Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 350474, 5 pages http://dx.doi.org/10.1155/2014/350474

Research Article

Approximation Theorems for Functions of Two Variables via ^-Convergence

Mohammed A. Alghamdi

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Correspondence should be addressed to Mohammed A. Alghamdi; proff-malghamdi@hotmail.com Received 23 October 2013; Accepted 13 December 2013; Published 10 February 2014 Academic Editor: M. Mursaleen

Copyright © 2014 Mohammed A. Alghamdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Cakan et al. (2006) introduced the concept of ^-convergence for double sequences. In this work, we use this notion to prove the Korovkin-type approximation theorem for functions of two variables by using the test functions 1, y, and x2 + y2 and construct an example by considering the Bernstein polynomials of two variables in support of our main result.

1. Introduction and Preliminaries

In [1], Pringsheim introduced the following concept of convergence for double sequences. A double sequence x = (xjk) is said to be convergent to the number L in Pringsheims sense (shortly, p-convergent to L) if for every e > 0 there exists an integer N such that \xjk - L\ < e whenever j, k > N. In this case L is called the p-limit of x.

A double sequence x = (xjk) of real or complex numbers is said to be bounded if ||x||TO = supjk\xjk\ < >x>. We denote the space of all bounded double sequences by Mu.

If x e Mu and is p-convergent to L, then x is said to be boundedly p-convergent to L (shortly, bp-convergent to L). In this case L is called the bp-limit of the double sequences (xjk). The assumption of being bp-convergent was made because a double sequence which is p-convergent is not necessarily bounded.

Assume that a is a one-to-one mapping from the set N (the set of natural numbers) into itself. A continuous linear functional f on the space lTO ofbounded single sequences is said to be an invariant mean or a o-mean if and only if (i) <p(x) > 0 when the sequence x = (xk) has xk > 0 for all k, (ii) f(e) = 1, where e = (1,1,1,...), and (iii) <p(x) = f(xa(k)) for all xe£m.

Throughout this paper we consider the mapping a which has no finite orbits; that is, ap(k) = k for all integer k > 0 and p > 1, where ap(k) denotes the pth iterate of a at k. Note

that a a-mean extends the limit functional on the space c of convergent single sequences in the sense that <p(x) = lim x for all x e c (see [2]). Consequently, c c Va the set of bounded sequences all of whose a-means are equal. We say that a sequence x = (xk) is a-convergent if and only if x e Va. Schaefer [3] defined and characterized the a-conservative, irregular, and a-coercive matrices for single sequences by using the notion of a-convergence. If a(n) = n + 1, then the set Va is reduced to the set f of almost convergent sequences due to Lorentz [4].

In 2006, Cakan et al. [5] presented the following definition of a-convergence for double sequences and established core theorem for a-convergence and later on this notion was studied by Mursaleen and Mohiuddine [6-8]. A double sequence x = (x:k) of real numbers is said to be a-convergent to a number L if and only if x e VB, where

Va =\xi Mu

. lim C

= L uniformly in s,t;L = a-limx} , (1)

^ (x)=(p+1)(q+1)

while here the limit means bp-limit. Let us denote by Va the space of a-convergent double sequences x = (xjk). If

o is translation mapping, then the set V a is reduced to the set F of almost convergent double sequences [9]. Note that

Chp c Va c Mu.

Example 1. Let w = (wmn) be a double sequence such that

1 if n is odd, -1 if n is even,

for all m. Then w is a-convergent to zero (for a(n) = n + 1) but not convergent.

Suppose that C[a,b] is the space of all functions f continuous on [a, b]. It is well known that C[a, b] is a Banach space with the norm

:= sup \f(x)\, f eC[a,b],

x][a,b]

The classical Korovkin approximation theorem is given as follows (see [10,11]).

Theorem 2. Let (Tn) be a sequence of positive linear operators from C[a,b] into C[a,b] and limn^Tn(fi, x) - /¡(x)\\m = 0,for i = 0,1,2, where f0(x) = 1, fi(x) = x, and f2(x) = x2. Then limn\\Tn(f,x)-f(x)\\m = 0,forallfe C[a,b].

In [12], Mohiuddine obtained the Korovkin-type approximation theorem through the notion of almost convergence for single sequences and proved some interesting results. Such type of approximation theorems for the function of two variables is proved in [13,14] through the concept of almost convergence and statistical convergence of double sequences, respectively. Recently, Mohiuddine et al. [15] determined the Korovkin-type approximation theorem by using the test functions 1, e-x, and e- x through the notion of statistical summability (C, 1). Quite recently, by using the concept of (X, -statistical convergence, Mohiuddine and Alotaibi [16] proved the Korovkin-type approximation theorem for functions of two variables. For more details and some recent work on this topic, we refer to [17-21] and references therein. In this work, we apply the notion of a-convergence to prove the Korovkin-type approximation theorem by using the test functions 1, x, y,andx + y .Weapply theclassical Bernstein polynomials of two variables to construct an example in support of our result.

2. Main Result

Now, we prove the classical Korovkin-type approximation theorem for the functions of two variables through o-convergence.

By C(I x I), we denote the set of all two dimensional continuous functions on I2 = I

T be a linear operator from C(I2) a linear

operator T is said to be positive provided that f(x, y) > 0 implies T(f; x, y) > 0.

Theorem 3. Suppose that (Tjk) is a double sequence of positive linear operators from C(I2) into C(I2) and Dmnpq(f;x, y) =

(1/pq)lpj- ZVo Ta> (m),oHn)(f';X,y) satisfying thefollowing

conditions:

lim \\Dmnp a (1;%,y)-1|| =0,

^LimJlDm,n,p,q (s; ^ f) - *|L = 0

P , q^j"

ÜmJIDm ,n,p,q (t; X, y) - y\]lj = °

P , q^j"

l^JK^p«(s2+t2;x,y)- (x2+^2)L =

which hold uniformly in m, n. Thenfor anyfunction f e C(I2) bounded on the whole plane, one has

a- lim \\Tjjc (fix,y)-f(x,y)\\ = 0. That is,

limJ\D™,n,p,q if; x,y)-f ^11 J = 0

uniformly in m, n.

Proof. Since f e C(I2) and f is bounded on the whole plane, we have

\f(x,y)\<M, -rn<x,y<rn. (6)

Therefore,

\ f (s,t) - f (x, y) \ <2M, -rn < s,t,x,y < <x>. (7) Also we have that f is continuous on I x I; that is,

\f (s,t) - f (x, y)\ < e, V|s-%|<<5, \t-y\<8. (8)

From (7) and (8), putting f1 = f1(s,x) = (s - x)2 and f2 = V2(t, y) = (t - y) , we obtain

\f(s,t)-f(x,y)\<e+2^ (f1 +f2), Vls-xl<S, \t-y\<8,

-e-2T +f2)<f(s,t)-f(x,y) 2M . .

< e + m +V2).

Now, we operate Ta>(m),ak(n)(1';x,y) on the above inequality since Taj(m) ak(n) (f; x, y) is monotone and linear. We obtain

1 a'(m),ak(n)

(m),ak (n)

il;x,y)if(s,t)-f(x,y)) (11)

(l;x,y)(e+^^T fa +^2)).

< To>(m),ok(n) (1;X

W = -mn

Therefore

- eTa'(m),ak(n) - —Tal(m)j(n) (f + W'X>y)

< Tai(m)^(n) (f; x,y)-f(x y) Tj,k (l;x y)

< eTa>{m),ak(n) (l';X,y) + ^^¡(rn^Hn) (fl +

Tai(rn),ak(n) (f;X,y)- f(x,y)

= Tai(rn),ak(n) (f X,y)-f (x' y) ToKm),<fi(n) (l'>X' y)

+ f (x, y) Tai(m)^(n) (l;x,y)-f (x, y)

= \T(?i(m),ak(n) (f; x,y)-f (x> y) Tai(rn),ak(n) (l'>X' y)]

+ f(x,y) \Ta> (rn),ak(n) (l;x,y) - l] •

From (12) and (13), we get

Tai(rn),ak(n) (fix y)-f(x y)

o>(m),ok(n) (l;x,y)

(n) (fl + f2 ;x y)

■ f(x'y)[To>(m),oHn) (l;x,y)-1)-

(n) (fi + W>x y)

= Ta> (rn),ok(n) {(s - x)2 + (t- yf'>x y)

o* (m)yOk (n)

o* (m),ok(n)

{s2 - 2sx + x2 +t2 - 2ty + y2; x, y)

(s2 + t2; x y) - 2xTaHm)fiHn) (s; x y) 22

- 2yTai(m),ak(n) (t;x y) + (x2 + y2) Ta'(rn),ak(n) (l; x' y) = \Ta>(m),ak(n) ^ + t2'; x> y) - + y2)]

-2x\Ta>(m),ak(n) (s;x y)-x]

- 2y \T(?i(m),ak(n) (t;x,y)-y]

+ {x2 + y2) \Tai(m)^(„) (Ux,y) - 1] •

Using (14), we obtain

Ta'(m),ak(n) (f;x,y)- f(x,y)

(m),cr(n)

(l;x y)

+ ^T {\Ta>(m),ak(«) {s2 + t2';x,y) - + /)] -2x\Ta>(m),ok(n) (s;x,y)-x]

-2y\Ta>(m),ok(n) (t;x y)-y] + {x2 + y2) \Tai(m)^(n) (l;x,y) - l]}

+ f(x,y){Tai(m),o*(n) (l;x,y)-l) = e\Ta>(m)^(n) (l;x,y)-l]+e

+ ~SM {\T°>M,°kw{s2 + t2;x,y)- {x2 + y2)]

-2x\Tai(rn),ak(n) (s;x> y) - x]

-2y\Ta>(m),ak(n) (t';x' y) - y] + {x2 + y2) \TaHmlaHn) (l;x,y) - l]}

+ f (x y) {Tai(m)^(„) (l;x y)-l) • Since e is arbitrary, we can write

Ttji(m),ok(n) (f;x,y)- f(x,y)

^ e\Ta>(m),ak(n) (l; x'y) - l]

+ ~M {\Ta>(rn),ak(n) ^ + t2 x, y) - {x + /) -2x\Tai(rn),ak(n) (s;x> y) - x]

-2y\Ta>(m),ak(n) (t'; x'y) - y] + {x2 + y2) \Tai(m)^(n) (l;x,y) - l]

+ f (x y) {TaHm)^(n) (l;x,y)-l) • Similarly,

(f;x y)-f(x y)

^e\Dm^q (l;x,y)-l]

+ 2M {\D^,M {s2 + t2;x,y)-{x2 +y2)]

-2x\Dm,n,p,q (s;x,y)-x]

(t;x y)- y]

+ {x2 +y2 )\Dm^M (l;x, y) - l]} + f(x,y) {Dm^M (l;x,y)-l)•

Thus, we have

WDm,n,p,q (f;X,y)- f(x,y)l 2M(a2 +b2)

< ( e +

+ M )\\Dm^M (l;x,y)-l

|Dmm,p,q (s;X,t)-x\\c Drn,n,p,q fax, y)-y\

+ -p\\Dm,n,p,q (s2 + t2;x, y) - (x2 + y2)\\

Taking the limit p,q ^ >x> and from (4), we obtain

pljmJK^ (fi; x>y)-f (x> ^L =0

uniformly in m, n.

Theorem 4. Suppose a double sequence (Tm n) of positive linear operators on C(I2) such that

1 m-1 n-1

^ sT -T<

s,t rnn j=Q k=0

a>(s),ak(t)

= 0. (21)

v-l™\K,n (^x)-^ =0 (] = 0>1>2>3)> (22)

where t0(x,y) = 1, tl(x,y) = x, t2(x,y) = y, andt3(x,y) = x2 + y2, then

mn\\Tmn (fix,y) - f (x' y)\\m = 0 (23)

for any function f e C(I2) bounded on the whole plane. Proof. From Theorem 3, we have that if (22) holds then

^lirn\\Tmn (fix,y)-f(x,y)\L = 0 (24) which is equivalent to

supDs,t,m,n (fiX,y)- f(x,y)

= 0. (25)

m-1 n-1

^s,t,m,n fyifi ^^ (t)

j=0 k=0

1 m-1 n-1 = ~XX (Tm,n - Ta> (s),a"(t)) ■

Therefore

1 m-1 n-1

Tm,n - supDs,t,mn = sup- XX (Tm,n - ^a>(s),ak(t)) ■

Hence, using the hypothesis, we get

Km\\Tm,n (fix,y)-f(x,y)\L

supDs,t,mn (fiX' y)-f(x' y)

That is, (23) holds.

3. Example and the Concluding Remark

In this section, we prove that our theorem is stronger than Theorem 2. Let us consider the following Bernstein polynomials (see [22]) of two variables:

Bm,n (fix,y)

0 < x,y < 1.

Let Amn : C(I2) ^ C(I2) be defined by

Am,n (fi x,y) = (l+ Wmn) Bm,n (fi x, y), (30)

where the double sequence (wmn) is defined by (2) in Section 1. Then

Bm,n (i;x,y) = i,

Bm,n (S;X,y) = X, Bm,n {t;X,y) = y,

22 2 2 *-* ,

o i 2 .2 \ 2 2 X - X

Bm,n (s +t;x,y) = x +y +-+

v ' m n

Also, (Amn) satisfies (4). Hence, we have

a-JmJAmn (fix'y)-f(x,y)\L = 0. (32)

Since Bmn(f;0,0) = /(0,0), we have Amn(fi;0,0) = (1 + wmn)f(0',0). Thus

IIA m,n (fi; x,y)-f (x, y)\L — \Am,n (fi; 0,0)-fi (0, o)|

M(0,0)\.

j=0 k=0

But Theorem 3 does not hold, since the limit wmn does not exist as m, n ^ >x>. Therefore we conclude that our Theorem 3 is stronger than the classical Korovkin theorem for functions of two variables due to Volkov [23].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, therefore, acknowledges with thanks DSR technical and financial support.

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