3_xxx_serenbay.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 5, No. 1, 2012, 25-29

ISSN 1307-5543 – www.ejpam.com

SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA

29 JUNE - 02 JULY 2011, ISTANBUL TURKEY

Rate of Convergence in Sobolev Space

Sevilay Kırcı Serenbay 1,∗, Hande Tanberkan2

1 Department of Mathematics, Faculty of Education, Başkent University, Ankara, Turkey
2 Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey

Abstract. In this paper, a new theorem on degree of approximation in L2
p
(Ω) Sobolev space of in-

tegrable functions of two variables by Bernstein-Chlodowsky polnomials on an unbounded triangular

domain is studied. Also by using the K- functional of Peetre the order of approximation are established.

2000 Mathematics Subject Classifications: 41A25, 41A35
Key Words and Phrases: Bernstein -Chlodowsky polynomials, convergence, absolutely continuous

function, L2
p
(Ω) Hardy-Littlewood majorante, K - functional of Peetre

1. Introduction

The aim of this paper is to study the problem on degree of the approximation of function

of two variables of f ∈ L2
p(Ω) by means of Bernstein - Chlodowsky polynomials in a triangular

domain extending infinity, where Ω = limn→∞∆bn
,∆bn

= {(x , y) : x ¶ 0, y ¾ 0, x + y ¶ bn}
and (bn) is a sequence of increasing positive number, such that:

lim
n→∞

bn =∞, lim
n→∞

bn

n
= 0. (1)

Some properties of approximation of functions of two variable by Bernstein -Chlodowsky poly-

nomials was proven in [1]-[5] and [7]. In addition, convergence of Bernstein-Chlodowsky

polynomials of two variables were investigated on a triangular domain in [6] and [7]. In

this paper we will use Bernstein-Chlodowsky polynomials on Ω which is introduced in [7].

Let, f ∈ L2
p(Ω),

Bn( f ; x , y) =

n∑

k=0

C k
n (1−

x + y

bn

)n−k
k∑

i=0

f (
k− i

n
bn,

i

n
bn)C

i
k
(

x

bn

)k−i(
y

bn

)i

∗Corresponding author.

Email addresses: kir
i�baskent.edu.tr ( S. Serenbay), handetanberkan�hotmail.
om (H. Tanberkan)

http://www.ejpam.com 25 c© 2012 EJPAM All rights reserved.



S. Serenbay, H. Tanberkan / Eur. J. Pure Appl. Math, 5 (2012), 25-29 26

for (x , y) ∈ ∆bn
. We note that formula (1) is the sequence of linear positive operators in

the space of integrable functions Lp of two variables, that is these linear positive operators

translate a positive function to an another positive one. But, in general, the function is not

necessarily a continuous one in Lp space. We can not use Korovkin’s Theorem. First, We give

certain results which are necessary to prove the main results.

Lemma 1. Suppose that ek,m(t, ) = tkm then

Bn(e0,0; x , y) = 1

Bn(e1,0; x , y) = x

Bn(e0,1; x , y) = y

Bn(e2,0; x , y) = x2+
x(bn− x)

n

Bn(e0,2; x , y) = y2 +
y(bn − y)

n

Simple calculations can be calculated above Lemma.

Theorem 1 ([7]). Let f ∈ Lp(Ω) and a be a fixed point in (0, bn). If, for every (x , y) ∈∆a and

(t, s) ∈∆bn

| f (t, s)− f (x , y)|

|(t, s)− (x , y)|
¶ M (2)

hold with the constant M,then

‖Bn( f )− f ‖Lp(∆a)
→ 0, n→∞.

Where a > 0.

2. Main Theorems

To simplify notation, we need the following.

L2
p(Ω1) = { f ∈ Lp(Ω1) :∆|α| f ∈ Lp(∆a), |α| = 2},Ω1 ⊂ [0, bn)× [0, bn).

We consider also the following K-functional of Peetre;

Kp( f ;δ) = inf
g∈L2

p(∆a)
[‖ f − g‖Lp(∆a)

+ δ(‖g‖L2
p(∆a)

)],δ ¾ 0.

for f ∈ Lp(∆a), we have limδ→0 K( f ;δ) = 0. Therefore the K-functional gives the degree of

approximation of a function f ∈ Lp(∆a) by smoother functions g ∈ L2
p(∆a). Remember that

the second order integral modulus of smoothness is given by

ω2,p( f ;δ) = sup
0¶h¶δ

‖ f (x + h)− 2 f (x)+ f (x − h)‖Lp(∆a)
(Ih)



S. Serenbay, H. Tanberkan / Eur. J. Pure Appl. Math, 5 (2012), 25-29 27

for an f ∈ Lp(∆a), where Ih indicates that the Lp-norm is taken over the interval [h, bn − h].

It is also know that there are constants a1 > 0, a2 > 0, independent of f and p such that

a1ω2,p( f ;δ1/2)¶ Kp( f ;δ)¶min(1,δ)‖ f ‖Lp(∆a)
+2a2ω2,p( f ;δ1/2) (3)

We prove the following theorems:

Theorem 2. Let f ∈ L2
p(Ω1), 1¶ p <∞ and a, M are constants, If the condition,

| f (t, s)− f (x , y)|

|(t, s)− (x , y)|
¶ M , t ∈ (a, bn], s ∈ (a, bn], (x , y) ∈∆a

is satisfied, then

‖Bn( f )− f ‖L2
p(∆a)
¶ Cp(‖ f ‖L2

p(∆a)
)δn,δn =

a(bn + a)

n

Cp =





p > 1,2(
p

p+ 1
)p

p = 1, a2

Proof. For f ∈ L2
p(Ω1) we can write that,

Bn( f (t, s)− f (x , y); x , y) = fx (x , y)Bn((t − x); x , y)+ f y (x , y)Bn((s− y); x , y)

+ Bn(

∫ t

x

fuu(u, y)(u− t)du; x , y)

+ Bn(

∫ s

y

fkk(x , k)(k− s)dk; x , y)

+ Bn(

∫ t

x

∫ s

y

fts(t, s)dsd t; x , y)

Now, we need the Hardy-Littlewood majorante of fx x at x , Which is defined as following:

ϕ fx x(x ,y) = sup
0¶t¶x ,t 6=x

(
1

t − x
)

∫ t

x

fuu(u, y)du

and using following inequality

∫

Ω

|ϕ fx y(x ,y)|
pd xd y ¶ 2(

p

p+ 1
)p
∫ a

0

∫ a

0

| fts(t, s)|
pdsd t

using Lp-norm, we get

|B1(x , y)|+ |B2(x , y)|+ |B3(x , y)| ¶ ϕ fx x(x ,y)δn +ϕ f y y(x ,y)δn+ϕ f x y(x ,y)

p
δn



S. Serenbay, H. Tanberkan / Eur. J. Pure Appl. Math, 5 (2012), 25-29 28

|B1|Lp(∆a)
+ |B2|Lp(∆a)

+ |B3|Lp(a)
¶ Cp(‖ fx x‖Lp(∆a)

+ ‖ f y y‖Lp(∆a)
+ ‖ fx y‖Lp(∆a)

)δn

< Cp(‖ f ‖L2
p(∆a)

)δn

‖Bn f − f ‖Lp(∆a)
¶ Cp(‖ f ‖L2

p(∆a)
)δn

where

Cp = 21/p(
p

p− 1
), (1< p <∞)

If p = 1,

∫

(∆a)

|B1(x , y)|d xd y ¶

∫ a

0

∫ a

0

|Bn(

∫ t

x

fuu(u, y)(u− t)du; x , y)|d xd y

¶

∫ a

0

∫ a

0

Bn(|t − x |

∫ t

x

fuu(u, y)du; x , y)d xd y

= ‖ fx x‖L1(∆a)
a2δn.

∫

(∆a)

|B2(x , y)|d xd y ¶

∫ a

0

∫ a

0

Bn(|s− y|

∫ t

x

fkk(x , k)dk; x , y)d xd y

= ‖ f y y‖L1(∆a)
a2δn.

∫

(∆a)

|B3(x , y)|d xd y ¶

∫ a

0

∫ a

0

|Bn(

∫ t

x

∫ s

y

fts(t, s)dsd t; x , y)|d xd y

= ‖ fx y‖L1(∆a)
a2δn.

then

‖B1‖Lp(∆a)
+ ‖B2‖Lp(∆a)

+ ‖B3‖Lp(a)
¶ a2(‖ fx x‖Lp(∆a)

+ ‖ f y y‖Lp(∆a)
+ ‖ fx y‖Lp(∆a)

)δn

‖Bn f − f ‖Lp(∆a)
¶ a2‖ f ‖L2

p(∆a)
δn.

Thus, the proof is completed.

Theorem 3. Let f ∈ L2
p(Ω

1) , 1 ≤ p < ∞ and f satisfies the condition (2) then the following

inequality

‖Bn f − f ‖Lp(∆a)
¶ Mp[‖ f ‖L2

p(∆a)
δn+ω2,p( f ;δ(1/2))] (4)

holds. Where a, M are constants.

Proof. For all sufficiently large n, from Theorem 2 we can write

‖Bnh− h‖Lp∆a
¶

¨
(ǫ+Mδna)‖h‖Lp(∆a)

,h ∈ Lp(∆a)

Cp‖ f ‖LP (∆a)
δn ,h ∈ L2

P(∆a)

where Cp is positive constant which independent of h,n and where h satisfies (2). When

f L2
p(Ω1) and g ∈ L2

p(∆a) the condition (2) is satisfied then

‖Bn f − f ‖Lp(∆a)
¶ ‖Bn( f − g)− ( f − g)‖Lp(∆a)

+ ‖Bn g − g‖Lp(∆a)



REFERENCES 29

¶ (ǫ+Mδna)‖ f − g‖Lp(∆a
+ Cp‖g‖L2

p(∆a
δn

¶ eM[‖ f − g‖Lp(∆a
+ ‖g‖L2

p(∆a
δn]

where eM =max{ǫ +Mδna, Cp}.
Using the K-functional we get,

‖Bn f − f ‖Lp(∆a)
¶ eM sup

g∈L2
p(∆a)

[‖ f − g‖Lp(∆a
+ ‖g‖L2

p(∆a)
δn]

since, for a sufficiently large n,δn and from (3),

Kp( f ;δ) ¶ δn‖ f ‖Lp(∆a )
+ 2a1ω2,p( f ;δ(1/2))

eMKp( f ;δ) ¶ eM[δn‖ f ‖Lp(∆a )
+ 2a1ω2,p( f ;δ(1/2))]

we obtain (4),

‖Bn f − f ‖Lp(∆a)
¶ Mp[‖ f ‖L2

p(∆a)
δn +ω2,p( f ;δ(1/2))].

Thus, the proof is completed.

References

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Rocky Mountain Journal of Mathematics, issue 2-3 , 1998.

[2] A Izgi, Order of Approximation of Functions of Two Variable by New Type Gamma Op-

erators. General Mathematics. Vol.17, No: 1 23-32 (2009).

[3] E Gadjieva and E Ibikli, On Generalization of Bernstein –Chlodowsky polynomials,

Hacettepe Bulletin of Natural Sciences and Engineering Volume 24/p.p.31-40 1995.

[4] E Ibikli and E Gadjieva, The Order of Approximation of Some Un bounded Functions by

the Sequence of Positive Linear Operators, Turkish J.of Math. V19 No.3 1995.

[5] E Ibikli, On approximation of Lp locally integrable functions by the sequences of linear

positive operators. Dokl. Nats. Akad. Nauk Azerb. 59. 2003.

[6] E Ibikli, On approximation for functions of two variables on a triangular domain. Rocky

Mountain J. Math. 35. 1523-1531, 2005.

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