A Note on Extremal Problem in Approximation by Zygmund Sums

C.Ü. Fen-Edebiyat Fakültesi

Fen Bilimleri Dergisi (2008)Cilt 29 Sayı 2

A Note on Extremal Problem in Approximation by Zygmund Sums

Uğur DEĞER

Department of Mathematics, Mersin University, Turkey degpar@hotmail.com

Received: 08.02.2008, Accepted: 27.10.2008

Abstract: In this paper we consider asymptotic representation about the approximation of functions from

the classes of  -integrals by the Zygmund sums. In particularly, we obtain asymptotic formula for the value n

(

, ns

)

sup ( ) ns( ); _C C _{f C} C Z f x Z f x ψ ∞ ψ ∞ ∈ = − E , where _C ( ) x max x ϕ = ϕ .

Key Words: Zygmund sums, the class of  -integrals

Zygmund Toplamları İle Yaklaşımda Ekstremal Problem Üzerine Bir Not

Özet: Bu makalede, Zygmund toplamları ile  -integrallerinin sınıflarından olan fonksiyonlara yaklaşım

ile ilgili asimptotik gösterimleri göz önünde tutarız. Özellikle, _C ( )

x max x ϕ = ϕ olmak üzere,

(

, s

)

sup ( ) s( ); n n n _C C f C C Z f x Z f x ψ ∞ ψ ∞ ∈ = −

E değeri için asimptotik formül elde ederiz.

Anahtar Kelimeler : Zygmund toplamı,  -integralleri sınıfı

1. Introduction.

Let L denote the space of integrable 2π -periodic functions, and let

[ ]

0

(

)

( )

cos sin ; a S f a kx b kx A f x ∞ ∞ = +

∑

+ ≡

∑

be the Fourier series of a function f ∈L, for any k =0,1, 2,L

( )

1

( )

cos k k a a f f t kt dt π π π ₋ = =

_∫

, b_k b_k

( )

f 1 f t

( )

sinkt dt π π π ₋ = =

_∫

.

The polynomials that have the form

( )

0 1

( )

1 ; 1 ; , 0 2 s n s n k k a k Z f x A f x s n − =  _{ }  = + _ −_{ } _ >    

∑

are called the Zygmund sums, and Fejér sums in case of s=1. In [1, Chpt.IV], C_∞ is a class of 2π - periodic continuous functions which represented in the form of convolution

(

) ( )

(

)

( )

0 1 0 ( ) 2 2 a a f x x t t dt f x π π ϕ π ₋ = +

∫

− Ψ = + ∗ Ψ

where Ψ

( )

x is a certain function that has the Fourier series

( )

( )

(

1 2

)

1 cos sin k k kx k kx ∞ = +

∑

  ,

(

1, 2

)

=

   is a pair of arbitrary fixed systems of numbers 1

( )

k and 2

( )

k

1, 2,

k = L,and the function ϕ is called  -derivative of function f , and is denoted by

( )

,

( )

1,

( )

0 f f t f t dt π π ∞ − ⋅ ≤

∫

=    _{. In [1], if}

( )

( )

1 cos 2 v = v βπ   and

( )

( )

2 sin 2 v = v βπ

  , then the classes C_∞ coincide with the classes

C

_β_,∞.

In this work we obtain some asymptotic results for

(

, s

)

n n

C

Cψ_∞ Z

E in the classes of

C_∞ that is larger than

C

_β_,∞,and we will give the asymptotic formula which gives relation between  and  for  _n

(

, _ns

)

C

C_∞ψ Z

E under the various conditions of functions

i

 and *

i

 , 1,2i= . Similar problems have been investigated by many mathematicians. A. I. Stepanets [1], and D. N. Busev, [2], was obtained some asymptotic results about

(

, ,

)

s n Cβ ∞ Zn _C



2. Some Auxiliary Results.

We will give two lemmas before the main theorem. But firstly, we will give some notations [1,Chpt.IV]:, M denotes the set of continuous positive functions 

( )

⋅ which are convex downwards for all v≥1 and with lim

( )

0

v→∞ v = and M denotes the ′ subset of functions  which from M that satisfy in addition the following

( )

⋅ condition:

( )

1 t dt t ∞ < ∞

∫

 .

Lemma 2.1. Let. M , _∈  M . Then, as n_∈ ′ → ∞ the following asymptotic equality holds:

(

, s

)

ˆ

( )

n n n n C C Z t dt ∞ ψ ∞ −∞ = τ

∫

+ γ E (2.1) where γ_n ≤0;

( )

2 ˆ n n t O t dt π γ τ ≥     =     

∫



and τˆ_n

( )

t is determined by the following equality:

( )

(

1

( )

2

( )

)

0 1 ˆ_n t v cosvt v sinvt dv ∞ τ = τ + τ π

∫

(2.2)

Here, τ_i

( )

v which is determined by

( )

( )

( )

( )

1 , 0 1 ,1 , i s s i i s i v v n v v v v n n v v n τ  _{≤ ≤}    =_ ≤ ≤   ≥     

for i=1, 2, are continuous functions for all v≥0, and their transformations

( )

( )

1 1 0 1 ˆ t v cosvtdv, τ τ π ∞ + =

∫

2

( )

1

( )

0 1 ˆ t v sinvtdv, τ τ π ∞ − =

∫

Proof. For any f ∈C_∞

( )

_ns

(

;

)

(

) ( )

ˆ_n . f x Z f x f x t τ t dt ∞ −∞ − =

∫

 − _(2.3)

is true, [1, Chp. IV], where τˆ_n

( )

t which is determined by (2.2).

According to the classes C_∞ are invariant under the shift of an argument, that is, if f ∈C_∞, then the function f1

( )

x = f x

(

+h

)

also belongs to C∞

 _{for any fixed h∈¡ ,}

from (2.3) we have

(

, s

)

sup

( ) ( )

ˆ ˆ

( )

n n n n C _{f C} Cψ Z f t τ t dt τ t dt ∞ ∞ ∞ ∞ ∈ _{−∞ −∞} =

∫

− ≤

∫

E   _(2.4)

On the other hand, since the inequality ϕ

( )

t ≤1 . .a e is true, there exist a function

( )

( )

;

f x = f ϕ x , for which f

( ) ( )

x =ϕ x , in classes C_∞. Hence, there exist a function

*( )

f t in C_∞ such that it has the form

( )

(

( )

)

* ˆ , 2 , 2 n sign t t f t periodicly t τ π π  ≤  =  _≥   _(2.5)

on the set

{

t t: ≤π 2

} {

∪ t t: ≥π 2

}

. Then, owing to (2.5), we have

( )

(

)

( )

( )

( ) ( )

* * * 2 2 ˆ ˆ ˆ 0 _ns ; 0 _{n n n} t t f Z f t dt t dt f t t dt π π τ τ τ ∞ −∞ ≥ ≥ − =

_∫

−

_∫

+

_∫

 − ≥

( )

( )

2 ˆn 2 ˆn . t t dt t dt π τ τ ∞ −∞ ≥ ≥

∫

−

∫

(2.6) Taking into account (2.4) and (2.6), as n→ ∞, we obtain (2.1).

Let 

( )

u be convex downwards with lim

( )

0

u→∞ u = for u≥ ≥a 1 and let ′′

( )

u be continuous on the interval

[

1,∞

)

and let g u

( )

=us

( )

u , s>0, u≥ ≥b 1, be convex ,i.e.,

( )

g u is convex downwards with lim

( )

0

u→∞g u = >c or limu→∞g u

( )

= ∞

( )

g u is convex upwards with lim

( )

0

u→∞g u = >c or limu→∞g u

( )

= ∞

If

( )

( )

( )

( )

( )

{ }

* ,1 , max , g M u g M Mg M u M g u g u u M a b ′ + − ′ ≤ ≤  =  _{≥ =}  and

( )

( )

( )

( )

( )

( )

1 * ,1 1 , s s s g M g M Mg M u M u g u u u u u u M − ′ − ′  + ≤ ≤  = _{= }  _≥    

then g u is convex on interval _*

( )

[

1,∞

)

, _

( )

u is continuous on

[

1,∞

)

, ′′_

( )

u is continuous on interval

[

1,∞

)

except for point u=M , which is first kind discontinuous point.

3. Main Result.

The following theorem has been given in the classes C_{β ∞}_, by D. N. Busev, [2].

Theorem. Let functions i

( ) ( )

v , g vi and

( )

*

i v

 , i=1, 2 satisfy the conditions of Lemma 2.2. Then as n→ ∞ the following asymptotic equality holds:

(

)

(

*

)

1 , s , s n n _C n n s C C Z C Z O n ψ ψ ∞ = ∞ +      E E (2.7) where n

(

*, ns

)

C C_∞ψ Z E is determined by (2.1).

Proof. In [1], we know that if f ∈C_∞ψ, we have

(

)

0

(

)

1

(

( )

( )

)

1 1 ; 1 cos sin s n s n k k Z f x A x t k kt k kt dt n π π ϕ π − = −   _{ }   = + +  _ −_{ } _ +     _{ }  

∑



∫

  (2.8) and

( )

0

(

)

( )

( )

1 1 cos sin k f x A x t k kt k kt dt π π ϕ π ∞ = −   = + + _ + _ 

∑



∫

  (2.9) where

( )

t 1, ( )t dt 0 π π ϕ _∞ ϕ − ≤

_∫

= .

From (2.8) and (2.9), we have

( )

( )

(

)

1

(

( )

( )

)

1 1 ; cos sin s n s n k k f x Z f x x t k kt k kt n π π ϕ π − = −    − = + _{  } + +   

∑

∫

 

( )

( )

(

)

1

(

) ( )

cos sin : k n k kt k kt dt x t A t dt π π ϕ π ∞ = −  + + = + 

∑

 

∫

Hence

(

)

2

(

) ( )

1 ₀ 1 ; s sup n n C C C Z x t A t dt ∞ π ψ ∞ ϕ ≤ = ϕ + ≤ π

∫

E

(

)

(

( )

( )

)

2 1 1 ₀ 1 1 1

sup cos sin

n s s k x t k k kt k kt n π ϕ_∞ π ϕ − ≤ ₌  ≤ + + + 

∑

∫

     

( )

( )

(

cos sin

)

k n _C k kt k kt dt ∞ =  +

∑

 +  _ +    

(

)

(

(

( )

( )

)

(

( )

( )

)

)

2 1 1 ₀ 1 1 1

sup cos sin

n s s k x t k k k kt k k kt n π ϕ_∞ π ϕ − ≤ ₌  +

∫

+ _

∑

 +  +        

( )

( )

(

)

(

( )

( )

)

(

cos sin

)

: k n C k k kt k k kt dt ∞ =  +

∑

 +  _ =        

(

) ( )

(

)

(

)

2 2 * 1 ₀ 1 ₀ 1 1 : sup sup , C C x t A t dt x t B t dt π π ϕ_∞≤ π ϕ ϕ_∞≤ π ϕ =

∫

+ +

∫

+ _ (2.10) Due to (2.10), we have

(

)

2

(

) ( )

* 2

(

)

(

)

1 ₀ 1 ₀ 1 1 ; s sup sup , n n C C C C Z x t A t dt x t B t dt ∞ ∞ π π ψ ∞ ϕ ≤ ϕ ≤ ≥ ϕ + − ϕ + π

∫

π

∫

 E (2.11) Since,

(

) ( )

* 2 * 1 ₀ 1 ; s sup n n C _C Cψ Z x t A t dt ∞ π ∞ ϕ ≤  _{ = ϕ +}   _π  

∫

E (2.12) owing to (2.10)-(2.12), we conclude

(

)

( )

*

( )

, s s n n n n n C C C_∞ψ Z = C Z_∞ψ +O F E E (2.13) where

(

)

(

)

2 1 ₀ 1 sup , n C F x t B t dt π ϕ_∞≤ π ϕ =

_∫

+ _ _.

Let’s estimate O F : In view of Lemma 2.2, for

( )

n k≥n, we have

( )

*

( )

, 1, 2 i k = i k i=   . Thus we obtain

(

)

(

(

( )

( )

)

(

( )

( )

)

)

2 1 1 _{0 1} 1 1

sup cos sin

n s n s k F x t k k k kt k k kt n π ϕ_∞ π ϕ − ≤ ₌  = + + + 

∑

∫

         

( )

( )

(

)

(

( )

( )

)

(

cos sin

)

k n C k k kt k k kt dt ∞ =  +

∑

 +  _ ≤        

(

( )

( )

( )

( )

)

§ ¨ 2 1 1 0 1 1 M s 1 s s k k k k k k dt O n n π π + =   _{ } ≤ _ + _ = _{ }   

∑



∫

          (2.14)

Consequently, we get (2.7) from (2.13) and (2.14). According to Lemma 2.1 we know that

(

*

)

*

( )

* ˆ , s n n n n C C Z t dt ∞ ψ ∞ −∞ = τ

∫

+ γ E .

Therefore the proof is completed.

References

[1] A. I. Stepanets, “Methods of Approximation Theory” Boston., VSP., 2005, p.880. [2] D. N. Bushev, “Approximation of Classes of Continuous Periodic Functions

Zygmund Sums [in Russian]”, Institute of Mathematics, Ukrainian Academy of