Approximation of functions by the sequence of integral operators

Applied Mathematics and Computation 219 (2012) 3863–3871

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Approximation of functions by the sequence of integral operators _ Sakaog˘lu 1 T. Yurdakadim ⇑, E. Tasß, I. Ankara University, Faculty of Science, Department of Mathematics, Tandog˘an, 06100 Ankara, Turkey

a r t i c l e

i n f o

a b s t r a c t In this paper, we study the approximation of the Lebesgue integrable functions at fixed characteristic points with the use of statistical convergence and matrix summability of the sequence of convolution type integral operators. Ó 2012 Elsevier Inc. All rights reserved.

Keywords: A-statistical convergence A-summability method Characteristic points Integral operators

1. Introduction In the present paper, we deal with convolution type approximation theorems at fixed characteristic points in the A-statistical sense and A-summability method. Convolution type approximation theory is a well established area of research which deals with the problem of approximating a function f by a sequence of integral operators. Let Uðf ; x; kÞ be the family of convolution type singular integral operators given by

Z p

Uðf ; x; kÞ ¼

f ðtÞKðt  x; kÞdt;

x 2 ðp; pÞ:

ð1:1Þ

p

In [18] Taberski studied the pointwise convergence at the Lebesgue points of f which belongs to L1 ðp; pÞ. Here Kðt; kÞ is the kernel satisfying suitable assumptions. Moreover, Taberski [18] also examined the approximation properties of the derivatives of these operators. Gadjiev [10] and Rydzewska [16] obtained the pointwise convergence of these operators at generalized Lebesgue points and l-generalized Lebesgue points of integrable functions in L1 ðp; pÞ, respectively. In 1984, Bardaro and Cocchieri [1] estimated the degree of pointwise approximation of Fejér-type singular integrals at the generalized Lebesgue points of functions f 2 L1 ðRÞ. Note that Fejér-type singular integrals are the special case of ( 1.1). Following their ideas, using the concept of A-statistical convergence and A-summability method, we study the convergence at characteristic points of convolution type integral operators of the form

T j ðf ; xÞ ¼

Z

1

f ðtÞK j ðx  tÞdt;

1 < x < 1

1

where K j is the kernel. Statistical convergence, while introduced over nearly sixty years ago, has only recently become an area of active research [3–6,11]. In classical convergence, all of the elements except finite number of elements of the sequence have to belong to arbitrarily small neighbourhood of the limit value while the main idea of statistical convergence is to relax this condition and to demand validity of the convergence condition only for a majority of elements. Because of this, we obtain various approximation theorems in A-statistical sense and A-summability method at fixed characteristic points for convolution type integral operators although the classical limit fails. ⇑ Corresponding author. 1

_ Sakaog˘lu). E-mail addresses: [email protected] (T. Yurdakadim), [email protected] (E. Tasß), [email protected] (I. The author was supported by the Scientific and Technological Research Council of Turkey (TUBITAK).

0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.10.020

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We first recall some notation and basic definitions used in the paper. Let A ¼ ðajn Þ be a nonnegative regular matrix. The A-density of K # N is given by

X dA ðKÞ :¼ lim ajn j

n2K

provided that the limit exists. A sequence x ¼ ðxn Þ is called A-statistically convergent to a number L if for every

dA ðfn 2 N : jxn  Lj P egÞ ¼ 0:

e > 0, ð1:2Þ

It is not difficult to see that (1.2) is equivalent to

lim j!1

X

ajn ¼ 0;

for every

e > 0:

n:jxn LjPe

This limit expression is denoted by st A  lim xn ¼ L [12,14]. It is known that x is A-statistically convergent to a number L if and n only if there exists a subset K of N such that dA ðKÞ ¼ 1 and lim xn ¼ L. The case in which A ¼ C 1 , the Cesàro matrix, reduces to n2K statistical convergence [7–9]. Let f 2 L1 ða; bÞ. A point x 2 R is called d-point of the function f if

1 h!0 h

lim

Z

h

½f ðx þ tÞ  f ðxÞdt ¼ 0

0

where d stands for differentiability; the d-points of an integrable function f are precisely those points at which the indefinite integral of f is differentiable to the value f ðxÞ. If

1 h!0 h

lim

Z

h

jf ðx þ tÞ  f ðxÞjdt ¼ 0

0

then x is called a Lebesgue point of f. Let Cðf Þ; Lðf Þ; Dðf Þ be the sets of all continuity, Lebesgue and d-points of f, respectively. So we have

Cðf Þ  Lðf Þ  Dðf Þ: It is well known that if f 2 L1 ða; bÞ, then almost all points of the interval ða; bÞ are Lebesgue points, and therefore they are the d-points of f (see [2,17]). p Let Lp ða; bÞ; 1 6 p < 1 be the class of functions f such that j f j are Lebesgue integrable in the interval ða; bÞ where ða; bÞ is an arbitrary interval in ð1; 1Þ. Let f 2 Lp ða; bÞ. A point x 2 R is called p-Lebesgue point of the function f if

1 lim h!0 h

Z

!1=p

h

p

jf ðx þ tÞ  f ðxÞj dt

¼ 0:

0

Let f 2 Lp ða; bÞ; 1 6 p < 1, we denote its modulus of continuity by wLp ðf Þ which is defined by

wLp ðf ; hÞ ¼ sup kf ðx þ tÞ  f ðxÞkLp : jt j6h

  Let T j ðf ; xÞ be a sequence of convolution type integral operators which is defined by

T j ðf ; xÞ ¼

Z

1

f ðtÞK j ðx  tÞdt;

1 < x < 1:

1

Here it is easy to see T j : Lp ðRÞ ! Lp ðRÞ by the suitable assumptions that are satisfied by the kernel K j . In the present paper, ( ) P ðnÞ using summability methods, we investigate pointwise convergence of akj T j ðf ; xÞ to f ðxÞ and pointwise statistical conj

  vergence of T j ðf ; xÞ to f ðxÞ. We note that the A-summability method includes both convergence and almost convergence [13]. Since almost convergence and statistical convergence methods are incompatible [15] we conclude that these methods can be used alternatively to get some approximation results. 2. Pointwise convergence with summability method In this section using the concept of A-summability method which includes both convergence and almost convergence, we ( ) P ðnÞ akj T j ðf ; xÞ to f ðxÞ. study pointwise convergence of the sequence j

n o ðnÞ Definition 1. Let A :¼ fAðnÞ g ¼ akj be a sequence of infinite matrices with nonnegative real entries. A sequence of kernels fK j ðtÞg is called Ad -shaped if the following conditions are satisfied;

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(i) K j ðtÞ are even, nonnegative functions such that

X ðnÞ X ðnÞ akj K j ð0Þ is finite for all n; k 2 N and limsup akj K j ð0Þ ¼ 1: k

j

(ii)

P j

ðnÞ akj

R1

n

j

K j ðtÞdt ¼ 1, for all n; k 2 N

1

(iii) There exists a number d > 0 such that lim sup k

n

P

ðnÞ

akj supjtjPd K j ðtÞ ¼ 0 and lim sup k

j

n

P j

ðnÞ

akj

R1 d

K j ðtÞdt ¼ 0.

With this terminology we have following. n o   ðnÞ Theorem 1. Let A :¼ fAðnÞ g ¼ akj be a sequence of infinite matrices with nonnegative real entries. Let K j ðtÞ be monotone decreasing Ad -shaped on ½0; 1Þ and x be a p-Lebesgue point of f. Then we have, for all f 2 Lp ð1; 1Þ, that

limsup k

n

X ðnÞ akj T j ðf ; xÞ ¼ f ðxÞ: j

Proof. Let f 2 Lp ð1; 1Þ. Using (i), we have

X ðnÞ Z X ðnÞ akj T j ðf ; xÞ ¼ akj j

X

ðnÞ

akj

j

It follows from (ii) that 2

X ðnÞ Z 2f ðxÞ akj j

P j

1

f ðtÞK j ðx  tÞdt ¼

1

j

¼

X ðnÞ Z akj

1

Z

j

0

f ðx þ tÞK j ðtÞdt þ

1 ðnÞ

akj

0

f ðx þ tÞK j ðtÞdt 1

X ðnÞ Z akj

1

f ðx þ tÞK j ðtÞdt ¼

0

j

R1

1

X ðnÞ Z akj j

1

½f ðx  tÞ þ f ðx þ tÞK j ðtÞdt:

0

K j ðtÞdt ¼ 1, hence

K j ðtÞdt ¼ f ðxÞ: 0

So we may write that

X ðnÞ Z X ðnÞ akj T j ðf ; xÞ  f ðxÞ ¼ akj j

j

1

½f ðx  tÞ þ f ðx þ tÞ  2f ðxÞK j ðtÞdt

0

and we get

  Z 1  X X   ðnÞ ðnÞ akj jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞjK j ðtÞdt:  akj T j ðf ; xÞ  f ðxÞ 6   j 0 j

ð2:1Þ

Applying the Hölder inequality to the right side of (2.1), we obtain

XZ

1 0

j

p

ðnÞ

akj jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞj K j ðtÞdt

" X ðnÞ Z 6 akj " X ðnÞ Z akj

0

jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞj K j ðtÞdt

1q

ðnÞ

akj K j ðtÞdt

#1p " X j

ðnÞ akj

Z

#1q

1

K j ðtÞdt

1

#1p

1

p

jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞj K j ðtÞdt :

0

j 1 p

p

1

0

j

6

1

1p Z

1 q

where þ ¼ 1. Then we have

p   X X ðnÞ Z 1   p ðnÞ akj jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞj K j ðtÞdt:  akj T j ðf ; xÞ  f ðxÞ 6   j 0 j Now define a function F by

FðtÞ :¼

Z

t

p

jf ðx  nÞ þ f ðx þ nÞ  2f ðxÞj dn;

ð2:2Þ

0

then we have p

dFðtÞ ¼ jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞj dt: By the definition of the p-Lebesgue point, we obtain limð1h FðhÞÞ ¼ 0 i.e., given h!0 jhj < d we have

e > 0 there exists a number d > 0 such that

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FðhÞ 6 eh:

ð2:3Þ

Then we split the above inequality into two terms,

p  Z d  X X   p ðnÞ ðnÞ a T ðf ; xÞ  f ðxÞ jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞj K j ðtÞdt  6 akj  j kj   j 0 j X ðnÞ Z 1 p þ akj jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞj K j ðtÞdt d

j

¼ I1 ðn; kÞ þ I2 ðn; kÞ; say:

ð2:4Þ

Now we have

I1 ðn; kÞ ¼

X ðnÞ Z akj

d

K j ðtÞdFðtÞ ¼

0

j

Z X ðnÞ  akj K j ðdÞFðdÞ þ

d

 FðtÞdðK j ðtÞÞ :

0

j

Considering (2.3) and evaluating the integral one can get

I1 ðn; kÞ 6

X

ðnÞ

akj

 Z K j ðdÞed þ e

d 0

j

 X Z ðnÞ tdðK j ðtÞÞ ¼ akj e

d

0

j

X ðnÞ Z K j ðtÞdt 6 e akj

1

K j ðtÞdt ¼ e:

1

j

Next we consider I2 ðn; kÞ. Since K j ðtÞ is monotone decreasing, we have

I2 ðn; kÞ 6

X

ðnÞ

akj

6

ðnÞ

akj

X

p

ðnÞ

akj

Z

1

Z

p

1

p

p

p

p

22p ðjf ðx  tÞj þ jf ðx þ tÞj þ jf ðxÞj ÞK j ðtÞdt

( X ðnÞ Z akj 62 2p

6 22p

p

2p ð2p ðjf ðx  tÞj þ jf ðx þ tÞj Þ þ 2p jf ðxÞj ÞK j ðtÞdt

d

j

"

p

2p ðjf ðx  tÞ þ f ðx þ tÞj þ 2p jf ðxÞj ÞK j ðtÞdt

d

j

6

1 d

j

X

Z

d

j

1

X ðnÞ Z jf ðx þ tÞj K j ðtÞdt þ akj p

j

1

d

X ðnÞ Z jf ðx  tÞj K j ðtÞdt þ akj p

j

#

X ðnÞ X ðnÞ X ðnÞ Z p akj K j ðdÞk f kp þ akj K j ðdÞk f kp þ 22p jf ðxÞj akj j

j

j

1

) p

jf ðxÞj K j ðtÞdt

d

1

K j ðtÞdt < e:

d

Thus, the proof is completed. h

3. Pointwise approximation via statistical convergence In this section, we consider the integral operators with the A-statistical convergence and study their approximation properties at fixed characteristic points. Definition 2. Let A be a nonnegative, regular summability method. A sequence of kernels fK j ðtÞg is called A-statistical dshaped if the following conditions are satisfied; (a) K j ðtÞ are even and nonnegative functions, K j ð0Þ is finite for all j 2 N and st A  lim K j ð0Þ ¼ 1. j

(b) There exists a number d > 0 such that st A  lim ðsup K j ðtÞÞ ¼ 0

and st A  lim j

j

Z

1

jt jPd

K j ðtÞdt ¼ 0: d

(c) There exists a subset M  N with dA ðMÞ ¼ 1 such that

R1 1

K j ðtÞdt ¼ 1, for all j 2 M.

Now we have   Theorem 2. Let K j ðtÞ be a sequence of kernels which is A-statistical d-shaped on ½0; 1Þ. Then for all f 2 Lp ð1; 1Þ, we have

stA  lim T j ðf ; xÞ  f ðxÞ p ¼ 0: j

Proof. Let f 2 Lp ð1; 1Þ. It is easy to see that

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T j ðf ; xÞ ¼

Z

Z

1

f ðtÞK j ðx  tÞdt ¼

1

1

f ðx þ tÞK j ðtÞdt ¼

Z

1

ðf ðx  tÞ þ f ðx þ tÞÞK j ðtÞdt:

0

1

According to (c), there exists M 1  N with dA ðM 1 Þ ¼ 1 such that for j 2 M 1 ,

f ðxÞ ¼ 2

Z

1

f ðxÞK j ðtÞdt:

0

Then we have for, j 2 M 1 , that

Z

  T j ðf ; xÞ  f ðxÞ 6

1

jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞjK j ðtÞdt:

ð3:1Þ

0

Now using generalized Minkowski’s inequality, we have

T j ðf ; xÞ  f ðxÞ 6 p

Z

1

Z

1

jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞjK j ðtÞdt

Z

1

6 0

¼

Z

1p dx

1p p jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞj dx K j ðtÞdt

1 d

Z

Z

1

0

þ

p

0

1

Z

1

1p p jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞj dx K j ðtÞdt

1

1

Z

d

1

1p p jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞj dx K j ðtÞdt

1

¼ V 1 ðjÞ þ V 2 ðjÞ; say: First we consider V 1 ðjÞ. We have

Z

1

p

1p

jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞj dxÞ

Z

1

6

1

1p Z p jf ðx þ tÞ  f ðxÞj dx þ

1

1

1p p jf ðx  tÞ  f ðxÞj dx

1

and

sup

Z

jtjPd

1

1p p jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞj dx 6 2wLp ðf ; dÞ:

1

Hence one can get, for j 2 M 1 , that

V 1 ðjÞ 6 2wLp ðf ; dÞ

Z

d

K j ðtÞdt:

ð3:2Þ

0

Next we consider V 2 ðjÞ. One can easily get

Z

1

p

jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞj dx 6

Z

1

1

1p Z p jf ðx  tÞj dx þ

1

1

1

1p

Z p jf ðx þ tÞj dx þ 2

1

1

p

jf ðxÞj dx

1p

6 4k f k p

and therefore, for j 2 M 1 , that

V 2 ðjÞ 6 4k f kp

Z

1

K j ðtÞdt:

ð3:3Þ

d

By (3.2) and (3.3) we have, for j 2 M 1 , that

T j ðf ; xÞ  f ðxÞ 6 2wL ðf ; dÞ þ 4k f k p p p

Z

1

K j ðtÞdt:

d

By (c) in Definition 2, there exists M 2  N with dA ðM 2 Þ ¼ 1 such that

lim

j2M 2

Z

1

K j ðtÞdt ¼ 0:

d

Then we have lim T j ðf ; xÞ  f ðxÞ p ¼ 0, where M :¼ M 1 \ M 2 , i.e.,

j2M stA  lim T j ðf ; xÞ  f ðxÞ p ¼ 0: j

This completes the proof. h

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  Theorem 3. Let K j ðtÞ be a sequence of kernels which is A-statistical d-shaped on ½0; 1Þ and x0 be a point of continuity of function f. Then, we have, for all f 2 L1 ð1; 1Þ, that

stA  limT j ðf ; x0 Þ ¼ f ðx0 Þ: j

Proof. Let f 2 L1 ð1; 1Þ. According to (c), there exists M 1  N with dA ðM 1 Þ ¼ 1 such that

R1 1

K j ðtÞdt ¼ 1, for j 2 M 1 .

Thus we have for j 2 M 1 , that

f ðx0 Þ ¼

Z

1

f ðx0 ÞK j ðtÞdt

1

and

T j ðf ; x0 Þ ¼

Z

1

f ðx0 ÞK j ðx0  tÞdt ¼ 1

Z

1

f ðx0 þ tÞK j ðtÞdt:

1

We get, for j 2 M 1 , that

  T j ðf ; x0 Þ  f ðx0 Þ 6

Z

1

jf ðx0 þ tÞ  f ðx0 ÞjK j ðtÞdt:

1

Since x0 is a point of continuity of f 2 L1 ð1; 1Þ, given

e > 0 there exists a number d > 0 such that jtj < d implies

jf ðx0 þ tÞ  f ðx0 Þj < e:

ð3:4Þ

Hence we have

jT n ðf ; x0 Þ  f ðx0 Þj 6

Z

d

jf ðx0 þ tÞ  f ðx0 ÞjK j ðtÞdt þ

1

Z

d

jf ðx0 þ tÞ  f ðx0 ÞjK j ðtÞdt þ

d

Z

1

jf ðx0 þ tÞ  f ðx0 ÞjK j ðtÞdt

d

¼ U 1 ðjÞ þ U 2 ðjÞ þ U 3 ðjÞ; say: First we consider U 2 ðjÞ. By (3.4) and (c), we have, for j 2 M 1

U 2 ðjÞ < e

Z

1

K j ðtÞdt < e:

1

Now we consider U 1 ðjÞ and U 3 ðjÞ together. For j 2 M 1 , we obtain

Z 1 jf ðx0  tÞ  f ðx0 ÞjK j ðtÞdt þ jf ðx0 þ tÞ  f ðx0 ÞjK j ðtÞdt d d Z 1 Z 1 Z 1 Z 1 K j ðtÞdt þ K j ðtÞdt 6 jf ðx0  tÞjK j ðtÞdt þ jf ðx0 Þj jf ðx0 þ tÞjK j ðtÞdt þ jf ðx0 Þj d d d d Z 1 Z 1 Z 1 K j ðtÞdt: 6 supK j ðtÞ jf ðx0  tÞjdt þ supK j ðtÞ jf ðx0 þ tÞjdt þ 2jf ðx0 Þj

U 1 ðjÞ þ U 3 ðjÞ ¼

Z

1

jt jPd

d

jt jPd

d

d

By (b), there exists M 2  N with dA ðM 2 Þ ¼ 1 such that

lim supK j ðtÞ ¼ 0

j2M 2 jt jPd

and, by (c), there exists M 3  N with dA ðM 3 Þ ¼ 1 such that

lim

j2M 3

Z

1

K j ðtÞdt ¼ 0:

d

Then we have lim T j ðf ; x0 Þ ¼ f ðx0 Þ where M :¼ M1 \ M 2 \ M 3 , i.e., j2M

stA  limT j ðf ; x0 Þ ¼ f ðx0 Þ: j

This completes the proof. h Theorem 4. Let K j ðtÞ be a monotone decreasing sequence which is A-statistical d-shaped on ½0; 1Þ. Then for every p-Lebesgue points of f 2 Lp ð1; 1Þ

stA  limT j ðf ; xÞ ¼ f ðxÞ: j

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Proof. Let f 2 Lp ð1; 1Þ; 1 < p < 1. According to (c), there exists M 1  N with dA ðM 1 Þ ¼ 1 such that for j 2 M 1 ,

f ðxÞ ¼ 2

Z

1

f ðxÞK j ðtÞdt:

0

By (3.1), we have, for j 2 M 1 , that

Z

  T j ðf ; xÞ  f ðxÞ 6

1

jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞjK j ðtÞdt:

0

Hence by the Hölder inequality and (c), we get

Z

  T j ðf ; xÞ  f ðxÞ 6

1

1

1

jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞjK j ðtÞp K j ðtÞq dt

0

Z

1

p

jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞj K j ðtÞdt

6

1p Z

0

1

1

K j ðtÞdt

1q

Z 6

1

p

jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞj K j ðtÞdt

1p

0

where 1p þ 1q ¼ 1. Hence we have

  T j ðf ; xÞ  f ðxÞp 6

Z

1

p

jf ðx  tÞ þ f ðx þ tÞ  2f ðxÞj K j ðtÞdt:

0

The remaining part of the proof follows easily from the definition of p-Lebesgue point as in the proof of Theorem 1 and the property of statistical convergence as in the proof of Theorems 2 and 3. h Theorem 5. Let K j ðtÞ be a monotone decreasing sequence which is A–statistical d  shaped on ½0; 1Þ. Then for every d points of f 2 Lp ð1; 1Þ, we have

stA  limT j ðf ; xÞ ¼ f ðxÞ: j

Proof. Let f 2 Lp ð1; 1Þ; 1 < p < 1. According to (c), there exists M 1  N with dA ðM 1 Þ ¼ 1 such that for j 2 M 1 ,

f ðxÞ ¼ 2

Z

1

f ðxÞK j ðtÞdt:

0

As we stated before, for j 2 M 1

T j ðf ; xÞ  f ðxÞ ¼

Z

1

½f ðx  tÞ þ f ðx þ tÞ  2f ðxÞK j ðtÞdt:

ð3:5Þ

0

From the definition of the d-point, we obtain

1 lim h!0 h i.e., given

Z

!

h

½f ðx  tÞ þ f ðx þ tÞ  2f ðxÞdt

¼0

0

e > 0 there exists a number d > 0 such that jhj < d we have

 Z   h   ½f ðx  tÞ þ f ðx þ tÞ  2f ðxÞdt  < eh:    0 Now we can split (3.5) into two parts as follows:

T j ðf ; xÞ  f ðxÞ ¼

Z

d

½f ðx  tÞ þ f ðx þ tÞ  2f ðxÞK j ðtÞdt þ

0

d

Let F be the function given in (2.2). Using the property of the d-point, we obtain

 1 lim FðhÞ ¼ 0: h!0 h First we consider Y 1 ðjÞ. Hence

Y 1 ðjÞ ¼ K j ðdÞFðdÞ þ

Z

Z 0

d

FðtÞdðK j ðtÞÞ:

1

½f ðx  tÞ þ f ðx þ tÞ  2f ðxÞK j ðtÞdt ¼ Y 1 ðjÞ þ Y 2 ðjÞ; say:

3870

T. Yurdakadim et al. / Applied Mathematics and Computation 219 (2012) 3863–3871

Then by (c) we get, for j 2 M 1 , that

jY 1 ðjÞj 6 K j ðdÞed þ edK j ðdÞ þ e

Z

Z

d

K j ðtÞdt 6 3e

0

1

K j ðtÞdt < 3e:

1

Now we can evaluate Y 2 ðjÞ. Hence

Y 2 ðjÞ 6 K j ðdÞ

Z

1

jf ðx  tÞ þ f ðx þ tÞjdt þ 2jf ðxÞj

Z

d

1

K j ðtÞdt:

d

According to (b), there exists M 2  N and dA ðM 2 Þ ¼ 1 such that

lim supK j ðtÞ ¼ 0

j2M 2 jt jPd

and, by (c), there exists M 3  N and dA ðM 3 Þ ¼ 1 such that lim j2M 3

R1 d

K j ðtÞdt ¼ 0.

Then we have lim T j ðf ; xÞ ¼ f ðxÞ where M :¼ M 1 \ M 2 \ M 3 , i.e., j2M

stA  limT j ðf ; xÞ ¼ f ðxÞ; j

which completes the proof. h We now exhibit a sequence of positive convolution operators for which our theorems hold and the classical theorems fail. Example 1. Let A ¼ ðajn Þ be the Cesàro matrix. We can choose a sequence ðdj Þ which is statistically null but nonconvergent.  Without loss of generality we may assume that ðdj Þ is nonnegative. Otherwise we would replace ðdj Þ by ðdj Þ. Now let

K j ðtÞ ¼

jð1 þ dj Þ j2 t2 pffiffiffiffi e ;

j2N

p

and

Z

T j ðf ; xÞ ¼

1

f ðtÞK j ðx  tÞdt ¼

1

jð1 þ dj Þ pffiffiffiffi

p

(i) K j ðtÞ are nonnegative and even, K j ð0Þ ¼ (ii)

supK j ðtÞ ¼ jt jPd

Z

1

f ðtÞej

2

ðxtÞ2

dt:

1 jð1þdj Þ pffiffiffi

p

is finite, for all j 2 N and st  lim K j ð0Þ ¼ 1. j

2 2 jð1 þ dj Þ jð1 þ d Þ 1 pffiffiffiffi sup ej t 6 pffiffiffiffi j 2 2 p jtjPd p ej d

and

lim j

j 2 2

ej

d

¼ 0;

since st  limð1 þ dj Þ ¼ 1, then we get j

!

st  lim supK j ðtÞ j

¼ 0:

jt jPd

(iii)

Z

1

K j ðtÞdt ¼

d

Z

1

jð1 þ dj Þ j2 t2 ð1 þ dj Þ pffiffiffiffi e dt ¼ pffiffiffiffi

p

d

Z

p

1

2 2

ej t jdt ¼

d

ð1 þ dj Þ pffiffiffiffi

p

Z

jd

and

st  lim j

Z

1

K j ðtÞdt ¼ 0:

d

(iv) Since

Z

1

K j ðtÞdt ¼ 2

Z 0

1

1

jð1 þ dj Þ j2 t2 ð1 þ dj Þ pffiffiffiffi e dt ¼ 2 pffiffiffiffi

p

p

and therefore

st  lim j

Z

1

1

K j ðtÞdt ¼ st  limð1 þ dj Þ ¼ 1 j

Z 0

1

2

1

eu du ¼ ð1 þ dj Þ

2

eu du

T. Yurdakadim et al. / Applied Mathematics and Computation 219 (2012) 3863–3871

3871

R1 so there exists a set M  N with dðMÞ ¼ 1 and 1 K j ðtÞdt ¼ 1, for j 2 M. From (i)–(iv) we see that K j ðtÞ satisfies our hypothessis. So our theorems hold but the classical theorems do not. References [1] C. Bardaro, C. Gori Cocchieri, On the degree of approximation for a class of singular integrals, Rend. Mat. 7 (4) (1984) 481–490 (Italian). [2] P.L. Butzer, R.J. Nessel, Fourier Analysis and Approximation, Academic Press, New York, London, 1971. [3] E. Erkusß-Duman, O. Duman, Statistical approximation properties of higher order operators constructed with the Chan–Chyan–Srivastava polynomials, Appl. Math. Comput. 218 (5) (2011) 1927–1933. [4] E. Erkusß-Duman, O. Duman, Statistical approximation of certain positive linear operators constructed with the Chan–Chyan–Srivastava polynomials, Appl. Math. Comput. 182 (1) (2006) 213–222. [5] O. Duman, M.K. Khan, C. Orhan, A-statistical convergence of approximating operators, Math. Inequal. Appl. 6 (2003) 689–699. [6] O. Duman, C. Orhan, Statistical approximation by positive linear operators, Studia Math. 161 (2004) 187–197. [7] H. Fast, Sur la convergence statistuque, Colloq. Math. 2 (1951) 241–244. [8] A.R. Freedman, J.J. Sember, Densities and summability, Pacific J. Math. 95 (1981) 293–305. [9] J.A. Fridy, On statistical convergence, Analysis 5 (1985) 301–313. [10] A.D. Gadjiev, On the order of convergence of singular integrals depending on two parameters. Special questions of functional analysis and its applications to the theory of differential equations and functions theory, Baku, 1968, pp. 40–44. [11] A.D. Gadjiev, C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32 (2002) 129–138. [12] E. Kolk, Matrix summability of statistically convergent sequences, Analysis 13 (1993) 77–83. [13] G.G. Lorentz, A contrubution to the theory of divergent sequences, Acta Math. 80 (1948) 167–190. [14] H.I. Miller, A measure theoritic subsequences characterization of statistical convergence, Trans. Amer. Soc. 347 (5) (1995) 1811–1819. [15] H.I. Miller, C. Orhan, On almost convergent and statistically convergent subsequences, Acta Math. Hungar. 93 (1–2) (2001) 135–151. [16] B. Rydzewska, Approximation des functions par des integrals singuleries ordinaires, Fasc. Math. No. 7 (1973) (1974) 71–81 (French). [17] E.M. Stein, G.M. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, NJ, 1971. [18] R. Taberski, Singular integrals depending on two parameters, Prace Mat. 7 (1962) 173–179.