Vol 6, No 7 (2015)

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ERROR ESTIMATE OF GENERALIZED SHANNON SAMPLING OPERATORS

IN WEIGHTED

_𝐋𝐋

SPACE

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_𝛂𝛂

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_𝟎𝟎

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_𝟎𝟎

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₎

SAHEB K. AL-SAIDY

_{, HUSSEIN A. AL-JUBOORI}

_{, ABDULSATTAR A. AL-DULAIMI}

1

_{Department of Mathematics,}

College of Science, University of Al-Mustansiry, Iraq.

2

_{Department of Mathematics,}

College of Science, University of Al-Mustansiry, Iraq.

3

_{Department of Mathematics,}

College Of Education of Pure Science, University of Al-Anbar, Iraq.

(Received On: 22-06-15; Revised & Accepted On: 26-07-15)

ABSTRACT

T

INTRODUCTION

The generalized sampling operators for the uniformly continuous and bounded functions 𝑓𝑓 ∈ 𝐶𝐶(𝑅𝑅) ([8], [11] are given by

(𝑆𝑆𝐺𝐺𝑓𝑓)(𝑡𝑡)≔ ∑∞𝑘𝑘=−∞𝑓𝑓 �𝑘𝑘_𝐺𝐺� 𝑠𝑠(𝐺𝐺𝑡𝑡 − 𝑘𝑘), (𝑡𝑡 ∈R;𝐺𝐺> 0) (1)

The operator 𝑆𝑆_𝐺𝐺:𝐶𝐶(𝑅𝑅)→ 𝐶𝐶(𝑅𝑅) to be well-defined where the condition

∑∞ |𝑠𝑠(𝑣𝑣 − 𝑘𝑘)| <∞ (𝑣𝑣 ∈ 𝑅𝑅)

𝑘𝑘=−∞ (2)

is satisfied. A systematic study of sampling operators (1) for arbitrary kernel functions s with (2) was initiated at WTH Aachen by P. L. Butzer and his students since 1977([12],[14]).

Definition 1.1 [14]: If 𝑠𝑠:𝑅𝑅 → 𝑅𝑅 is a bounded function such that (2) the absolute convergence being uniform on compact subsets of

∑∞ |𝑠𝑠(𝑣𝑣 − 𝑘𝑘)| = 1 (𝑣𝑣 ∈ 𝑅𝑅)

𝑘𝑘=−∞ (3)

Then s is said to be a kernel for sampling operators (1).

If the kernel function is 𝑠𝑠(𝑡𝑡) =𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝑡𝑡)≔𝑠𝑠𝑠𝑠𝑠𝑠 𝜋𝜋𝑡𝑡

𝜋𝜋𝑡𝑡 , which do not satisfy (2), we get the classical Shannon operator

�𝑆𝑆𝐺𝐺𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑓𝑓�(𝑡𝑡)≔ ∑∞𝑘𝑘=−∞𝑓𝑓 �_𝐺𝐺𝑘𝑘� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝐺𝐺𝑡𝑡 − 𝑘𝑘). (𝑡𝑡 ∈ 𝑅𝑅 ;𝐺𝐺 > 0) (4)

In this paper we estimate the order of approximation in terms of modulus of smoothness in weighted L_P,α space

(α> 0, 0 <𝑝𝑝< 1).

2. PRELIMINARY RESULTS

2.1The modulus of smoothness in 𝐋𝐋_{𝐏𝐏,𝛂𝛂} space (𝛂𝛂> 0, 0 <𝐏𝐏< 1)

Definition 2.1.1(Weight function) [1]: An integrable function 𝐰𝐰 is called a weight function on the interval [a, b] if w(x) > 0 for all x∈[a, b]. For example w(x) = eαx, α> 0.

Consider the space 𝐿𝐿_{𝑝𝑝,𝛼𝛼}(𝑋𝑋), 0 <𝑝𝑝< 1 of all unbounded functions f on X such that |𝑓𝑓(𝑥𝑥)|≤ 𝑀𝑀𝑒𝑒𝛼𝛼𝑥𝑥, where M is a positive real number, which are equipped with the following quasi norm

‖𝑓𝑓‖𝑝𝑝,𝛼𝛼𝑝𝑝 =�∫ �𝑓𝑓(𝑥𝑥)_𝑒𝑒𝛼𝛼𝑥𝑥�

𝑝𝑝 𝑋𝑋 𝑑𝑑𝑥𝑥�

1 𝑝𝑝

<∞ (5) .

Let 𝑓𝑓 be unbounded function on R, and 𝛿𝛿 ≥0, the modulus of smoothness 𝜔𝜔𝑘𝑘(𝑓𝑓;𝛿𝛿)𝑝𝑝,𝛼𝛼 in 𝐿𝐿𝑝𝑝,𝛼𝛼(𝑋𝑋), 𝛼𝛼> 0, 0 <𝑝𝑝< 1 is defined exactly as in case 1≤p≤ ∞:[22]

𝜔𝜔𝑘𝑘(𝑓𝑓;𝛿𝛿)𝑝𝑝,𝛼𝛼 = 𝑠𝑠𝑠𝑠𝑝𝑝� 0≤ℎ<𝛿𝛿

�∫ �∆ℎ𝑘𝑘�𝑓𝑓(𝑥𝑥)_𝑒𝑒𝛼𝛼𝑥𝑥��

𝑝𝑝 𝑏𝑏−𝑘𝑘ℎ 𝑎𝑎 � 1 𝑝𝑝 (6)

2.2 The space 𝚲𝚲𝐩𝐩,𝛂𝛂

Definition 2.2.1 ([10]):

(a) A sequence ∑ ≔ �xj�_j∈Z⊂R is called an admissible partition of R or an admissible sequence, if it satisfies 0 < infj∈Z∆j≤supj∈Z∆j<∞ .

(b) Let ∑ ≔ �xj�_j∈Z be an admissible partition of R, and let ∆j= xj−xj−1. The discrete ℓp(∑)−norm of a sequence of function values f∑ on the partition (∑) of the function f: R→C is defined for 1≤p <∞ by

‖f‖ℓp(∑) =�∑ �f�xj��

p

∆j

j∈Z �

1

p ₍₇₎

(c) The space Λp _{for 1≤p <∞ is defined by}

Λp _{≔ �f;‖f‖}

ℓp(∑) <∞� for each admissible sequence (∑).

We can defined Λp,α _{α> 0, 0 <𝑝𝑝< 1 the space of all unbounded functions 𝑓𝑓 on admissible sequence (∑) of R,} such that

‖𝑓𝑓‖_ℓ𝑝𝑝_𝑝𝑝(∑),𝛼𝛼 =�∑ �𝑓𝑓(𝑥𝑥)_𝑒𝑒𝛼𝛼𝑥𝑥�

𝑝𝑝

∆𝑗𝑗

𝑗𝑗 ∈𝑍𝑍 �

1 𝑝𝑝

(8)

Theorem 2.1.2: For 𝑓𝑓 ∈ 𝛬𝛬𝑝𝑝,𝛼𝛼 , (𝛼𝛼> 0, 0 <𝑝𝑝< 1), and 𝑘𝑘 ∈ 𝑅𝑅 then

�𝑆𝑆𝐺𝐺𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑓𝑓 − 𝑓𝑓�_{𝑝𝑝,𝛼𝛼}𝑝𝑝 ≤ 𝐶𝐶𝜔𝜔𝑘𝑘�𝑓𝑓;1_𝐺𝐺�_{𝑝𝑝,𝛼𝛼} (9)

Proof:

�𝑆𝑆𝐺𝐺𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑓𝑓 − 𝑓𝑓�_{𝑝𝑝,𝛼𝛼}𝑝𝑝 =�∑ �𝑆𝑆𝐺𝐺

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠_{𝑓𝑓(𝑥𝑥)−𝑓𝑓(𝑥𝑥)}

𝑒𝑒𝛼𝛼𝑥𝑥 �

𝑝𝑝

∆𝑗𝑗

𝑗𝑗 ∈𝑍𝑍 �

1 𝑝𝑝

=�∑ �∑∞𝑘𝑘=−∞𝑓𝑓�𝑘𝑘𝐺𝐺�𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝐺𝐺𝑡𝑡−𝑘𝑘)−𝑓𝑓(𝑥𝑥)

𝑒𝑒𝛼𝛼𝑥𝑥 �

𝑝𝑝

∆𝑗𝑗

𝑗𝑗 ∈𝑍𝑍 �

1 𝑝𝑝

≤ 𝑠𝑠𝑠𝑠𝑝𝑝 �∑ �∑∞𝑘𝑘=−∞𝑓𝑓�𝐺𝐺𝑘𝑘�𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝐺𝐺𝑡𝑡−𝑘𝑘)−𝑓𝑓(𝑥𝑥)

𝑒𝑒𝛼𝛼𝑥𝑥 �

𝑝𝑝

∆𝑗𝑗

𝑗𝑗 ∈𝑍𝑍 �

1 𝑝𝑝

≤sup�∑ �f�Gk� ∑∞k=−∞sinc (Gt−k)−f(x)

eαx �

p ∆j j∈Z � 1 p

By using (3) we have

�𝑆𝑆𝐺𝐺𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑓𝑓 − 𝑓𝑓�_{𝑝𝑝,𝛼𝛼}𝑝𝑝 ≤ 𝑠𝑠𝑠𝑠𝑝𝑝 �∑ �𝑓𝑓�

𝑘𝑘 𝐺𝐺�−𝑓𝑓(𝑥𝑥)

𝑒𝑒𝛼𝛼𝑥𝑥 �

𝑝𝑝

∆𝑗𝑗

𝑗𝑗 ∈𝑍𝑍 �

1 𝑝𝑝

≤ 𝐶𝐶(𝑃𝑃)�∑ �∆ℎ𝑘𝑘𝑓𝑓 �1_𝐺𝐺� 𝑒𝑒−𝛼𝛼𝑥𝑥� 𝑝𝑝

∆𝑗𝑗

𝑗𝑗 ∈𝑍𝑍 �

1 𝑝𝑝

≤ 𝐶𝐶(𝑃𝑃)�∆ℎ𝑘𝑘𝑓𝑓 �1_𝐺𝐺��_{𝑃𝑃,𝛼𝛼} 𝑃𝑃

≤ 𝐶𝐶(𝑃𝑃)𝜔𝜔𝑘𝑘�𝑓𝑓;1_𝐺𝐺� 𝑝𝑝,𝛼𝛼

2.3 Band-limited kernels

s(t)≔sλ(t)≔ ∫₀1λ(u)cos(πtu)du. (10)

We studied the generalized sampling operators S_W: C(R)→C(R) with the kernels in form (9) in ([2], [3], [5], [6], and [7]).

Many the band-limited kernel have been used in applications ([15], [16], [17], [18]). Many kernels can be defined by (9), e.g.

1) λ_(r)(u) = 1−u2, r≥1 defines the Zygmund (or Riesz) kernel, denoted by Z_r = Z_r(t), which special case r = 1, the Fej´er kernel (see[19])

sF(t) =1₂sinc2�₂t� (11)

2) λ_(j)(u)≔cosπ(j + 1 2⁄ )u, j = 0,1,2, … defines the Rogosinski-type kernel (see [4]) in the form

rj(t)≔ �sinc(t + j + 1 2⁄ ) + sinc(t−j−1 2⁄ )� (12)

λ_(H)(u)≔cos2_�πu 2�=

1

2(1 + cosπu) defines the Hann kernel (see[5])

s_H(t) =1

2 sinc t

1−t2. (13)

Powers of the Hann window (see [15])

λ_H,m(u)≔cosm�πu

2�= 1

2m∑mk=0�m_k�cos��k−m₂�πu� (14)

3) Give the general Hann kernel in the form

sH,m(t) = 2−m_Γ�1+mΓ(1+m) 2−t�Γ�1+m2+t�

(15)

From ([5], Prorposition 2) we have that for m = 0,1,2, … , and ℓ≤m

sH,m(t) =₂m1−ℓ∑k=0m−ℓ�m_k−ℓ�sH,ℓ�t + k−m−₂ℓ� (16)

4) The Blackman-Harris window function

𝜆𝜆𝐶𝐶𝑎𝑎(𝑠𝑠) =∑𝑚𝑚𝑘𝑘=0𝑎𝑎𝑘𝑘𝑠𝑠𝑐𝑐𝑠𝑠 𝑘𝑘𝜋𝜋𝑠𝑠=∑� 𝑎𝑎2𝑘𝑘𝑠𝑠𝑐𝑐𝑠𝑠(2𝑘𝑘𝜋𝜋𝑠𝑠)

𝑚𝑚 2�

𝑘𝑘=0 +∑� 𝑎𝑎2𝑘𝑘+1𝑠𝑠𝑐𝑐𝑠𝑠�(2𝑘𝑘+ 1)𝜋𝜋𝑠𝑠�

𝑚𝑚 −1 2 �

𝑘𝑘=0 (17)

Where ( ⌊x⌋ is the largest integer less than or equal to x∈R )

∑ a_2k=∑ a_2k+1=1

2 �m₂−1�

k=0 �m₂�

k=0 (18)

Defines through (9) the Blackman-Harris kernel (see [7])

𝑠𝑠𝐶𝐶𝑎𝑎(𝑡𝑡) =1₂∑𝑚𝑚𝑘𝑘=0𝑎𝑎𝑘𝑘�𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝑡𝑡 − 𝑘𝑘) +𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝑡𝑡+𝑘𝑘)� (19)

Proposition 2.3.1([20])

For m∈N, 1≤ℓ≤m the kernel

s(t) = sinc(t)−₂21ℓ+1∑k=om−ℓ(−1)k+ℓqk�∆12ℓsinc(t−k) +∆12ℓsinc(t + k)� (20)

With q∈Rm−ℓ+1,∑_k=0m−ℓq_k = 1 is a Blackman Harris kernel s_C,a(q) with parameter vector a(q)∈Rm+1.

3. MAIN RESULTS

In this suction we shall estimates error approximation of some sampling operators S_Gf: C(R)→C(R) which is linear combinations of translated sinc-functions.

3.1 Rogosinski-type sampling operators

Let consider the Rogosinski-type sampling operators R_G,j defined by the kernel functions r_j in (9). These kernel functions are deduced by the window functions λ_(j)(u)≔cosπ(j + 1 2⁄ )u, j∈N. (see[20])

Theorem 3.1.1: Assume that a Rogosinski-type sampling operator R_G,j (j = 0,1,2, … ), G > 0 defined by (1) with the kernel (10). Then for f∈Λp,α , (α> 0, 0 <𝑝𝑝< 1) we have

�RG,jf−f�_p,p_α≤Cjω2�f;1_G�_P,α. (20)

Proof: Since the Rogosinski-type kernel in (10) is a linear componation of translated sinc-function. Then we give this operator R_G,j which is representation

�RG,jf�(t) =∑j∈Zf�_Gj��1₂�sinc�t + j +1₂�+ sinc�t−j−1₂���

=1

2�∑ f� j G�

j∈Z sinc�t + j +1₂�+∑j∈Zf�_Gj�sinc�t−j−1₂��

=1

2��SGsincf� �t + 2j+1

2G �+�SGsincf� �t− 2j+1

2G ��

We obtain

�RG,jf�(t)−f(t) =1₂��SGsincf� �t +2j+1_2G �+�SGsincf� �t−2j+1_2G �� −f(t)

=1

2�

�SGsincf� �t +2j+1_2G �+�SsincG f� �t−2j+1_2G � −f�t +2j+1_2G � +f�t +2j+1_2G � −f�t−2j+1_2G �+ f�t−2j+1_2G � −2f(t) �

=1

2�

�RG,jf� �t +2j+1_2G � −f�t +2j+1_2G �+�SGsincf� �t−2j+1_2G �

− f�t−2j+1_2G �+�f�t +2j+1_2G � −2f(t) + f�t−2j+1_2G ���

Since 0 <𝑝𝑝< 1 then by properties of trigonometric inequality for quasi norm we have

��RG,jf� −f�_p,p_α≤2p��1₂� ��SGsincf−f�_p,p_α+�SGsincf−f�_p,p_α+�∆2j+1 2G

2 _f� p,α p

��

≤2p��S_Gsincf−f�

p,α p

+1₂ω2�f;_G1� P,α�

By using theorem (2.1.2) we have

��RG,jf� −f�_p,αp ≤2p�ω2�f,_G1�_P,α+1₂�1 +2j+1_2G � ω2�f;1_G�_P,α� ≤ ω2�f;1_G�_P,α�2p+1₂�1 +2j+1_2G ��

≤Cjω2�f;_G1�

P,α.

3.2 Hann sampling operators

Consider Hann sampling operators H_W,m (m = 0,1,2, … ). The Hann kernel (see [21] s_H,m(t) = O(|t|−m−1) as|t|→∞. from (15) if ℓ= 0 we have aliner combination of sinc –function because H_W,0= sinc.

Theorem 3.2.1: For the Hann sampling operator H_G,m(m = 1,2, … ) defined by (1) with the kernel (14). Then for f∈Λp,α _{, (α> 0, 0 <𝑝𝑝< 1) we have}

�HG,mf−f�_p,p_α≤Cmω2�f;1_G�_P,α. (21)

where the constant C_m is independent of f and G.

Proof: According to ([20] equation (9)) we give this operator �H_G,m� which has the form

�HG,mf�(t) =1₂��HG,m−1f� �t−_2G1�+�HG,m−1f� �t +_2G1�� Hence

�HG,mf�(t)−f(t) =1₂��HG,m−1f� �t−_2G1�+�HG,m−1f� �t +_2G1�� −f(t)

=1₂��HG,m−1f� �t−

1

2G� −f�t− 2j+1

2G �+ f�t− 2j+1

2G �+�HG,m−1f� �t + 1 2G�

−f�t +2j+1_2G �+ f�t +2j+1_2G � −2f(t) �

=1₂��HG,m−1f� �t−

1

2G� −f�t− 2j+1

2G �+�HG,m−1f� �t + 1 2G�

−f�t +2j+1_2G �+�f�t +2j+1_2G � −2f(t) + f�t−2j+1_2G �� �

Since 0 <𝑝𝑝< 1 we have from properties of quasi norm,

��HG,mf� −f�_p,αp ≤2p��1₂� ��HG,m−1f−f�_p,αp +�HG,m−1f−f�_p,αp +�∆1 2G

2 _f� p,α p

��

≤2p_��H

By using induction the proof give

�HG,mf−f�p_p,α ≤2p��HG,0f−f�_p,αp +m₂ω2�f;_2G1�_P,α�

From (15) if we take ℓ= 0 then we have a linear combination of sinc-function because S_H,0= sinc . hence HG,0= SGsinc ,

Therefore by using theorem (2.1.2) we have

�HG,mf−f�_p,αp ≤2p�C2ω2�f;1_G� P,α+

m 2ω2�f;

1 2G�P,α� ≤Cm�ω2�f;_2G1�_P,α� .

3.3 Blackman-Harris sampling operators

Before over 52 years ([15], [16], [17], [18]) The Blackman window has been used in signal analysis. Recently Lasser and Obermaier [13] studied the role of the Blackman window for defining approximative identities in Fourier approximation.

Theorem 3.3.1: consider C_G,a be the blackman-harris sampling operator defined by (1) with the kernel (18). Then for f∈Λp,α , (α> 0, 0 <𝑝𝑝< 1) we have

�CG,af−f�_p,p_α≤Taω2�f;_G1�_P,α (22)

where the constant T_a is independent of f and G.

Proof: from (18) we show that the blackman-harris kernel is a linear combination of translated sinc-functions.

Therefore the operator C_G,a can be representation by

�CG,af�(t) =1₂∑j∈Zf�_Gj� ∑k=om ak�sinc(Gt−j + k) + sinc(Gt−j−k)�

=1

2∑mk=oak�∑ f� j G�

j∈Z �sinc(Gt−j + k) +∑j∈Zf�_Gj�sinc(Gt−j−k)��

=1

2∑mk=oak��SGsincf� �t + k

G�+�SGsincf� �t− k G��

Therefore

�CG,af�(t)−f(t) =₂1∑mk=oak��SGsincf� �t +k_G�+�SGsincf� �t−k_G�� −f(t)

=1

2∑mk=oak�

�SGsincf� �t +_Gk� −f�t +_Gk�+�SGsincf� �t−_Gk� −f�t−_Gk� +�f�t +k_G� −2f(t) + f�t−k_G�� �

Since 0 <𝑝𝑝< 1 , by properties of quasi norm we obtain

�CG,af−f�_p,αp ≤2p 1₂∑mk=o|ak|��CG,af−f�_p,αp +�CG,af−f�_p,αp +�∆k G

2_f� p,α p

�

≤2p 1

2∑mk=o|ak|��CG,af−f�p,α p

+1₂ω2�f;_Gk� P,α�

By using theorem (2.1.2) we have

�CG,af−f�_p,p_α≤2p∑mk=o|ak|�Cω2�f;_G1� P,α+

k2

2ω2�f; 1 G�_P,α� ≤ω₂�f;1

G�_P,α�2

p_{∑ |a} k| m

k=o �C +k

2

2��

≤M_aω₂�f;1

G�_P,α.

Theorem 3.3.2: For C_G,a (a∈Rm+1) let ℓ, 1≤ℓ≤m be fixed. For a parameter vector q∈Rm−ℓ+1, such that we have for the kernel (13) a representation via central differences(4) inform (14),

Then for f∈Λp,α, (α> 0, 0 <𝑝𝑝< 1) we have

�CG,af−f�_p,p_α≤Da,ℓω2ℓ�f;1_G�

Proof: For kernel (20) we get operator C_G,a which is representation

�CG,af�(t) =∑j∈Zf�_Gj��sinc(t)−₂21ℓ+1∑m−ℓj=o (−1)j+ℓqj�∆12ℓsinc(t−j) +∆12ℓsinc(t−j)��

=∑j∈Zf�_Gj�sinc(Gt−k)−∑j∈Zf�_Gj��₂21ℓ+1∑m−ℓj=o (−1)j+ℓqj�∆2ℓ1 sinc(t−j) +∆12ℓsinc(t−j)�� =�SGsincf�(t)−₂21ℓ+1∑m−ℓj=o(−1)j+ℓqj∑j∈Zf�_Gj��∆2ℓ1 sinc(t−j) +∆12ℓsinc(t−j)�

=�SGsincf�(t)−₂2ℓ1+1∑m−ℓj=0(−1)j+ℓqj�∆2ℓ1 ∑j∈Zf�_Gj�sinc(Gt−j) +∆12ℓ∑j∈Zf�_Gj�sinc(Gt−j)�

Therefore

�CG,af�(t) =�SGsincf�(t)−₂21ℓ+1∑m−ℓj=0 (−1)j+ℓqj�∆12ℓ�SGsincf� �t−_Gj�+∆12ℓ�SGsincf� �t +_Gj��

We obtain

�CG,af�(t)−f(t) =�SGsincf�(t)−₂21ℓ+1∑m−ℓj=0 (−1)j+ℓqj�∆12ℓ�SGsincf� �t−

j

G�+∆12ℓ�SGsincf� �t + j

G�� −t(t)

=�SGsincf�(t)−f(t)−₂21ℓ+1∑m−ℓj=0 (−1)j+ℓqj�

∆12ℓ��SGsincf� �t−_Gj� − f�t−_Gj�+ f�t−_Gj��

+∆12ℓ��SGsincf� �t +_Gj� −f�t +_Gj�+ f�t +_Gj��

�

=�SGsincf�−₂2ℓ1+1∑m−ℓj=0 (−1)j+ℓqj�

∆1 G

2ℓ_��S G

sinc_{f� �t−}j

G� − f�t− j G�� +∆1

G

2ℓ_��S

Gsincf� �t +_Gj� −f�t +_Gj��

�

−₂21ℓ+1∑m−ℓj=0 (−1)j+ℓqj��∆1 G

2ℓ_f�t−j

G��+�∆1_G2ℓf�t + j G���

Since 0 <𝑝𝑝< 1 from properties of quasi norm we obtain

�CG,af−f�_p,αp ≤2p��SGsincf−f�_p,αp +∑ �qj��SGsincf−f�p_p,α+₂12ℓ∑ �qj� �∆1 G

2ℓ_f� p,α p m−ℓ

j=0 m−ℓ

j=0 �

≤2p_{��1 +∑ �q} j� m−ℓ

j=0 � ��SGsincf−f�_p,αp �+₂12ℓ∑m−ℓj=0�qj��ω2ℓ�f;1_G� P,α��

By using theorem (2.1.2) we have

�CG,af−f�_p,αp ≤2p�1 +∑j=0m−ℓ�qj�+₂12ℓ∑m−ℓj=0 �qj�� ω2ℓ�f;

1 G�_P,α ≤Da,ℓω2ℓ�f;_G1�_P,α.

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Source of support: Nil, Conflict of interest: None Declared

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