SQU Journal for Science, 2017, 22(2), 89-95  DOI: http://dx.doi.org/10.24200/squjs.vol22iss2pp89-95 

Sultan Qaboos University  

89 

Approximation Properties of de la Vallée-
Poussin sums in Morrey spaces 

Ahmed Kinj*, Mohammad Ali and Suleiman Mahmoud
 

Department of Mathematics, Faculty of Science, Tishreen University, Lattakia, Syria. *Email: 
A.Kinj@tishreen.edu.sy. 

ABSTRACT: In this paper, we investigate the problem of the deviation of a function 𝑓 from its de la Vallée-Poussin 
sums of Fourier series in Morrey spaces defined on the unite circle in terms of the best approximation to 𝑓. Moreover, 
approximation properties of de la Vallée-Poussin sums of Faber series in Morrey-Smirnov classes of analytic functions, 

defined on a simply connected domain bounded by a curve satisfying Dini's smoothness condition are obtained. 

 

Keywords: de la Vallée-Poussin; Faber polynomials; modulus of smoothness; Morrey Smirnov classes. 

 دي ال فالي بواسون في فضاءات موريتقريب الدوال بواسطة مجاميع خواص 

 محمد علي وسليمان محمود، أحمد كنج 

في فضاءات موري المعرفة على دائرة  𝑓عن مجاميع دي ال فالي بواسون لمتسلسلة فورييه للدالة  𝑓بحثنا في هذه الورقة مسألة انحراف دالة  :صلخمال
حصلنا على خواص تقريب الدوال من صفوف موري سميرنوف المعرفة على منطقة بسيطة . وعالوة على ذلك، 𝑓الواحدة في ضوء أفضل تقريب للدالة 

 الترابط )االتصال( ومحاطة بمنحٍن يحقق شرط ديني للملوسة بمجاميع دي ال فالي بواسون لمتسلسلة فابير.

 

 .دي فالي بواسون، كثيرات حدود فابير، معامل الملوسة، صفوف سميرنوف موري: مفتاحيةالكلمات ال

1.  Introduction 

ain approximation problems in Lebesgue spaces have been studied by several authors [1, 2]. The approximation 

of functions of Lebesgue spaces by partial sum of Faber-Laurent series was obtained by Israfilov [3]. These 

results are generalized to Muckenhoupt weighted Lebesgue's spaces [4]. Approximation properties of Faber series in 

weighted and non-weighted Orlicz spaces were dealt with by Jafarov and Israfilov [5-7]. 

The concept of Morrey space, introduced by C. Morrey [8] in 1938, has been studied intensively by various authors 

and plays an important role in many areas such as applied mathematics, the theory of differential equations, potential 

theory, and maximal and singular operator theory. Currently there are several investigations relating to the fundamental 

problems in this space [9-14]. Therefore, the investigation into the approximation of functions by means of Fourier 

trigonometric series in Morrey spaces is also important in these areas of research. 

In the present paper, we investigate the problems of estimating the deviation of functions from their de la Vallée -

Poussin sums in Morrey spaces. Similar results in weighted Smirnov spaces and weighted Smirnov Orlicz spaces can 

be found in the papers [15-17]. 

2. Notation and Basic definitions 

Let 𝐺 be a finite simply connected domain in the complex plane ℂ bounded by a rectifiable Jordan curve Γ and 
𝐺 − ∶= 𝑒𝑥𝑡 𝛤. Without loss of generality, we suppose that 0 ∈ 𝐺. Further, let 𝛾0 ∶=  {𝑤 ∈ ℂ ∶  |𝑤| = 1}, 𝐷: =  𝑖𝑛𝑡 𝛾0, 
𝐷− ∶=  𝑒𝑥𝑡 𝛾0. We denote by 𝑤 = 𝜑(𝑧) the conformal mapping of 𝐺

− onto domain 𝐷− normalized by the conditions 
 

𝜑(∞) = ∞, 𝑙𝑖𝑚
𝑧→∞

𝜑(𝑧)

𝑧
> 0,   

and let 𝜓 be the inverse mapping of 𝜑. 

We begin with the following definitions:  

M 

mailto:A.Kinj@tishreen.edu.sy


AHMED KINJ ET AL 

90 

 

Definition 2.1 [18] For 0 ≤ 𝛼 ≤ 2 and 1 ≤ 𝑝 < ∞, we denote by 𝐿𝑝,𝛼 (Γ) the Morrey space, as the set of locally 
integrable function 𝑓, with a finite norm: 

‖𝑓‖𝐿𝑝,𝛼(Γ) ≔ {sup
𝐵

1

|𝐵 ∩ Γ|
Γ

1−
𝛼
2

∫ |𝑓(𝑧)|𝑝|𝑑𝑧|
𝐵∩Γ

}

1
𝑝

< ∞, 

where 𝐵 is an arbitrary disk centered on Γ and |𝐵 ∩ Γ|Γ is the linear Lebesgue measure of the set 𝐵 ∩ Γ. 
In the case of Γ = 𝛾0 ≔ {𝑤 ∈ ℂ:  |𝑤| = 1} we obtain the space 𝐿

𝑝,𝛼 (γ0). 
Under this definition 𝐿𝑝,𝛼 (Γ) is a Banach space. If 𝛼 = 2 then the class 𝐿𝑝,2(Γ) coincides with the class 𝐿𝑝 (Γ), and for 
𝛼 = 0 the class 𝐿𝑝,0(Γ) coincides with the class 𝐿∞(Γ). Moreover, 𝐿𝑝,𝛼1 (Γ) ⊂ 𝐿𝑝,𝛼2 (Γ) for 0 ≤ 𝛼1 ≤ 𝛼2. Thus, 
𝐿𝑝,𝛼 (Γ) ⊂ 𝐿1(Γ), ∀𝛼 ∈ [0,2]. 
For given 𝑓 ∈ 𝐿1(𝛾0), let 

 
𝑎0

2
+ ∑ 𝑎𝑘 (𝑓) cos 𝑘𝑥 + 𝑏𝑘 (𝑓) sin 𝑘𝑥

∞
𝑘=0  (1) 

be the Fourier series of 𝑓, where 𝑎𝑘 (𝑓) and 𝑏𝑘 (𝑓) are Fourier coefficients of the function 𝑓. Further, let 

𝑆𝑛(𝑥, 𝑓) =
𝑎0
2

+ ∑ 𝑎𝑘 (𝑓) cos 𝑘𝑥 + 𝑏𝑘 (𝑓) sin 𝑘𝑥

𝑛

𝑘=0

 

be the 𝑛 𝑡ℎ partial sums of series (1). 
We define the 𝑛 − 𝑡ℎ de la Vallée-Poussin sums of series (1) as 

𝑉𝑛,𝑚(𝑥, 𝑓) =
1

𝑚 + 1
∑ 𝑆𝑘 (𝑥, 𝑓)

𝑛

𝑘=𝑛−𝑚

,   0 ≤ 𝑚 ≤ 𝑛, 𝑚, 𝑛 = 1,2,3, …  . 

Definition 2.2 [19] We define the 𝑟 modulus of smoothness of a function 𝑓 ∈ 𝐿𝑝,𝛼 (𝛾0) for 𝑟 = 1,2,3, . ..by the relation 

𝜔𝑝,𝛼
𝑟 (𝑓, 𝑡) ≔ sup

|ℎ|≤𝑡
‖∆ℎ

𝑟 (𝑓, . )‖𝐿𝑝,𝛼(γ0), 𝑡 > 0, 

where 

∆ℎ
𝑟 (𝑓, 𝑥) = ∑ (

𝑟
𝑘

) (−1)𝑟−𝑘 𝑓(𝑥 + 𝑘ℎ)𝑟𝑘=0 . 

The best approximation to 𝐿𝑝,𝛼 (γ0) in the class 𝒯𝑛 of trigonometric polynomials of degree not greater than 𝑛 is defined 
by 

𝐸𝑛(𝑓)𝐿𝑝,𝛼(γ0) ≔ inf{‖𝑓 − 𝑇𝑛‖𝐿𝑝,𝛼(γ0): 𝑇𝑛 ∈ 𝒯𝑛 }. 

Let 𝑇∗ ∈ 𝒯𝑛 be a trigonometric polynomial such that 

𝐸𝑛 (𝑓)𝐿𝑝,𝛼(γ0) ≔ ‖𝑓 − 𝑇
∗‖𝐿𝑝,𝛼(γ0). 

If 𝑚, 𝑛 ∈ ℕ such that 𝑚 ≥ 𝑛, then we get 

 𝐸𝑚(𝑓)𝐿𝑝,𝛼(γ0)  ≤ 𝐸𝑛(𝑓)𝐿𝑝,𝛼(γ0). (2) 

Using the boundedness of operator 𝑓 → 𝑆𝑛 (. , 𝑓) in the Morrey spaces 𝐿
𝑝,𝛼 (𝛾0) we get 



APPROXIMATION PROPERTIES OF DE LA VALLÉE-POUSSIN SUMS IN MORREY SPACES 

 

91 
 

‖𝑓 − 𝑆𝑛(. , 𝑓)‖𝐿𝑝,𝛼(𝛾0) ≤ ‖𝑓 − 𝑇
∗‖𝐿𝑝,𝛼(𝛾0) + ‖𝑇

∗ − 𝑆𝑛(. , 𝑓)‖𝐿𝑝,𝛼(𝛾0)

= 𝐸𝑛(𝑓)𝐿𝑝,𝛼(γ0) + ‖𝑆𝑛(. , 𝑓 − 𝑇
∗)‖𝐿𝑝,𝛼(𝛾0) ≤ 𝐸𝑛 (𝑓)𝐿𝑝,𝛼(γ0) + 𝐶‖𝑓 − 𝑇

∗‖𝐿𝑝,𝛼(𝛾0)

= (𝐶 + 1)𝐸𝑛(𝑓)𝐿𝑝,𝛼(γ0) = 𝑐𝐸𝑛 (𝑓)𝐿𝑝,𝛼(γ0), 

where C is a positive constant and  𝑐 = 𝐶 + 1, i.e. there exists a constant 𝑐 such the following relation holds 

 ‖𝑓 − 𝑆𝑛(. , 𝑓)‖𝐿𝑝,𝛼(𝛾0) ≤ 𝑐𝐸𝑛(𝑓)𝐿𝑝,𝛼(γ0). (3) 

Definition 2.3 [9] We define the Morrey-Smirnov classes 𝐸𝑝,𝛼 (𝐺), 0 ≤ 𝛼 ≤ 2 and 1 ≤ 𝑝 < ∞, of analytic functions in 
𝐺 as 

𝐸𝑝,𝛼 (𝐺) ≔ {𝑓 ∈ 𝐸1(𝐺) ∶ 𝑓 ∈ 𝐿𝑝,𝛼 (Γ)}. 

If we define ‖𝑓‖𝐸𝑝,𝛼(G) ≔ ‖𝑓‖𝐿𝑝,𝛼(Γ), then 𝐸
𝑝,𝛼 (𝐺) becomes a Banach space. 

 

Definition 2.4 [20] A smooth curve Γ: σ(s) is called Dini-smooth if it satisfies the condition 

∫
Ω(𝜎′(𝑠), 𝑠)

𝑠
𝑑𝑠

𝛿

0

< ∞, 𝛿 > 0, 

where Ω(𝜎 ′(𝑠), 𝑠) modulus of continuity of function 𝜎 ′(𝑠). By 𝒟 we denote the set of all Dini-smooth curves. 

If Γ ∈ 𝒟, then [21]  

 0 < 𝑐1 ≤ |𝜓
′ (𝑤)| ≤ 𝑐2 < ∞,   0 < 𝑐3 ≤ |𝜑′(𝑧)| ≤ 𝑐4 < ∞ (4) 

for some constants, 𝑐1, 𝑐2, 𝑐3 and , 𝑐4. 

Hence, if Γ ∈ 𝒟 and using (4), then by [9]  

 𝑓 ∈ 𝐿𝑝,𝛼 (Γ) ⟺ 𝑓0 ≔ 𝑓 ∘ 𝜓 ∈ 𝐿
𝑝,𝛼 (𝛾0) (5) 

and the function 𝑓0
+: 𝐷 → ℂ defined by 

 𝑓0
+(𝑤) =

1

2𝜋𝑖
∫

𝑓0(𝜏)

𝜏−𝑤
𝑑𝜏

𝛾0
 , 𝑤 ∈ 𝐷 (6) 

is analytic in 𝐷 and 𝑓0
+ ∈ 𝐸𝑝,𝛼 (𝐷) [9] . 

If Γ ∈ 𝒟 and 𝑟 = 1,2,3, …, we define the 𝑟 − modulus of smoothness of 𝑓 ∈ 𝐿𝑝,𝛼 (Γ) by the relation (see, [9]) 

 ΩΓ,𝑝,𝛼
𝑟 (𝑓, 𝑡) ≔ ω𝑝,𝛼

𝑟 (𝑓0
+, t),   𝑡 > 0 .               (7) 

The Faber polynomials Φ𝑘 (𝑡) of degree 𝑘 are defined by the relation [22] 

 
𝜓′(𝑤)

𝜓(𝑤)−𝑡
= ∑

Φ𝑘(𝑡)

𝑤𝑘+1
∞
𝑘=0 , 𝑡 ∈ G, 𝑤 ∈ 𝐷

−. (8) 



AHMED KINJ ET AL 

92 

 

If 𝑓 ∈ 𝐸𝑝,𝛼 (𝐺), then by the definition 2.3,  𝑓 ∈ 𝐸1(𝐺) and hence 

𝑓(𝑧) =
1

2𝜋𝑖
∫

𝑓(𝑠)

𝑠 − 𝑧
𝑑𝑠

Γ

=
1

2𝜋𝑖
∫

𝜓′(𝑤)

𝜓(𝑤) − 𝑧
𝑓0(𝑤)𝑑𝑤

𝛾0

 , z ∈ G. 

From the last formula and the relation (8), for every 𝑧 ∈ 𝐺 we have 
 

 𝑓(𝑧)~ ∑ 𝑎𝑘 (𝑓)Φ𝑘(𝑧)
∞
𝑘=0 ,   𝑧 ∈ 𝐺, (9) 

where 

𝑎𝑘 (𝑓) ≔
1

2𝜋𝑖
∫

𝑓0(𝑤)

𝑤 𝑘+1
𝑑𝑤

γ0

, 𝑘 = 0,1,2, …   . 

 

The 𝑛 − 𝑡ℎ de la Vallée-Poussin sums of the series (9) are defined as 

𝑉𝑛,𝑚(𝑥, 𝑓) =
1

𝑚 + 1
∑ 𝑆𝑘 (𝑥, 𝑓)

𝑛

𝑘=𝑛−𝑚

,   0 ≤ 𝑚 ≤ 𝑛, 𝑚, 𝑛 = 1,2,3, …  , 

where 

𝑆𝑛(𝑧, 𝑓) = ∑ 𝑎𝑘 (𝑓)Φ𝑘 (𝑧)

𝑛

𝑘=0

. 

We define the operator 𝑇 as follows: 

𝑇: 𝐸𝑝,𝛼 (𝐷) → 𝐸𝑝,𝛼 (𝐺) 
 

 𝑇(𝑓)(𝑧) ≔
1

2𝜋𝑖
∫

𝑓(𝑤)𝜓′(𝑤)

𝜓(𝑤)−𝑧
𝑑𝑤

𝛾0
, 𝑧 ∈ 𝐺. (10) 

 

In order to prove our main results, we need the following theorems. 

Theorem 2.1 [10] If Γ ∈ 𝒟, then the operator 𝑇 defined by (10) is linear, bounded, one to one and onto. Moreover 
𝑇(𝑓0

+) = 𝑓 for 𝑓 ∈ 𝐸𝑝,𝛼 (𝐺). 
Theorem 2.2 [9] Let 𝑔 ∈ 𝐸𝑝,𝛼 (D) with 0 < 𝛼 ≤ 2 and 1 < 𝑝 < ∞. Then for a given 
 𝑟 = 1,2,3, … the inequality 

𝐸𝑛(𝑔)𝐿𝑝,𝛼(γ0) ≤ 𝑐5 𝜔𝑝,𝛼
𝑟 (𝑔,

1

𝑛 + 1
)  , 𝑛 = 1,2,3, ..  

holds with a constant 𝑐5 > 0 independent of 𝑛. 

3. Main Results 

In this section, we present the main results.  

Theorem 3.1 Let 𝐿𝑝,𝛼 (𝛾0) be a Morrey space with 0 < 𝛼 ≤ 2 and 1 < 𝑝 < ∞, then there exists a positive constant 𝑐6 
such that for any 𝑓 ∈ 𝐿𝑝,𝛼 (𝛾0), 0 ≤ 𝑚 ≤ 𝑛, 𝑚, 𝑛 = 1,2, …  the inequality  

 

 ‖𝑓 − 𝑉𝑛,𝑚(. , 𝑓)‖𝐿𝑝,𝛼(𝛾0)
≤

𝑐6

𝑚+1
∑ 𝐸𝑘 (𝑓)𝐿𝑝,𝛼(𝛾0)

𝑛
𝑘=𝑛−𝑚                          (11) 

is true. 

 

Proof. Let us chose the integer 𝑗 such that  2𝑗 ≤ 𝑚 + 1 ≤ 2𝑗+1. Then 

𝑓(𝑥) − 𝑉𝑛,𝑚(𝑥, 𝑓) =
1

𝑚 + 1
[𝑓(𝑥) − 𝑆𝑛−𝑚(𝑥, 𝑓)] 

+
1

𝑚 + 1
{∑ ∑ [𝑓(𝑥) − 𝑆𝑖 (𝑥, 𝑓)]

𝑛−𝑚+2𝑘−1

𝑖=𝑛−𝑚+2𝑘−1

𝑗

𝑘=1

} +
1

𝑚 + 1
{ ∑ [𝑓(𝑥) − 𝑆𝑘 (𝑥, 𝑓)]

𝑛

𝑘=𝑛−𝑚+2𝑗

}. 

 

And from this, we get  



APPROXIMATION PROPERTIES OF DE LA VALLÉE-POUSSIN SUMS IN MORREY SPACES 

 

93 
 

‖𝑓 − 𝑉𝑛,𝑚(. , 𝑓)‖𝐿𝑝,𝛼(𝛾0)

≤  
1

𝑚 + 1
‖𝑓 − 𝑆𝑛−𝑚(. , 𝑓)‖𝐿𝑝,𝛼(𝛾0) +

1

𝑚 + 1
{∑ ∑ ‖𝑓 − 𝑆𝑖 (. , 𝑓)‖𝐿𝑝,𝛼(𝛾0)

𝑛−𝑚+2𝑘−1

𝑖=𝑛−𝑚+2𝑘−1

𝑗

𝑘=1

}

+
1

𝑚 + 1
{ ∑ ‖𝑓 − 𝑆𝑘 (. , 𝑓)‖𝐿𝑝,𝛼(𝛾0)

𝑛

𝑘=𝑛−𝑚+2𝑗

}. 

 

From the relation (3), we get 

 ‖𝑓 − 𝑉𝑛,𝑚(. , 𝑓)‖𝐿𝑝,𝛼(𝛾0)
≤  

c7

𝑚+1
𝐸𝑛−𝑚(𝑓)𝐿𝑝,𝛼(𝛾0) +

c8

𝑚+1
{∑ ∑ 𝐸𝑖 (𝑓)𝐿𝑝,𝛼(𝛾0)

𝑛−𝑚+2𝑘−1
𝑖=𝑛−𝑚+2𝑘−1

𝑗
𝑘=1 } +

c9

𝑚+1
{∑ 𝐸𝑘 (𝑓)𝐿𝑝,𝛼(𝛾0)

𝑛
𝑘=𝑛−𝑚+2𝑗

}.                                 (12) 

 

From                                 (12) and using (2), we get 

 

 ‖𝑓 − 𝑉𝑛,𝑚(. , 𝑓)‖𝐿𝑝,𝛼(𝛾0)
≤

𝑐10

𝑚+1
{𝐸𝑛−𝑚(𝑓)𝑋,𝜔 + ∑ 2

𝑘−1𝐸𝑛−𝑚+2𝑘−1 (𝑓)𝐿𝑝,𝛼(𝛾0)
𝑗
𝑘=1 } + 𝑐11

1

𝑚+1
(𝑚 −

2𝑗 + 1)𝐸𝑛−𝑚+2𝑗 (𝑓)𝐿𝑝,𝛼(𝛾0).                                       (13) 
 

On the other hand, we have 

 ∑ 2𝑘−1𝐸𝑛−𝑚+2𝑘−1 (𝑓)𝐿𝑝,𝛼(𝛾0)
𝑗
𝑘=1 ≤ 𝐸𝑛−𝑚+1(𝑓)𝐿𝑝,𝛼(𝛾0) + 2 ∑ ∑ 𝐸𝑖 (𝑓)𝐿𝑝,𝛼(𝛾0)

𝑛−𝑚+2𝑘−1−1
𝑖=𝑛−𝑚+2𝑘−2

𝑗
𝑘=2 ≤

𝑐12 ∑ 𝐸𝑘 (𝑓)𝐿𝑝,𝛼(𝛾0)
𝑛−𝑚+2𝑗−1

𝑘=𝑛−𝑚 .                                      (14) 

Since  2𝑗 ≤ 𝑚 + 1 < 2𝑗+1, we get 2𝑗 > 𝑚 − 2𝑗 + 1. Hence 

 (𝑚 − 2𝑗 + 1)𝐸𝑛−𝑚+2𝑗 (𝑓)𝐿𝑝,𝛼(𝛾0) ≤ ∑ 𝐸𝑘 (𝑓)𝐿𝑝,𝛼(𝛾0).
𝑛−𝑚+2𝑗−1
𝑘=𝑛−𝑚                        (15) 

From                                       (13),                                     (14) and                        (15) we obtain 

‖𝑓 − 𝑉𝑛,𝑚(. , 𝑓)‖𝐿𝑝,𝛼(𝛾0)

≤
𝑐13

𝑚 + 1
{𝐸𝑛−𝑚(𝑓)𝐿𝑝,𝛼(𝛾0) + ∑ 𝐸𝑘 (𝑓)𝐿𝑝,𝛼(𝛾0)

𝑛−𝑚+2𝑗−1

𝑘=𝑛−𝑚

+ ∑ 𝐸𝑘 (𝑓)𝐿𝑝,𝛼(𝛾0)

𝑛−𝑚+2𝑗−1

𝑘=𝑛−𝑚

}

≤
𝑐6

𝑚 + 1
∑ 𝐸𝑘 (𝑓)𝐿𝑝,𝛼(𝛾0)

𝑛

𝑘=𝑛−𝑚

 

and the inequality                         (11) is true. 

 

Corollary 3.1 Let 𝐿𝑝,𝛼 (𝛾0) be a Morrey space with 0 < 𝛼 ≤ 2 and 1 < 𝑝 < ∞, then there exists a positive constant 𝑐14 
such that for any 𝑓 ∈ 𝐿𝑝,𝛼 (𝛾0), 0 ≤ 𝑚 ≤ 𝑛, 𝑚, 𝑛 = 1,2, … the inequality  
 

 ‖𝑓 − 𝑉𝑛,𝑚(. , 𝑓)‖𝐿𝑝,𝛼(𝛾0)
≤

𝑐14

𝑚+1
∑ 𝜔𝑝,𝛼

𝑟 (𝑓,
1

𝑘+1
)𝑛𝑘=𝑛−𝑚                  (16) 

is true. 

 

Proof. From Theorem 3.1 we have 

‖𝑓 − 𝑉𝑛,𝑚(. , 𝑓)‖𝐿𝑝,𝛼(𝛾0)
≤

𝑐6
𝑚 + 1

∑ 𝐸𝑘 (𝑓)𝐿𝑝,𝛼(𝛾0)

𝑛

𝑘=𝑛−𝑚

 

and from Theorem 2.2 we get 

𝐸𝑛 (𝑓)𝐿𝑝,𝛼(γ0) ≤ 𝑐5 𝜔𝑝,𝛼
𝑟 (𝑓,

1

𝑛 + 1
)  , 𝑛 = 1,2,3, …  . 

 

 



AHMED KINJ ET AL 

94 

 

We reach 

‖𝑓 − 𝑉𝑛,𝑚(. , 𝑓)‖𝐿𝑝,𝛼(𝛾0)
≤

𝑐14
𝑚 + 1

∑ 𝜔𝑝,𝛼
𝑟 (𝑓,

1

𝑘 + 1
)

𝑛

𝑘=𝑛−𝑚

 , 𝑛 = 1,2, …  . 

 

Theorem 3.2 Let 𝐺 be a simply connected domain in the complex plane, bounded by a curve Γ ∈ 𝒟. If 𝑓 ∈ 𝐸𝑝,𝛼 (𝐺) 
with 0 < 𝛼 ≤ 2 and 1 < 𝑝 < ∞, then for every 0 ≤ 𝑚 ≤ 𝑛,  𝑛, 𝑚 ∈ ℕ the estimate 

‖𝑓 − 𝑉𝑛(. , 𝑓)‖𝐿𝑝,𝛼(Γ) ≤ 𝑐15 ∑ ΩΓ,𝑝,𝛼
𝑟 (𝑓,

1

𝑘 + 1
)

𝑛

𝑘=𝑛−𝑚

  

holds, where 𝑐15 is a positive constant. 
 

Proof. Since 𝑓 ∈ 𝐸𝑝,𝛼 (𝐺) and Γ is a Dini – smooth curve, then the boundary function of 𝑓 belongs to 𝐿𝑝,𝛼 (Γ) and from 
the relation (5) we get 𝑓0 ∈ 𝐿

𝑝,𝛼 (𝛾0), and the function 𝑓0
+ which defined by (6) belongs to 𝐸𝑝,𝛼 (𝐷). Since 𝐸𝑝,𝛼 (𝐷) ⊂

𝐸1(𝐷), we obtain 𝑓0
+ ∈ 𝐸1(𝐷) which has the following Taylor expansion 

 𝑓0
+(𝑤) = ∑ 𝑎𝑘 (𝑓0

+)𝑤 𝑘∞𝑘=0 ,   𝑤 ∈ 𝐷. (17) 

Let {𝑐𝑘 } be the Fourier coefficients of the boundary function of 𝑓0
+, then by [23] we get 𝑐𝑘 = 𝑎𝑘 (𝑓0

+) for 𝑘 ≥ 0 and 
𝑐𝑘 = 0 for 𝑘 < 0, and then by substitution in (17) we obtain 

𝑓0
+(𝑤) = ∑ 𝑐𝑘 𝑤

𝑘

∞

𝑘=0

,   𝑤 ∈ 𝐷. 

Note that for the function 𝑓 ∈ 𝐸𝑝,𝛼 (𝐺) the following Faber series holds 

𝑓(𝑧)~ ∑ 𝑎𝑘 (𝑓)Φ𝑘 (𝑧)

∞

𝑘=0

,   𝑧 ∈ 𝐺, 

where 𝑎𝑘 (𝑓), 𝑘 = 0,1,2, … are the Taylor coefficients of the function 𝑓0
+, and by Theorem 2.1 we obtain 

 

𝑇 (∑ 𝑎𝑘 (𝑓0
+)𝑤𝑘

𝑛

𝑘=0

) = ∑ 𝑎𝑘 (𝑓)Φ𝑘 (𝑧)

n

𝑘=0

 

and 

𝑇 (𝑉𝑛,𝑚(w, 𝑓0
+)) = 𝑉𝑛,𝑚(𝑧, 𝑓), 0 ≤ 𝑚 ≤ 𝑛, 𝑛, 𝑚 = 0,1,2, …   . 

 

Hence, using the boundedness of operator 𝑇 defined by (10) and the relation                         (11), we reach 
 

‖𝑓 − 𝑉𝑛,𝑚(. , 𝑓)‖𝐿𝑝,𝛼(Γ)
= ‖𝑇(𝑓0

+) − 𝑇 (𝑉𝑛,𝑚(. , 𝑓0
+))‖

𝐿𝑝,𝛼(Γ)
≤ 𝑐16‖𝑓0

+ − 𝑉𝑛,𝑚(. , 𝑓0
+)‖

𝐿𝑝,𝛼(𝛾0)

≤
𝑐17

𝑚 + 1
∑ 𝐸𝑘 (𝑓0

+)𝐿𝑝,𝛼(𝛾0).

𝑛

𝑘=𝑛−𝑚

 

 

Using the Theorem 2.2 we get 

‖𝑓 − 𝑉𝑛,𝑚(. , 𝑓)‖𝐿𝑝,𝛼(Γ)
≤

𝑐15
𝑚 + 1

∑ 𝜔𝑝,𝛼
𝑟 (𝑓0

+,
1

𝑘 + 1
) .

𝑛

𝑘=𝑛−𝑚

 

And by the relation               (7) we reach 

‖𝑓 − 𝑉𝑛,𝑚(. , 𝑓)‖𝐿𝑝,𝛼(Γ)
≤

𝑐15
𝑚 + 1

∑ ΩΓ,𝑝,𝛼
𝑟 (𝑓,

1

𝑘 + 1
) .

𝑛

𝑘=𝑛−𝑚

 

Consequently, we have proved the Theorem 3.2. 

 

 



APPROXIMATION PROPERTIES OF DE LA VALLÉE-POUSSIN SUMS IN MORREY SPACES 

 

95 
 

4. Conclusion 

A method was developed to estimate the deviation of functions from their de la Vallée-Poussin sums in Morrey 

spaces in terms of the best approximation. 

References 

1. Guven, A. Trignometric approximation of function in weighted Lp spaces. Sarajevo Journal of Math,  2009, 5(17),  
99-108. 

2. Andersson, J. On the degree of polynomial approximation in E
p
(D). Journal of Approximation Theory, 1977, 

19(1), 61-68. 

3. Çavu, A. and Israfilov, D. Approximation by Faber-Laurent rational functions in the mean of functions of class 

Lp(Γ) with 1< p<+∞. Approximation Theory and its Applications, 1995, 11(1), 105-118. 

4. Israfilov, D. Approximation by p-Faber-Laurent  rational functions. Czechoslovak Mathematical Journal,  2004, 

54(129), 751-765. 

5. Jafarov, S. Approximation by polynomials and rational functions in Orlicz spaces. Journal of computational 

analysis and applications, 2011, 13(5), 953-962. 

6. Jafarov, S. On approximation in weighted Smirnov Orlicz classes. Complex Variables and Elliptic Equations,  

2012, 57(5), 567-577. 

7. Israfilov, D. and Akgun, R. Approximation in weighted Smirnov- Orlicz classes. Journal of Math. Kyoto 

University., 2006, 46(4), 755-770. 

8. Morrey, C. On the solutions of quasi-linear elliptic partial differential equations. Transactions of the American 

Mathematical Society,  1938, 43(1), 126-166. 

9. Israfilov, D. and Tozman, N. Approximation by polynomials in Morrey-Smirnov classes. East Journal on, 

Approximation, 2008, 14(3), 225-269. 

10. Israfilov, D. and Tozman, N. Approximation in Morrey-Smirnov classes. Azerbaijan Journal of Math., 2011, 1(1),   

99-113. 

11. Bilalov, B. and Quliyeva, A. On basicity of exponential systems in Morrey-type spaces. International Journal of 

Mathematics, 2014, 25(06), 1450054 (10 pages). 

12. Sadigova, S. On approximation by Shift Operators in Morrey Type Spaces. Caspian Journal of Applied 

Mathematics, Ecology and Economics, 2015, 3(2). 

13. Bilalov, B., Gasymov, T., and Guliyeva, A. On the solvability of the Riemann boundary value problem in Morrey-

Hardy classes. Turkish Journal of Mathematics, 2016, 40(5), 1085-1101. 

14. Guliyeva, F., Abdullayeva, R., and Cetin, S. On Morrey type Spaces and Some Properties. Caspian Journal of 

Applied Mathematics, Ecology and Economics, 2016, 4(1), 3-16. 

15. Kokilasvili, V. On approximation of analytic functions from Ep classes. Trudy Tbiliss. Razmadze Mathematical 

Institute, 1968, 34, 82-102. 

16. Guven, A. and Israfilov, D. Approximation by mean of Fourier trignometric series in weighted Orlicz spaces. 

Advanced Studies in Contemporary Mathematics, 2009, 19(2), 283-295. 

17. Jafarov, S. Approximation of functions by de la Vallee-Poussin sums in weighted Orlicz spaces. Arabian Journal 

of Mathematics, 2016, 5(3), 125-137. 

18. Fucík, S., John, O., Kufner, A., and Pick, L. Function spaces: De Gruyter Series in Nonlinear Analysis and 

Applications 14 (2nd Edition), De Gruyter, Berlin/Boston, 2013. 

19. Devore, R.A. and Lorentz, G.G. Constructive approximation: A Series of Comprehensive Studies in Mathematics, 

Springer, New York, 1993. 

20. Pommerenke, C. Boundary Behavior of Conformal Maps: A Series of Comprehensive Studies in Mathematics, 

Springer, Berlin, 1992. 

21. Warschawski, S.E. Uber das ranverhalten der Ableitung der Abildunggsfunktion bei Konfermer Abbildung, Math. 

Z. 1932, 35(German). 

22. Markushevich, A. Theory of Analytic Functions, Izdatelstvo Nauka, Moscow, 1968. 

23. Duren, P.L. Theory of H
p
 spaces: A Series of Monographs and Textbooks, Academic Press, New York, 1970. 

 

 

Received   18 October 2016   

Accepted   1
st
 August 2017