1)-statistical convergencecoincides with the notion of statistical convergence for double sequence, which was introduced in [18, 19]. Finally, if wereplace the matrix A by the identity matrix for four-dimensional matrices, then A-statistical convergence reduces to thePringsheim convergence.

2. A Korovkin-type approximation theorem

In [8–10], Bögel introduced a new concept of continuity and periodicity as follows. Let X and Y be compact subsets ofthe real numbers, and let D = X × Y . A function f : D → R is called Bögel-type continuous (or, briefly, B-continuous)at a point (x, y) ∈ D if, for every ε > 0, there exists a positive number δ = δ(ε) such that

∣∣∆x,y [f (u, v)]∣∣ < ε for

any (u, v) ∈ D with |u − x| < δ and |v − y| < δ where the symbol ∆x,y [f (u, v)] denotes the mixed difference of fdefined by ∆x,y [f (u, v)] = f(u, v) − f(u, y) − f(x, v) + f(x, y). A function f : R2 → R is called Bögel-type 2π-periodic(or, B-2π-periodic) if ∆x,y [f (u + 2π, v + 2π)] = ∆x,y [f (u, v)] holds for each (x, y) , (u, v) ∈ R2. Then, by B2π we denotethe space of all real valued B-continuous and B-2π-periodic functions on R2. Recall that C2π,2π and B(R2) denote thespace of all continuous (in the usual sense) functions which are 2π-periodic with respect to both variables and thespace of all bounded function on R2, respectively. Then, notice that C2π,2π ⊂ B2π . Moreover, one can find an unboundedfunction belonging to B2π , which follows from the fact that, for any function of the type f(u, v) = g(u) + h(v),we have∆x,y [f (u, v)] = 0 for all (x, y), (u, v) ∈ R2.The usual supremum norm on the spaces B(R2) is given by

‖f‖ := sup(x,y)∈R2

|f (x, y)| for f ∈ B(R2).

We recall that the following lemma for B-continuous functions was proved by Badea et. al. [4].

Lemma 2.1 ([4]).If f ∈ B2π , then, for every ε > 0, there are two positive numbers A(ε) = A(ε, f) and B(ε) = B(ε, f) such that

∣∣∆x,y [f(u, v)]∣∣ ≤ ε

3 + A(ε) sin2(u − x

2

)+ B(ε) sin2

( v − y2

)holds for all (x, y), (u, v) ∈ R2.

Let L be a linear operator from B2π into B(R2). Then, as usual, we say that L is positive linear operator provided that

f ≥ 0 implies L(f) ≥ 0. Also, we denote the value of L(f) at a point (x, y) ∈ R2 by L(f(u, v); x, y) or, briefly, L(f ; x, y).Throughout the paper, for fixed (x, y) ∈ R2 and f ∈ B2π , we use the function Fx,y defined as follows:

Fx,y(u, v) = f(u, y) + f(x, v) − f(u, v) for (u, v) ∈ R2. (1)

Since ∆x,y[Fx,y(u, v)

]= −∆x,y [f(u, v)] holds for all (x, y), (u, v) ∈ R2, the B-continuity and B-2π-periodicity of f imply

the B-continuity and B-2π-periodicity of Fx,y for every fixed (x, y) ∈ R2.In this paper, we consider the following five test functions:

f0(u, v) = 1, f1(u, v) = sin u, f2(u, v) = sin v , f3(u, v) = cos u and f4(u, v) = cos v.

With this terminology we have the following main result.

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Theorem 2.1.Let {Lm,n} be a sequence of positive linear operators mapping B2π into B

(R2) , and let A = [aj,k,m,n] be a non-negative

RH-regular summability matrix. Assume that the following conditions hold:

P − limj,k

∑(m,n)∈K

aj,k,m,n = 1, (2)

whereK :=

{(m, n) ∈ N2 : Lm,n(f0; x, y) = 1 for all (x, y) ∈ R2} , (3)

andst(2)

A − limm,n

‖Lm,n(fi) − fi‖ = 0 for i = 1, 2, 3, 4. (4)

Then, for all f ∈ B2π , we havest(2)

A − limm,n

∥∥Lm,n(Fx,y) − f∥∥ = 0,

where Fx,y is given by (1).

Proof. Let (x, y) ∈ R2 and f ∈ B2π be fixed. Then, we get from (2) and (3) that

P − limj,k

∑(m,n)∈N2\K

aj,k,m,n = 0. (5)

Since Fx,y given by (1) belongs to B2π , Lemma 2.1 implies that, for every ε > 0, there exist two positive numbers A(ε)and B(ε) such that ∣∣∆x,y

[Fx,y(u, v)

]∣∣ ≤ ε3 + A(ε) sin2

(u − x2

)+ B(ε) sin2

( v − y2

)(6)

holds for every (u, v) ∈ R2. Also, by (1), observe that

Lm,n(Fx,y; x, y

)− f(x, y) = Lm,n

(∆x,y

[Fx,y(u, v)

]; x, y

)(7)

holds for all (m, n) ∈ K. Then, using the monotonicity and the linearity of the operators Lm,n, for all (m, n) ∈ K, itfollows from (6) and (7) that

∣∣Lm,n(Fx,y; x, y

)− f(x, y)

∣∣ =∣∣Lm,n

(∆x,y

[Fx,y(u, v)

]; x, y

)∣∣≤ Lm,n

(∣∣∆x,y[Fx,y(u, v)

]∣∣ ; x, y)

≤ ε3 + A(ε)Lm,n

(sin2

(u − x2

); x, y

)+B(ε)Lm,n

(sin2

( v − y2

); x, y

)≤ ε

3 + C (ε)2 {2 − sin xLm,n(f1; x, y)

− sin yLm,n(f2; x, y) − cos xLm,n(f3; x, y)− cos yLm,n(f4; x, y)},

where C (ε) = max{A(ε), B(ε)}. From the last inequality, we have

∣∣Lm,n(Fx,y; x, y

)− f(x, y)

∣∣ ≤ ε3 + C (ε)

2

4∑i=1

|Lm,n (fi; x, y) − fi(x, y)| (8)

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holds for all (m, n) ∈ K. Taking supremum over (x, y) ∈ R2 on the both-sides of inequality (8) we obtain, for all(m, n) ∈ K, that ∥∥Lm,n(Fx,y) − f

∥∥ ≤ ε3 + C (ε)

2

4∑i=1

‖Lm,n(fi) − fi‖ . (9)

Now for a given r > 0, choose ε > 0 such that ε < 3r. Define the following sets:

U : ={

(m, n) ∈ N2 :∥∥Lm,n(Fx,y) − f

∥∥ ≥ r}

,

Ui : ={

(m, n) ∈ N2 : ‖Lm,n(fi) − fi‖ ≥ 3r − ε6C (ε)

}, i = 1, 2, 3, 4.

Hence, it follows from (9) that

U ∩ K ⊆4⋃

i=1

(Ui ∩ K ),

which gives, for all (j, k) ∈ N2,

∑(m,n)∈U∩K

aj,k,m,n ≤4∑

i=1

∑(m,n)∈Ui∩K

aj,k,m,n ≤4∑

i=1

∑(m,n)∈Ui

aj,k,m,n (10)

Letting j, k → ∞ (in any manner) in (10) and also using (4), we conclude that

P − limj,k

∑(m,n)∈U∩K

aj,k,m,n = 0. (11)

Furthermore, if we use the inequality

∑(m,n)∈U

aj,k,m,n =∑

(m,n)∈U∩K

aj,k,m,n +∑

(m,n)∈U∩(N2\K )

aj,k,m,n

≤∑

(m,n)∈U∩K

aj,k,m,n +∑

(m,n)∈(N2\K )

aj,k,m,n

and if we take limit as j, k → ∞, then it follows from (5) and (11) that

P − limj,k

∑(m,n)∈U

aj,k,m,n = 0,

which meansst(2)

A − limm,n

∥∥Lm,n(Fx,y

)− f

∥∥ = 0.

The proof is completed.

Remark 2.1.We know that, for some f ∈ B2π , the function f may be unbounded on R2. However, we can say from the conditions (2),(6) and (7) that the norm

∥∥Lm,n(Fx,y) − f∥∥ is meaningful for each (m, n) ∈ K , where K is given by (3).

Remark 2.2.If we replace the matrix A in Theorem 2.1 by the identity double matrix, then we immediately get the following classicalresult which was first introduced by Badea et. al. [4].

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Corollary 2.1 ([4]).Let {Lm,n} be a sequence of positive linear operators mapping B2π into B

(R2). Assume that the following conditions

hold:

(i) Lm,n(f0; x, y) = f0(x, y) for all (x, y) ∈ R2 and (m, n) ∈ N2,

(ii) Lm,n(f1; x, y) = f1(x, y) + um,n(x, y),

(iii) Lm,n(f2; x, y) = f2(x, y) + vm,n(x, y),

(iv) Lm,n(f3; x, y) = f3(x, y) + tm,n(x, y),

(v) Lm,n(f4; x, y) = f4(x, y) + wm,n(x, y),

where {um,n(x, y)}, {vm,n(x, y)}, {tm,n(x, y)} and {wm,n(x, y)} converge to zero uniformly on R2 as m, n approach infinityin any manner whatsoever. Then, for all f ∈ B2π , the sequence {Lm,n

(Fx,y; x, y

)} converges uniformly to f(x, y) with

respect to (x, y) ∈ R2, where Fx,y is given by (1).

Remark 2.3.We now show that our result Theorem 2.1 is stronger than its classical version Corollary 2.1. To see this we first considerthe following sequence of positive linear operators {Km,n} introduced by Badea et. al. [4]:

Km,n(f ; x, y) = 4(Nm + 2) (Nn + 2)

Nm+2∑k=1

Nn+2∑l=1

Fx,y(tk,m, tl,n

)φm

(tk,m − x

)φn

(tl,n − y

)(12)

where Fx,y is given by (1), ts,t = 2sπNt + 2 , s = 1, 2, ..., Nt + 2, and φn is a nonnegative cosine polynomial of the form

φn (x) = 12 +

Nn∑v=1

ρv,n. cos vx with limn→∞

ρ1,n = 1.

Then, by Corollary 2.1, we know [4, 7] that, for any f ∈ B2π ,

P − limm,n

∥∥Km,n(Fx,y) − f∥∥ = 0. (13)

Now take A = C (1, 1) and define a double sequence {um,n} by

um,n ={

1, if m and n are squares0, otherwise. (14)

In this case, observe thatst(2)

C (1,1) − limm,n

um,n = 0. (15)

However, it is easy to see that the sequence {um,n} is not P-convergent. Now using (12) and (14), we define thefollowing positive linear operators on B2π as follows:

Lm,n(f ; x, y) = (1 + um,n)Km,n(f ; x, y). (16)

Then, observe that the sequence of positive linear operators {Lm,n} satisfy all hypotheses of Theorem 2.1 defined by(16). So, by (13) and (15), we have

st(2)C (1,1) − lim

m,n

∥∥Lm,n(Fx,y) − f∥∥ = 0.

However, since {um,n} is not P-convergent, the sequence {Lm,n(f ; x, y)} given by (16) does not converge uniformly (inthe usual sense) to the function f ∈ B2π . So, we conclude that Corollary 2.1 does not work for the operators Lm,n in (16)while our Theorem 2.1 still works.

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3. Rates of A-statistical convergence in Theorem 2.1

In this section we study the rates of A-statistical convergence of a sequence of positive linear operators mapping B2π

into B(R2) with the help of the mixed modulus of smoothness.

We first present two different ways to compute the corresponding rates in A-statistical convergence in Theorem 2.1.

Definition 3.1.Let A = [aj,k,m,n] be a non-negative RH-regular summability matrix and let {αm,n} be a positive non-increasing doublesequence. A double sequence x = {xm,n} is A-statistically convergent to a number L with the rate of o(αm,n) if for everyε > 0,

P − limj,k→∞

1αj,k

∑(m,n)∈K (ε)

aj,k,m,n = 0,

whereK (ε) :=

{(m, n) ∈ N2 : |xm,n − L| ≥ ε

}.

In this case, it is denoted byxm,n − L = stA − o(αm,n) as m, n → ∞.

Definition 3.2.Let A = [aj,k,m,n] and {αm,n} be the same as in Definition 3.1. Then, a double sequence x = {xm,n} is A-statisticallyconvergent to a number L with the rate of om,n(αm,n) if for every ε > 0,

P − limj,k→∞

∑(m,n)∈M(ε)

aj,k,m,n = 0,

whereM(ε) :=

{(m, n) ∈ N2 : |xm,n − L| ≥ ε αm,n

}.

In this case, it is denoted byxm,n − L = stA − om,n(αm,n) as m, n → ∞.

We see from the above statements that, in Definition 3.1 the rate sequence {am,n} directly effects the entries of the matrixA = [aj,k,m,n] although, according to Definition 3.2, the rate is more controlled by the terms of the sequence x = {xm,n}.Using these definitions we obtain the following auxiliary result.

Lemma 3.1.Let {xm,n} and {ym,n} be double sequences. Assume that let A = [aj,k,m,n] be a non-negative RH-regular summabilitymatrix and let {αm,n} and {βm,n} be positive non-increasing sequences. If xm,n − L1 = stA − o(αm,n) and ym,n − L2 =stA − o(βm,n), then we have

(i) (xm,n − L1) ∓ (ym,n − L2) = stA − o(γm,n) as m, n → ∞ , where γm,n := max {αm,n, βm,n} for each (m, n) ∈ N2,

(ii) λ(xm,n − L1) = stA − o(αm,n) as m, n → ∞ for any real number λ.

Furthermore, similar conclusions hold with the symbol “o” replaced by “om,n”.

Proof. (i) Assume that xm,n − L1 = stA − o(αm,n) and ym,n − L2 = stA − o(βm,n). Also, for ε > 0, define

K : ={

(m, n) ∈ N2 : |(xm,n − L1) ∓ (ym,n − L2)| ≥ ε}

,

K1 : ={

(m, n) ∈ N2 : |xm,n − L1| ≥ ε2

},

K2 : ={

(m, n) ∈ N2 : |ym,n − L2| ≥ ε2

}.

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Then observe thatK ⊂ K1 ∪ K2

which gives, for all (j, k) ∈ N2, ∑(m,n)∈K

aj,k,m,n ≤∑

(m,n)∈K1

aj,k,m,n +∑

(m,n)∈K2

aj,k,m,n. (17)

Since γm,n = max {αm,n, βm,n}, by (17), we get

1γj,k

∑(m,n)∈K

aj,k,m,n ≤ 1αj,k

∑(m,n)∈K1

aj,k,m,n + 1βj,k

∑(m,n)∈K2

aj,k,m,n. (18)

Now by taking limit as j, k → ∞ (in any manner) in (18) and using the hypotheses, we conclude that

P − limj,k→∞

1γj,k

∑(m,n)∈K

aj,k,m,n = 0,

which completes the proof of (i). Since the proof of (ii) is similar, we omit it.

Now we remind the concept of mixed modulus of smoothness. For f ∈ B2π , the mixed modulus of smoothness of f ,denoted by ωmixed (f ; δ1, δ2), is defined to be

ωmixed (f ; δ1, δ2) = sup{∣∣∆x,y [f(u, v)]

∣∣ : |u − x| ≤ δ1, |v − y| ≤ δ2}

for δ1, δ2 > 0. In order to obtain our result, we will make use of the elementary inequality

ωmixed (f ; λ1δ1, λ2δ2) ≤ (1 + λ1) (1 + λ2) ωmixed (f ; δ1, δ2) (19)

for λ1, λ2 > 0. The modulus ωmixed has been used by several authors in the framework of “Boolean sum type” approxi-mation (see, for example, [11]). Elementary properties of ωmixed can be found in [22] (see also [2]) and in particular forthe case of B-continuous functions in [3].Then we have the following result.

Theorem 3.1.Let {Lm,n} be a sequence of positive linear operators mapping B2π into B

(R2) and let A = [aj,k,m,n] be a non-negative

RH-regular summability matrix. Let {αm,n} and {βm,n} be a positive non-increasing double sequence. Assume that thefollowing conditions hold:

P − limj,k→∞

1αj,k

∑(m,n)∈K

aj,k,m,n = 1, (20)

where K ={

(m, n) ∈ N2 : Lm,n(f0; x, y) = 1 for all (x, y) ∈ R2} ; and

ωmixed(f ; γm,n, δm,n

)= stA − o(βm,n) as m, n → ∞, (21)

where γm,n :=√

‖Lm,n(φ)‖ and δm,n :=√

‖Lm,n(Ψ)‖ with φ(u, v) = sin2(u − x

2

), Ψ(u, v) = sin2

( v − y2

). Then we have,

for all f ∈ B2π , ∥∥Lm,n(Fx,y) − f∥∥ = stA − o(cm,n) as m, n → ∞,

where Fx,y is given by (1) and cm,n := max {αm,n, βm,n} for each (m, n) ∈ N2. Furthermore, similar results holds when thesymbol “o” is replaced by “om,n”.

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Proof. Let (x, y) ∈ R2 and f ∈ B2π be fixed. It follows from (20) that

P − limj,k→∞

1αj,k

∑(m,n)∈N2\K

aj,k,m,n = 0. (22)

Also, for each (u, v) ∈ R2, we choose (u, v) = (u + 2lπ, v + 2kπ), l, k ∈ Z, such that |u − x| ≤ π, |v − y| ≤ π. Then wehave ∣∣∆x,y [f (u, v)]

∣∣ =∣∣∆x,y [f (u, v)]

∣∣ ,

and

|u − x| ≤ π∣∣∣sin

(u − x2

)∣∣∣ ,

|v − y| ≤ π∣∣∣sin

( v − y2

)∣∣∣ .

Then, we obtain ∣∣∆x,y [f (u, v)]∣∣ ≤ ωmixed (f ; |u − x| , |v − y|)

≤(

1 + πδ1

∣∣∣sin(u − x

2

)∣∣∣)×

(1 + π

δ2

∣∣∣sin( v − y

2

)∣∣∣) ωmixed (f ; δ1, δ2) .

(23)

Hence, using the monotonicity and the linearity of the operators Lm,n, for all (m, n) ∈ K , it follows from (23) that∣∣Lm,n(Fx,y; x, y) − f (x, y)∣∣ =

∣∣−Lm,n(∆x,y [f(u, v)] ; x, y

)∣∣≤ Lm,n

(∣∣∆x,y [f(u, v)]∣∣ ; x, y

)≤ Lm,n

((1 + π

δ1

∣∣∣sin(u − x

2

)∣∣∣) (1 + π

δ2

∣∣∣sin( v − y

2

)∣∣∣) ; x, y)

×ωmixed (f ; δ1, δ2)

={

1 + πδ1

Lm,n

(∣∣∣sin(u − x

2

)∣∣∣ ; x, y)

+ πδ2

Lm,n

(∣∣∣sin( v − y

2

)∣∣∣ ; x, y)

+ π2

δ1δ2Lm,n

(∣∣∣sin(u − x

2

)sin

( v − y2

)∣∣∣ ; x, y)}

×ωmixed (f ; δ1, δ2) .

Using the Cauchy-Schwarz inequality, we have

∣∣Lm,n(Fx,y; x, y) − f (x, y)∣∣ ≤

{1 + π

δ1

√Lm,n (φ; x, y) + π

δ2

√Lm,n (Ψ; x, y)

π2

δ1δ2

√Lm,n (φ (u, v) .Ψ (u, v) ; x, y)

}ωmixed (f ; δ1, δ2) .

for all (m, n) ∈ K . Taking supremum over (x, y) ∈ R2 on the both-sides of the above inequality, we obtain, for all(m, n) ∈ K, that ∥∥Lm,n(Fx,y) − f

∥∥ ≤ (1 + π)2ωmixed(f ; γm,n, δm,n

)(24)

where δ1 = γm,n =√

‖Lm,n(φ)‖ and δ2 = δm,n =√

‖Lm,n(Ψ)‖. Now, given ε > 0, define the following sets:

D : ={

(m, n) ∈ N2 :∥∥Lm,n(Fx,y) − f

∥∥ ≥ ε}

,

D1 : ={

(m, n) ∈ N2 : ωmixed(f ; γm,n, δm,n

)≥ ε

(1 + π)2

}.

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Hence, it follows from (24) thatD ∩ K ⊆ D1 ∩ K,

which gives, for all (j, k) ∈ N2,

1cj,k

∑(m,n)∈D∩K

aj,k,m,n ≤ 1cj,k

∑(m,n)∈D1∩K

aj,k,m,n

≤ 1cj,k

∑(m,n)∈D1

aj,k,m,n

≤ 1βj,k

∑(m,n)∈D1

aj,k,m,n. (25)

where cm,n = max {αm,n, βm,n}. Letting j, k → ∞ (in any manner) in (25) and from (22), we conclude that

P − limj,k→∞

1βj,k

∑(m,n)∈D∩K

aj,k,m,n = 0. (26)

Furthermore, if we use the inequality

∑(m,n)∈D

aj,k,m,n =∑

(m,n)∈D∩K

aj,k,m,n +∑

(m,n)∈D∩(N2\K )

aj,k,m,n

≤∑

(m,n)∈D∩K

aj,k,m,n +∑

(m,n)∈(N2\K )

aj,k,m,n

which gives,1

cj,k

∑(m,n)∈D

aj,k,m,n ≤ 1cj,k

∑(m,n)∈D∩K

aj,k,m,n + 1αj,k

∑(m,n)∈(N2\K )

aj,k,m,n. (27)

Letting j, k → ∞ (in any manner) in (27) and using (22), (26), we conclude that

P − limj,k→∞

1cj,k

∑(m,n)∈D

aj,k,m,n = 0.

The proof is completed.

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Statistical approximation to Bögel-type continuous and periodic functions

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