Approximation of Signals (Functions) by Trigonometric Polynomials in 𝐿_𝑝-Norm

Abstract

Mittal and Rhoades (1999, 2000) and Mittal et al. (2011) have initiated a study of error estimates 𝐸_𝑛(𝑓) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix 𝑇 does not have monotone rows. In this paper, the first author continues the work in the direction for 𝑇 to be a 𝑁_𝑝-matrix. We extend two theorems on summability matrix 𝑁_𝑝 of Deger et al. (2012) where they have extended two theorems of Chandra (2002) using 𝐶_𝜆-method obtained by deleting a set of rows from Cesàro matrix 𝐶₁. Our theorems also generalize two theorems of Leindler (2005) to 𝑁_𝑝-matrix which in turn generalize the result of Chandra (2002) and Quade (1937).

“In memory of Professor K. V. Mital, 1918 - 2010.”

1. Introduction

Let 𝑓 be a 2𝜋 periodic signal (function) and let 𝑓∈𝐿_𝑝∶=𝐿_𝑝[0,2𝜋],𝑝≥1. Let 𝑠_𝑛𝑓∶=𝑠_𝑛=𝑎𝑓;𝑥⁰2+^𝑛_𝑘=1𝑎_𝑘cos𝑘𝑥+𝑏_𝑘=sin𝑘𝑥_𝑛𝑘=0𝑢_𝑘𝑓;𝑥 (1) denote the partial sums, called trigonometric polynomials of degree (or order) 𝑛, of the first (𝑛+1) terms of the Fourier series of 𝑓 at a point 𝑥.

The integral modulus of continuity of 𝑓 is defined by 𝜔_𝑝𝛿;𝑓∶=sup_{0<|ℎ|≤𝛿}12𝜋₀^2𝜋𝑓(𝑥+ℎ)−𝑓(𝑥)^𝑝𝑑𝑥^1/𝑝. (2)

If, for 𝛼>0, 𝜔_𝑝𝛿;𝑓=𝑂𝛿^𝛼, (3) then 𝑓∈𝐿𝑖𝑝(𝛼,𝑝)(𝑝≥1). Throughout ‖⋅‖_𝑝 will denote the 𝐿_𝑝-norm, defined by 𝑓_𝑝1∶=2𝜋₀^2𝜋𝑓(𝑥)^𝑝𝑑𝑥^1/𝑝,𝑓∈𝐿_𝑝𝑝≥1. (4)

A positive sequence 𝐜∶={𝑐_𝑛} is called almost monotone decreasing (increasing) if there exists a constant 𝐾∶=𝐾(𝐜), depending on the sequence 𝐜 only, such that, for all 𝑛≥𝑚, 𝑐_𝑛≤𝐾𝑐_𝑚𝐾𝑐_𝑛≥𝑐_𝑚. (5)

Such sequences will be denoted by 𝐜∈AMDS and 𝐜∈AMIS, respectively. A sequence which is either AMDS or AMIS is called almost monotone sequence and will be denoted by 𝐜∈AMS. Let 𝔽 be an infinite subset of ℕ and 𝔽 as the range of strictly increasing sequence of positive integers; say 𝔽={𝜆(𝑛)}^∞_𝑛=1. The Cesàro submethod 𝐶_𝜆 is defined as 𝐶_𝜆𝑥_𝑛=1𝜆(𝑛)^𝜆(𝑛)_𝑘=1𝑥_𝑘,(𝑛=1,2,3,…), (6) where {𝑥_𝑘} is a sequence of real or complex numbers. Therefore, the 𝐶_𝜆-method yields a subsequence of the Cesàro method 𝐶₁, and hence it is regular for any 𝜆. 𝐶_𝜆 is obtained by deleting a set of rows from Cesàro matrix. The basic properties of 𝐶_𝜆-method can be found in [1, 2]. In the present paper, we will consider approximation of 𝑓∈𝐿_𝑝 by trigonometric polynomials 𝑁^𝜆_𝑛(𝑓;𝑥) and 𝑅^𝜆_𝑛(𝑓;𝑥) of degree (or order) 𝑛, where 𝑁^𝜆_𝑛=1𝑓;𝑥𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=0𝑝_{𝜆(𝑛)−𝑚}𝑠_𝑚,𝑅𝑓;𝑥_𝜆𝑛=1𝑓;𝑥𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=0𝑝_𝑚𝑠_𝑚,𝑠𝑓;𝑥_𝑛=1𝑓;𝑥𝜋₀^2𝜋𝑓(𝑥+𝑡)𝐷_𝑛(𝑡)𝐷𝑑𝑡,_𝑛(𝑡)=(sin(𝑛+1/2)𝑡)(),𝑃2sin𝑡/2_𝜆(𝑛)=𝑝₀+𝑝₁+⋯+𝑝_𝜆(𝑛)≠0(𝑛≥0), (7) and by convention 𝑝₋₁=0=𝑃₋₁.

The case 𝑝_𝑛=1 for all 𝑛(≥0) of either 𝑁^𝜆_𝑛(𝑓;𝑥) or 𝑅^𝜆_𝑛(𝑓;𝑥) yields 𝜎^𝜆_𝑛=1𝑓;𝑥𝜆(𝑛)+1^𝜆(𝑛)_𝑚=0𝑠_𝑚𝑓;𝑥. (8)

We also use Δ𝑎_𝑛=𝑎_𝑛−𝑎_𝑛+1,Δ_𝑚𝑔(𝑛,𝑚)=𝑔(𝑛,𝑚)−𝑔(𝑛,𝑚+1). (9)

Mittal and Rhoades [3, 4] have initiated the study of error estimates 𝐸_𝑛(𝑓) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix 𝑇 does not have monotone rows. In this paper, the first author continues the work in the direction for 𝑇 to be a 𝑁_𝑝-matrix. Recently, Chandra [5] has proved three theorems on the trigonometric approximation using 𝑁_𝑝-matrix. Some of them give sharper estimates than the results proved by Quade [6], Mohapatra and Russell [7], and himself earlier [8]. These results of Chandra [5] are improved in different directions by different investigators such as Leindler [9] who dropped the monotonicity on generating sequence {𝑝_𝑛} and Mittal et al. [10, 11] who used more general matrix while very recently Deger et al. [12] used more general 𝐶_𝜆-method in view of Armitage and Maddox [1].

2. Known Results

Leindler [9] proved the following.

Theorem 1 (see [9]). If 𝑓∈𝐿𝑖𝑝(𝛼,𝑝) and {𝑝_𝑛} be positive. If one of the conditions(i)𝑝>1,0<𝛼<1, and {𝑝_𝑛}∈𝐴𝑀𝐷𝑆,(ii)𝑝>1,0<𝛼<1, and {𝑝_𝑛}∈𝐴𝑀𝐼𝑆, (𝑛+1)𝑝_𝑛=𝑂𝑃_𝑛ℎ𝑜𝑙𝑑𝑠, (10)(iii)𝑝>1,𝛼=1, and ∑^𝑛−1_𝑘=1𝑘|Δ𝑝_𝑘|=𝑂(𝑃_𝑛),(iv)𝑝>1,𝛼=1,∑^𝑛−1_𝑘=0|Δ𝑝_𝑘|=𝑂(𝑃_𝑛/𝑛), and (10) holds,(v)𝑝=1,0<𝛼<1, and ∑^𝑛−1_𝑘=−1|Δ𝑝_𝑘|=𝑂(𝑃_𝑛/𝑛)
maintains, then 𝑓−𝑁_𝑛(𝑓)_𝑝=𝑂𝑛^−𝛼. (11)

Theorem 2 (see [9]). Let 𝑓∈𝐿𝑖𝑝(𝛼,1),0<𝛼<1. If the positive {𝑝_𝑛} satisfies conditions (10) and ∑^𝑛−1_𝑘=0|Δ𝑝_𝑘|=𝑂(𝑃_𝑛/𝑛) hold, then 𝑓−𝑅_𝑛(𝑓)₁=𝑂𝑛^−𝛼. (12)

Deger et al. [12] proved.

Theorem 3 (see [12]). Let 𝑓∈𝐿𝑖𝑝(𝛼,𝑝) and let {𝑝_𝑛} be positive such that (𝜆(𝑛)+1)𝑝_𝜆(𝑛)=𝑂𝑃_𝜆(𝑛). (13)
If either (i)  𝑝>1,0<𝛼≤1, and {𝑝_𝑛} is monotonic or (ii)  𝑝=1,0<𝛼<1, and {𝑝_𝑛} is nondecreasing, then 𝑓−𝑁^𝜆_𝑛(𝑓)_𝑝=𝑂𝑛^−𝛼. (14)

Theorem 4 (see [12]). Let 𝑓∈𝐿𝑖𝑝(𝛼,1),0<𝛼<1. If the positive {𝑝_𝑛} satisfies condition (13) and is nondecreasing, then 𝑓−𝑅^𝜆_𝑛(𝑓)₁=𝑂𝑛^−𝛼. (15)

3. Main Results

In this paper we generalize Theorems 3 and 4 of Deger et al. [12], by dropping monotonicity on the elements of the matrix rows which in turn generalize Theorems 1 and 2, respectively, of Leindler [9] to a more general 𝐶_𝜆-method. We prove the following.

Theorem 5. If 𝑓∈𝐿𝑖𝑝(𝛼,𝑝) and {𝑝_𝑛}is positive and if one of the following conditions(i)𝑝>1,0<𝛼<1, and {𝑝_𝑛}∈𝐴𝑀𝐷𝑆,(ii)𝑝>1,0<𝛼<1, {𝑝_𝑛}∈𝐴𝑀𝐼𝑆, and (13) holds,(iii)𝑝>1,𝛼=1, and ∑^{𝜆(𝑛)−1}_𝑘=1𝑘|Δ𝑝_𝑘|=𝑂(𝑃_𝜆(𝑛)),(iv)𝑝>1,𝛼=1,∑^{𝜆(𝑛)−1}_𝑘=0|Δ𝑝_𝑘|=𝑂(𝑃_𝜆(𝑛)/𝜆(𝑛)), and (13) holds,(v)𝑝=1,0<𝛼<1, and ∑^{𝜆(𝑛)−1}_𝑘=−1|Δ𝑝_𝑘|=𝑂(𝑃_𝜆(𝑛)/𝜆(𝑛))maintains, then 𝑓−𝑁^𝜆_𝑛(𝑓)_𝑝=𝑂(𝜆(𝑛))^−𝛼. (16)

Theorem 6. Let 𝑓∈𝐿𝑖𝑝(𝛼,1),0<𝛼<1. If the positive {𝑝_𝑛} satisfies (13) and the condition ∑^{𝜆(𝑛)−1}_𝑘=0|Δ𝑝_𝑘|=𝑂(𝑃_𝜆(𝑛)/𝜆(𝑛)) holds, then 𝑓−𝑅^𝜆_𝑛(𝑓)₁=𝑂(𝜆(𝑛))^−𝛼. (17)

Remarks. (1) If 𝜆(𝑛)=𝑛, then our Theorems 5 and 6 reduce to Theorems 1 and 2, respectively.

(2) Deger et al. [12] have used monotone sequences {𝑝_𝑛} in Theorems 3 and 4, while our Theorems 5 and 6 claim less than the requirements of their theorems. For example, the condition of the sum in (iii) of Theorem 5 is always satisfied if the sequence {𝑝_𝑛} is nonincreasing; that is, ^{𝜆(𝑛)−1}_𝑘=1𝑘Δ𝑝_𝑘=^{𝜆(𝑛)−1}_𝑘=1𝑘𝑝_𝑘−𝑝_𝑘+1=𝑃_{𝜆(𝑛)−1}−(𝜆(𝑛)−1)𝑝_𝜆(𝑛)=𝑂𝑃_𝜆(𝑛), (18) while if sequence {𝑝_𝑛} is nondecreasing and condition (13) holds, then the condition in (iv) of Theorem 5 is also satisfied; that is, ^{𝜆(𝑛)−1}_𝑘=0Δ𝑝_𝑘=^{𝜆(𝑛)−1}_𝑘=0𝑝_𝑘+1−𝑝_𝑘=𝑝_𝜆(𝑛)−𝑝₀≤𝑝_𝜆(𝑛)𝑃=𝑂^𝜆(𝑛)𝜆(𝑛). (19)

Thus our theorems generalize the two theorems of Deger et al. [12] under weaker assumptions and give sharper estimates because all the estimates of Deger et al. [12] are in terms of 𝑛, while our estimates are in terms of 𝜆(𝑛) and (𝜆(𝑛))^−𝛼≤𝑛^−𝛼 for 0<𝛼≤1.

4. Lemmas

We will use the following lemmas in the proof of our theorems.

Lemma 1 (see [6]). If 𝑓∈𝐿𝑖𝑝(𝛼,𝑝), for 0<𝛼≤1 and 𝑝>1, then 𝑓−𝑠_𝑛(𝑓)_𝑝=𝑂𝑛^−𝛼. (20)

Lemma 2 (see [6]). If 𝑓∈𝐿𝑖𝑝(1,𝑝), for 𝑝>1, then 𝜎_𝑛(𝑓)−𝑠_𝑛(𝑓)_𝑝=𝑂𝑛⁻¹. (21)

Lemma 3 (see [6]). If 𝑓∈𝐿𝑖𝑝(𝛼,1),0<𝛼<1, then 𝑓−𝜎_𝑛(𝑓)₁=𝑂𝑛^−𝛼. (22)

Lemma 4. Let {𝑝_𝑛}∈𝐴𝑀𝐷𝑆 or let {𝑝_𝑛}∈𝐴𝑀𝐼𝑆 and satisfy (13). Then, for 0<𝛼<1, ^𝜆(𝑛)_𝑚=1𝑚^−𝛼𝑝_{𝜆(𝑛)−𝑚}=𝑂(𝜆(𝑛))^−𝛼𝑃_𝜆(𝑛) (23) holds.

Proof. Let 𝑟 denote the integral part of (𝜆(𝑛)/2). Then, if {𝑝_𝑛}∈AMDS, ^𝜆(𝑛)_𝑚=1𝑚^−𝛼𝑝_{𝜆(𝑛)−𝑚}=^𝑟_𝑚=1𝑚^−𝛼𝑝_{𝜆(𝑛)−𝑚}+^𝜆(𝑛)_𝑚=𝑟+1𝑚^−𝛼𝑝_{𝜆(𝑛)−𝑚}≤𝐾𝑝^𝑟_{𝜆(𝑛)−𝑟𝑚=1}𝑚^−𝛼+(𝑟+1)^{−𝛼𝜆(𝑛)}_𝑚=𝑟+1𝑝_{𝜆(𝑛)−𝑚}≤𝐾𝑝_{𝜆(𝑛)−𝑟}^𝜆(𝑛)_𝑚=1𝑚^−𝛼+(𝑟+1)^{−𝛼𝜆(𝑛)}_𝑚=0𝑝_{𝜆(𝑛)−𝑚}=𝐾𝑝_{𝜆(𝑛)−𝑟}(𝜆(𝑛))^1−𝛼+𝑂(𝜆(𝑛))^−𝛼𝑃_𝜆(𝑛)=𝑂(𝜆(𝑛))^−𝛼𝑃_𝜆(𝑛). (24)
If {𝑝_𝑛}∈AMIS and (13) holds, then ^𝜆(𝑛)_𝑚=1𝑚^−𝛼𝑝_{𝜆(𝑛)−𝑚}=^𝑟_𝑚=1𝑚^−𝛼𝑝_{𝜆(𝑛)−𝑚}+^𝜆(𝑛)_𝑚=𝑟+1𝑚^−𝛼𝑝_{𝜆(𝑛)−𝑚}≤𝐾𝑝^𝑟_{𝜆(𝑛)𝑚=1}𝑚^−𝛼+(𝑟+1)^{−𝛼𝜆(𝑛)}_𝑚=𝑟+1𝑝_{𝜆(𝑛)−𝑚}𝑃=𝑂^𝜆(𝑛)𝜆(𝑛)+1^𝜆(𝑛)_𝑚=1𝑚^−𝛼+𝑂(𝜆(𝑛))^−𝛼𝑃_𝜆(𝑛)=𝑂(𝜆(𝑛))^−𝛼𝑃_𝜆(𝑛). (25)
This completes the proof of Lemma 4.

5. Proof of the Main Results

Proof of Theorem 5. We prove cases (i) and (ii) together. Since 𝑁^𝜆_𝑛(𝑥)=1𝑓;𝑥−𝑓𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=0𝑝_{𝜆(𝑛)−𝑚}𝑠_𝑚𝑓;𝑥−𝑓(𝑥), (26) thus in view of Lemmas 1 and 4 and condition (13), we have 𝑓−𝑁^𝜆_𝑛(𝑓)_𝑝≤1𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=0𝑝_{𝜆(𝑛)−𝑚}𝑓−𝑠_𝑚(𝑓)_𝑝=1𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=1𝑝_{𝜆(𝑛)−𝑚}𝑓−𝑠_𝑚𝑓_𝑝+𝑝_𝜆(𝑛)𝑃_𝜆(𝑛)𝑠₀−𝑓_𝑝=1𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=1𝑝_{𝜆(𝑛)−𝑚}𝑂𝑚^−𝛼𝑝+𝑂^𝜆(𝑛)𝑃_𝜆(𝑛)=𝑂(𝜆(𝑛))^−𝛼. (27)
Next we consider case (iv).
Let 𝑝>1 and 𝛼=1. By Abel’s transformation, we get 𝑁^𝜆_𝑛=1𝑓;𝑥𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=0𝑃_{𝜆(𝑛)−𝑚}𝑢_𝑚𝑓;𝑥, (28) and thus 𝑠^𝜆_𝑛𝑓;𝑥−𝑁^𝜆_𝑛=1𝑓;𝑥𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=1𝑃_𝜆(𝑛)−𝑃_{𝜆(𝑛)−𝑚}𝑢_𝑚𝑓;𝑥. (29)
Hence again by Abel’s transformation, we get 𝑠^𝜆_𝑛𝑓;𝑥−𝑁^𝜆_𝑛=1𝑓;𝑥𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=1Δ_𝑚𝑚⁻¹𝑃_𝜆(𝑛)−𝑃_{𝜆(𝑛)−𝑚}×^𝑚_𝑘=1𝑘𝑢_𝑘+1𝑓;𝑥(𝜆(𝑛))×+1_{𝜆(𝑛)𝑘=1}𝑘𝑢_𝑘𝑓;𝑥. (30)
Thus 𝑠^𝜆_𝑛(𝑓)−𝑁^𝜆_𝑛(𝑓)_𝑝≤1𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=1Δ_𝑚𝑚⁻¹𝑃_𝜆(𝑛)−𝑃_{𝜆(𝑛)−𝑚}⋅^𝑚_𝑘=1𝑘𝑢_𝑘𝑓_𝑝+1(𝜆(𝑛)+1)^𝜆(𝑛)_𝑘=1𝑘𝑢_𝑘(𝑓)_𝑝. (31)
Since 𝑠_𝑛𝑓;𝑥−𝜎_𝑛=1𝑓;𝑥𝑛+1^𝑛_𝑘=1𝑘𝑢_𝑘𝑓;𝑥, (32) thus by Lemma 2^𝑛_𝑘=1𝑘𝑢_𝑘𝑝=(𝑛+1)𝜎_𝑛(𝑓)−𝑠_𝑛(𝑓)_𝑝=𝑂(1). (33)
In view of (31) and (33), we obtain 𝑠^𝜆_𝑛𝑓−𝑁^𝜆_𝑛𝑓_𝑝1=𝑂𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=1Δ_𝑚𝑚⁻¹𝑃_𝜆(𝑛)−𝑃_{𝜆(𝑛)−𝑚}+𝑂(𝜆(𝑛))⁻¹. (34)
Now Δ_𝑚𝑃_𝜆(𝑛)−𝑃_{𝜆(𝑛)−𝑚}𝑚=1𝑚Δ_𝑚𝑃_𝜆(𝑛)−𝑃_{𝜆(𝑛)−𝑚}+𝑃_𝜆(𝑛)−𝑃_{𝜆(𝑛)−𝑚−1}𝑚()=𝑃𝑚+1_{𝜆(𝑛)−𝑚−1}−𝑃_{𝜆(𝑛)−𝑚}𝑚+𝑃_𝜆(𝑛)−𝑃_{𝜆(𝑛)−𝑚−1}𝑚()=𝑃𝑚+1_𝜆(𝑛)−𝑃_{𝜆(𝑛)−𝑚−1}𝑚()−𝑝𝑚+1𝑚^{𝜆(𝑛)−𝑚}=1𝑚()𝑃𝑚+1^𝜆(𝑛)−𝑃_{𝜆(𝑛)−𝑚−1}−(𝑚+1)𝑝_{𝜆(𝑛)−𝑚},Δ_𝑚𝑃_𝜆(𝑛)−𝑃_{𝜆(𝑛)−𝑚}𝑚=1𝑚()×𝑚+1_{𝜆(𝑛)(𝑛)𝑘=𝜆−𝑚}𝑝_𝑘−(𝑚+1)𝑝_{𝜆(𝑛)−𝑚}. (35)(36)
Next we will verify by the induction that ^𝜆(𝑛)_{(𝑛)𝑘=𝜆−𝑚}𝑝_𝑘−(𝑚+1)𝑝_{𝜆(𝑛)−𝑚}≤^𝑚_𝑘=1𝑘𝑝_{𝜆(𝑛)−𝑘+1}−𝑝_{𝜆(𝑛)−𝑘}. (37)
If 𝑚=1, ^𝜆(𝑛)_{(𝑛)𝑘=𝜆−1}𝑝_𝑘−2𝑝_{𝜆(𝑛)−1}=𝑝_𝜆(𝑛)−𝑝_{𝜆(𝑛)−1}. (38)
Thus (37) holds for 𝑚=1. Now let us assume that (37) is true for 𝑚=𝑗 and we verify 𝑚=𝑗+1(≤𝜆(𝑛)). Since ^𝜆(𝑛)_{(𝑛)−𝑘=𝜆𝑗+1}𝑝_𝑘−𝑗+2𝑝_{𝜆(𝑛)−(𝑗+1)}=^𝜆(𝑛)_{(𝑛)𝑘=𝜆−𝑗}𝑝_𝑘−𝑗+1𝑝_{𝜆(𝑛)−(𝑗+1)}=^𝜆(𝑛)_{𝑘=𝜆(𝑛)−𝑗}𝑝_𝑘−𝑗+1𝑝_{𝜆(𝑛)−𝑗}+𝑗+1𝑝_{𝜆(𝑛)−𝑗}−𝑗+1𝑝_{𝜆(𝑛)−(𝑗+1)}≤^𝜆(𝑛)_{(𝑛)𝑘=𝜆−𝑗}𝑝_𝑘−𝑗+1𝑝_{𝜆(𝑛)−𝑗}+𝑗+1𝑝_{𝜆(𝑛)−𝑗}−𝑗+1𝑝_{𝜆(𝑛)−(𝑗+1)}≤^𝑗_𝑘=1𝑘𝑝_{𝜆(𝑛)−𝑘+1}−𝑝_{𝜆(𝑛)−𝑘}+𝑗+1𝑝_{𝜆(𝑛)−𝑗}−𝑝_{𝜆(𝑛)−(𝑗+1)}=^𝑗+1_𝑘=1𝑘𝑝_{𝜆(𝑛)−𝑘+1}−𝑝_{𝜆(𝑛)−𝑘}, (39) thus (37) is proved for 𝑚=𝑗+1; that is, (37) is true for any 1≤𝑚≤𝜆(𝑛). Using (36) and (37) and interchanging the order of summation, we get ^𝜆(𝑛)_𝑚=1Δ_𝑚𝑃_𝜆(𝑛)−𝑃_{𝜆(𝑛)−𝑚}𝑚≤^𝜆(𝑛)_𝑚=11𝑚()×𝑚+1_𝑚𝑘=1𝑘𝑝_{𝜆(𝑛)−𝑘+1}−𝑝_{𝜆(𝑛)−𝑘}≤^𝜆(𝑛)_𝑘=1𝑘𝑝_{𝜆(𝑛)−𝑘+1}−𝑝_{𝜆(𝑛)−𝑘}×^∞_𝑚=𝑘1𝑚()=𝑚+1_{𝜆(𝑛)−1𝑘=0}Δ𝑝_𝑘. (40)
Now combining this with the assumption ∑^{𝜆(𝑛)−1}_𝑘=0|Δ𝑝_𝑘|=𝑂(𝑃_𝜆(𝑛)/𝜆(𝑛)), we get from (34) 𝑠^𝜆_𝑛(𝑓)−𝑁^𝜆_𝑛(𝑓)_𝑝=𝑂(𝜆(𝑛))⁻¹. (41)
This and Lemma 1 with 𝛼=1 yield 𝑓−𝑁^𝜆_𝑛(𝑓)_𝑝=𝑂(𝜆(𝑛))⁻¹. (42)
In the proof of case (iii), we first verify that the condition ∑^{𝜆(𝑛)−1}_𝑘=1𝑘|Δ𝑝_𝑘|=𝑂(𝑃_𝜆(𝑛)) implies that ^𝜆(𝑛)_𝑚=1Δ_𝑚𝑃_𝜆(𝑛)−𝑃_{𝜆(𝑛)−𝑚}𝑚𝑃=𝑂^𝜆(𝑛)𝜆(𝑛). (43)
In view of (36) and (37) ^𝜆(𝑛)_𝑚=1Δ_𝑚𝑃_𝜆(𝑛)−𝑃_{𝜆(𝑛)−𝑚}𝑚≤^𝜆(𝑛)_𝑚=11𝑚(𝑚+1)^𝑚_𝑘=1𝑘Δ_𝑘𝑝_{𝜆(𝑛)−𝑘}=^𝑟₁+^𝜆(𝑛)_𝑟+1∶=𝐵₁+𝐵₂,say. (44)
Denoting again by 𝑟 the integral part of (𝜆(𝑛)/2), then, by Abel’s transformation, we have 𝐵₁=^𝑟_𝑚=11𝑚(𝑚+1)^𝑚_𝑘=1𝑘Δ_𝑘𝑝_{𝜆(𝑛)−𝑘}≤^𝑟_𝑘=1Δ_𝑘𝑝_{𝜆(𝑛)−𝑘}≤^{𝜆(𝑛)−1}_{𝑖=𝑟−2}Δ𝑝_𝑖𝑃=𝑂^𝜆(𝑛)𝜆(𝑛), (45) at the last step; we have used the condition ∑^{𝜆(𝑛)−1}_𝑘=1𝑘|Δ𝑝_𝑘|=𝑂(𝑃_𝜆(𝑛)). Consider the following: 𝐵₂=^𝜆(𝑛)_𝑚=𝑟+11𝑚(𝑚+1)^𝑚_𝑘=1𝑘Δ𝑝_{𝜆(𝑛)−𝑘}=^𝜆(𝑛)_𝑚=𝑟+11𝑚()×𝑚+1_𝑟𝑘=1𝑘Δ𝑝_{𝜆(𝑛)−𝑘}+^𝑚_𝑘=𝑟𝑘Δ𝑝_{𝜆(𝑛)−𝑘}∶=𝐵₂₁+𝐵₂₂. (46)
Furthermore, using again our assumption, we get 𝐵₂₁≤^𝜆(𝑛)_𝑚=𝑟1𝑚+1^{𝜆(𝑛)−1}_{𝑖=𝑟−2}Δ𝑝_𝑖𝑃=𝑂^𝜆(𝑛)𝜆(𝑛),𝐵₂₂≤^𝜆(𝑛)_𝑚=𝑟1𝑚+1^𝑚_𝑘=𝑟Δ𝑝_{𝜆(𝑛)−𝑘}1=𝑂𝜆(𝑛)Δ𝑝₀+2Δ𝑝₁+⋯+(𝑟+1)Δ𝑝_𝑟+1𝑃=𝑂^𝜆(𝑛)𝜆(𝑛). (47)
Summing up our partial results, we verified (43). Thus (34) and Lemma 1 again yield 𝑓−𝑁^𝜆_𝑛(𝑓)_𝑝=𝑂(𝜆(𝑛))⁻¹. (48)
Now, we prove case (v), by using (26), 𝑝₋₁=0, and Abel’s transformation 𝑁^𝜆_𝑛(𝑥)=1𝑓;𝑥−𝑓𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=0𝑝_{𝜆(𝑛)−𝑚}𝑠_𝑚(𝑥=1𝑓;𝑥−𝑓)𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=0Δ_𝑚𝑝_{𝜆(𝑛)−𝑚}^𝑚_𝑘=0𝑠_𝑘(𝑥=1𝑓;𝑥−𝑓)𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=0(𝑚+1)Δ_𝑚𝑝_{𝜆(𝑛)−𝑚}×𝜎_𝑚𝑓;𝑥−𝑓(𝑥). (49)
Hence in view of Lemma 3𝑁^𝜆_𝑛𝑓−𝑓₁≤1𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=0(𝑚+1)Δ_𝑚𝑝_{𝜆(𝑛)−𝑚}𝑓−𝜎_𝑚(𝑓)₁1=𝑂𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=0(𝑚+1)^1−𝛼Δ_𝑚𝑝_{𝜆(𝑛)−𝑚}𝜆(𝑛)=𝑂^1−𝛼𝑃_𝜆(𝑛)^{𝜆(𝑛)−1}_𝑚=−1Δ𝑝_𝑚𝜆(𝑛)=𝑂^1−𝛼𝑃_𝜆(𝑛)𝑃⋅𝑂^𝜆(𝑛)𝜆(𝑛)=𝑂(𝜆(𝑛))^−𝛼. (50)
Herewith case (v) is also verified and thus the proof of Theorem 5 is complete.

Proof of Theorem 6. Since 𝑅^𝜆_𝑛(𝑓;𝑥)=(1/𝑃_𝜆(𝑛))∑^𝜆(𝑛)_𝑚=0𝑝_𝑚𝑠_𝑚(𝑓;𝑥), so, in view of the assumptions of Theorem 6, we get 𝑓−𝑅^𝜆_𝑛(𝑓)₁=1𝑃_𝜆(𝑛)^𝜆(𝑛)_𝑚=0𝑝_𝑚{𝑓−𝑠_𝑚(𝑓)}₁≤1𝑃_𝜆(𝑛)^{𝜆(𝑛)−1}_𝑚=0(𝑚+1)Δ𝑝_𝑚⋅𝑓−𝜎_𝑚(𝑓)₁+(𝜆(𝑛)+1)𝑝_𝜆(𝑛)𝑃⁻¹_𝜆(𝑛)𝑓−𝜎^𝜆_𝑛(𝑓)₁=𝑂(𝜆(𝑛))^1−𝛼𝑃_𝜆(𝑛)^{−1𝜆(𝑛)−1}_𝑚=0Δ𝑝_𝑚+𝑂(𝜆(𝑛))^−𝛼=𝑂(𝜆(𝑛))^−𝛼. (51)
This proves Theorem 6.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are thankful to the learned referee Dr. A. I. Zayed for his valuable comments and suggestions to improve the presentation of the paper.