Product (N, p _n ) (C, 1) summability of a sequence of Fourier coefficients

Abstract

Purpose

The purpose of the present paper is to study the product (N, p _n ) (C, 1) summability of a sequence of Fourier coefficients which extends a theorem of Prasad.

Methods

We use N _p . C ¹ summability methods with dropping monotonicity on the generating sequence {p_n − k} (that is, by weakening the conditions on the filter, we improve the quality of digital filter).

Results

Let B _n (x) denote the nth term of conjugate series of a Fourier series. Mohanty and Nanda were the first to establish a result for C₁ summability of the sequence {n B _n (x)}. Varshney improved the result for H₁. C₁ summability which was generalized by various investigators using different summability methods with different sets of conditions. In this paper, we extend a result of Prasad by dropping the monotonicity on the sequence {p_n − k}.

Conclusions

Various results pertaining to the C₁ and H₁. C₁ summabilities of the sequence {n B _n (x)} have been reviewed and the condition of monotonicity on the means generating the sequence {p_n − k} has been relaxed. Moreover, a proper set of conditions have been discussed to rectify the errors pointed out in Remark 3.2 (1) and (2).

Introduction

Let ${\displaystyle {\sum }_{n=0}^{\infty }{u}_n}$ be a given infinite series with sequence of its nth partial sums {s _n }. If the {p _n } be a nonnegative and nondecreasing, which generates sequences of constants, real or complex, let us write

$$\begin{array}{l}{P}_n={\displaystyle \sum _{k=0}^n{P}_k\ne 0\forall n\ge 0,P_{-1}=0=P_{-1}\text{and}}\\ P_n\to \mathit{\infty }\text{as}n\to \infty .\end{array}$$

The condition for regularity of Nörlund summability are easily seen to be

$$\underset{n\to \infty }{{\displaystyle \text{lim}}}\frac{P_n}{P_n}\to 0\mathit{\text{and}}$$

(1.)

$${\displaystyle {\sum }_{k=0}^{\infty }\left|p_k\right|}=O\left(P_n\right),\mathrm{\text{as}}n\to \infty .$$

(2.)

The sequence-to-sequence transformation

$$t_n^N=\frac{1}{P_n}{\displaystyle {\sum }_{k=0}^n{p}_{n-k}{s}_k.}$$

(1.1)

defines the sequence {t _n ^N} of Nörlund means of the sequence {s _n }, as generated by the sequence of coefficients {p _n }. The series ∑ _n = 0^∞u _n is said to be summable (N, p _n ) to the sum s if $\underset{n\to \infty }{\mathit{\text{lim}}}{t}_n^N$ exists and equal to s.

In the special case in which

$$p_n=\left(\begin{array}{c}\hfill n+\alpha -1\hfill \\ \hfill \alpha -1\hfill \end{array}\right)=\frac{Г\left(n+\alpha \right)}{Г\left(n+1\right)Г\left(\alpha \right)};\left(\alpha >-1\right)$$

the Nörlund summability (N, p _n ) reduces to the familiar (C, α) summability.

The product of N _p summability with a C¹summability defines N _p . C ¹ summability. Thus the N _p . C ¹ mean is given by ${t^{NC}}_n\left(x\right)=P_n^{-1}{\displaystyle {\sum }_{k=1}^n{p}_{n-k}{C}_k\left(x\right)}.$

If t _n ^NC → s as n → ∞ , then the infinite series ∑ _n = 0^∞u _n is said to be the summable N _p .C¹ to the sum s if $\underset{n\to \infty }{\mathit{\text{lim}}}{t_n}^{NC}$ exists and is equal to s.

Let f (x) be a 2π-periodic function and Lebesgue integrable. The Fourier series of f (x) at any point x is given by

$$f\left(x\right)\sim \frac{a_0}{2}+{\displaystyle {\sum }_{k=1}^{\infty }\left(a_k\text{cos}kx+b_k\mathit{sin}kx\right)}\equiv {\displaystyle {\sum }_{k=0}^{\infty }{A}_k\left(x\right)}.$$

(1.2)

With nth partial sum, s _n (f; x) is called trigonometric polynomial of degree (order) n of the first (n + 1) terms of the Fourier series of f.

The conjugate series of Fourier series (1.2) is given by

$${\displaystyle {\sum }_{k=1}^{\infty }\left(b_k\text{cos}kx-a_{k}sinkx\right)}\equiv {\displaystyle {\sum }_{k=1}^{\infty }{B}_k\left(x\right)}$$

(1.3)

The regularity conditions of N _p  . C¹ are as follows: n B  _n  → s ⇒ C ¹(n B  _n ) = t _n ^C = n^− 1 ∑ _k = 1ⁿk B _k  (x) → s, as n → ∞, C ¹ method is regular, ⇒ N _p  {C¹(nB _n )} = t _n ^NC = P _n ^− 1 ∑ _k = 1ⁿP_n −k(k^− 1 ∑ _r = 1^kr B _r (x)) → s,  as n → ∞,  N _p method is also regular, and ⇒ C¹.N _p method is regular. We note that t _n ^N and t _n ^NC are also trigonometric polynomials of degree (order) n.

Abel’s transformation

The formula

$${\displaystyle {\sum }_{k=m}^n{u}_k{v}_k}={\displaystyle {\sum }_{k=m}^{n-1}{U}_k\left(v_k-v_{k+1}\right)}-U_{m-1}{v}_m+U_n{v}_n,$$

(1.4)

where 0 ≤ m ≤ n, U _k  = u₀ + u₁ + u₂+.. + u _k , if k ≥ 0, U_− 1 = 0, which can be verified, is known as Abel’s transformation and will be used extensively in the succeeding discussion.

If v _m , v_m + 1, …, v _n are nonnegative and nonincreasing, the left hand side of (1.4) does not exceed $2v_m\underset{m-1\le k\le n}{\text{max}}\left|U_k\right|$ in the absolute value. In fact,

$$\begin{array}{l}\left|{\displaystyle {\sum }_{k=m}^n{u}_k{v}_k}\right|\le \text{max}\left|U_k\right|\left\{{\displaystyle {\sum }_{k=m}^{n-1}\left(v_k-v_{k+1}\right)+v_m+v_n}\right\}\\ =2v_m\text{max}\left|U_k\right|.\end{array}$$

Throughout in this paper, we use the following notations

$$\begin{array}{l}\psi \left(t\right)={\psi }_x\left(t\right)=f\left(x+t\right)-f\left(x-t\right)-l,\hfill \\ \Psi \left(t\right)={\displaystyle {\int }_0^t\left|\psi \left(u\right)\right|\mathit{\text{du}},}\hfill \\ \text{Q}\left(n,t\right)=\frac{1}{\pi P_n}{\displaystyle \sum _{k=1}^n{p}_{n-k}\left\{\frac{\mathit{\text{sin}}kt}{k{t}^2}-\frac{\mathit{\text{cos}}kt}{t}\right\}},\hfill \\ {\Delta }_k{p}_{n-k}=p_{n-k}-p_{n-k-1},0\le k\le n,\hfill \end{array}$$

and τ = [1/t] is the largest integer contained in 1/t, where l is a constant.

The (C, 1) and (H, 1) denotes the Cesàro and harmonic summabilities respectively of order one. The product summability (N, p _n ) (C, 1) is obtained by superimposing (N, p _n ) summability on (C, 1) summability, and the product summability (N, p _n ) (C, 1) plays an important role in signal theory as a double digital filter in finite impulse response in particular [1].

Methods

Known theorems

The theory of summability is a very extensive field. Mohanty and Nanda [2] proved the following theorem on C₁ summability of the sequence {n B _n (x)}.

Theorem 2.1 [[2]] If

$$\Psi \left(t\right)=o\left(t/\mathrm{\text{log}}\left(1/t\right)\right),\mathrm{\text{as}}t\to +0$$

(2.1)

and

$$a_n={\rm O}\left(n^{-\delta }\right);b_n={\rm O}\left(n^{-\delta }\right),\mathrm{\text{as}}t\to +0,$$

(2.2)

then the sequence {n B _n (x)} is the summable C ₁ to the value of l/π.

Varshney [3] improved Theorem 2.1 by extending it to product H₁. C₁ summability. He has proved that

Theorem 2.2 [[3]] if

$$\Psi \left(t\right)=o\left(t/\mathrm{\text{log}}\left(1/t\right)\right),\mathrm{\text{as}}t\to +0,$$

(2.3)

then the sequence {n B _n (x)} is the summable H₁. C₁to the value of l/π.

Various investigators such as Sharma [4], Rhoades [5] (cor. 19, p. 533), Pandey [6], Rai [7], Dwivedi [8], Mittal and Prasad [9], Prasad [10], Mittal [11], Chandra [12], Mittal et al. [13, 14], and Mittal and Singh [1] used different summability methods with different sets of conditions. In particular, Prasad [10] has proved the following:

Theorem 2.3 [[6]] Let p(u) be monotonically decreasing and strictly positive value with u ≥ 0. Let p _n = p(n) and

$$P\left(u\right)={\displaystyle {\int }_0^{u}p\left(x\right)\mathit{\text{dx}}}\to \infty ,\mathrm{\text{as}}u\to \infty .$$

(2.4)

Let α(t) be a positive and nondecreasing function of t. If

$$\Psi \left(t\right)=o\left(t/\alpha \left(1/t\right)\right),\mathrm{\text{as}}t\to +0,$$

(2.5)

then a sufficient condition that the sequence {n B _n (x)} be a summable N _P . C₁to the value of l/π is that

$${\displaystyle {\int }_1^n\frac{P\left(x\right)}{x\mathrm{\alpha }\left(x\right)}\mathit{\text{dx}}}={\rm O}\left(P\left(n\right)\right),\mathrm{\text{as}}n\to \infty .$$

(2.6)

Results and discussion

Main theorem

In the present paper, we extend Theorem 2.3 by dropping the monotonicity on the generating sequence {P_n − k} (that is, by weakening the conditions on the filter, we improve the quality of the digital filter). More precisely, we prove in Theorem 3.1:

Theorem 3.1 Let {p _k } be a nonnegative value such that

$$\text{(i.)}{\displaystyle {\sum }_{k=r}^n\left|{\Delta }_k{p}_{n-k}\right|}={\rm O}\left(p_{n-r}\right),\text{(ii.)}n{p}_n={\rm O}\left(p_n\right)\text{.}$$

(3.1)

Let α(t) be a positive and increasing function of t such that

$$\Psi \left(t\right)=o\left(t/\alpha (1/t)\right)\text{,}\text{as}t\to +0$$

(3.2)

and

$$\alpha \left(n\right)\to \infty ,\text{as}n\to \infty$$

(3.3)

then a sufficient condition for the sequence {nB _n (x)} to be as the summable (N, p _n ) (C, 1) to the value of l/π is

$${\displaystyle {\int }_{1/\delta }^n\frac{\mathrm{P}\left(x\right)}{x\mathrm{\alpha }\left(x\right)}\mathit{\text{dx}}=}{\rm O}\left(\mathrm{P}\left(n\right)\right),\mathrm{\text{as}}n\to \infty .$$

(3.4)

Remark 3.2 (1) If p_n − k ≤ p_{n − k − 1}, ∀ 0 ≤ k < n, as used in Theorem 2.3, then both the conditions (3.1) holds. Thus Theorem 3.1 extends Theorem 2.3. (2) In the proof of Theorem 2.3, author in [10] has used the condition (3.3) but did not mention in his statement.

Lemmas. For the proof of our Theorem 3.1, we require the following lemmas.

Lemma 4.1[10]If 0 ≤ t ≤ 1/n, then

$$\left|Q\left(n,t\right)\right|=0\left(n\right)$$

Lemma 4.2[15]For all values of n and t

$$\left|{\displaystyle {\sum }_{k=0}^n\frac{\mathrm{\text{sin}}\left(k+1\right)t}{k+1}}\right|\le 1+\frac{\pi }{2}.$$

(4.2)

Lemma 4.3 Under the regularity conditions of matrix (N, p _n ) in satisfying (3.1), we get Q(n, t) = Ο(t^− 1 P(τ)/P _n ) + Ο(t^− 2 p₁/P _n ),   for

$$1/n\le t\le \delta .$$

(4.3)

Proof We have$\begin{array}{l}\text{Q}\left(n,t\right)={\displaystyle {\sum }_{k=0}^{n-1}{p}_{n-k}/\pi P_n\left\{\frac{\mathrm{\text{sin}}\left(n-k\right)t}{\left(n-k\right)t^2}-\frac{\mathrm{\text{cos}}\left(n-k\right)t}{t}\right\}}\\ ={\mathrm{Q}}_1\left(n,t\right)+{\mathrm{Q}}_2\left(n,t\right),\text{as we say}\text{.}\end{array}$

By using Abel’s transformation, Lemma 4.2, and condition (3.1), we have

$$\begin{array}{l}\left|Q_1\left(n,t\right)\right|=\left|{\displaystyle {\sum }_{k=0}^{n-1}{p}_{n-k}/\pi P_n\frac{\mathrm{\text{sin}}\left(n-k\right)t}{\left(n-k\right)t^2}}\right|\\ \le \left|{\displaystyle {\sum }_{k=0}^{\tau -1}{p}_{n-k}/\pi P_n\frac{\mathrm{\text{sin}}\left(n-k\right)t}{\left(n-k\right)t^2}}\right|+\left|{\displaystyle {\sum }_{k=\tau }^{n-1}{p}_{n-k}/\pi P_n\frac{\mathrm{\text{sin}}\left(n-k\right)t}{\left(n-k\right)t^2}}\right|\end{array}$$

$$\begin{array}{l}\le \left[t^{-1}{\displaystyle {\sum }_{k=0}^{\tau -1}{p}_{n-k}\left|\frac{\mathrm{\text{sin}}\left(n-k\right)t}{\left(n-k\right)t}\right|}+t^{-2}\left|{\displaystyle {\sum }_{k=\tau }^{n-1}{p}_{n-k}\frac{\mathrm{\text{sin}}\left(n-k\right)t}{n-k}}\right|\right]/\pi P_n\\ \le \left[t^{-1}{\displaystyle {\sum }_{k=0}^{\tau -1}{p}_{n-k}}+t^{-2}\left|{\displaystyle {\sum }_{k=\tau }^{n-2}\left({\Delta }_k{p}_{n-k}{\displaystyle {\sum }_{r=0}^k\frac{\mathrm{\text{sin}}\left(n-r\right)t}{n-r}}\right)}\right|\right]/\pi P_n\\ +t^{-2}\left|p_{n-\tau }/\pi P_n{\displaystyle {\sum }_{k=0}^{\tau -1}\frac{\mathrm{\text{sin}}\left(n-k\right)t}{n-k}}\right|+t^{-2}\left|p_1/\pi P_n{\displaystyle {\sum }_{k=0}^{n-1}\frac{\mathrm{\text{sin}}\left(n-k\right)t}{n-k}}\right|\end{array}$$

$$\begin{array}{l}\le [t^{-1}{\displaystyle {\sum }_{k=0}^{\tau }{p}_{n-k}}+t^{-2}\left(1+\frac{\pi }{2}\right)\left({\displaystyle {\sum }_{k=\tau }^{n-2}\left|{\Delta }_k{p}_{n-k}\right|}+p_{n-\tau }+p_1\right)]/\pi P_n\\ ={\rm O}\left(t^{-1}\right)\left(\left(\mathrm{P}\left(\tau \right)+\left(\tau +1\right)p_{n-\tau }\right)/P_n\right)+{\rm O}\left(t^{-2}{p}_1/P_n\right)\\ ={\rm O}\left(t^{-1}\mathrm{P}\left(\tau \right)/P_n\right)+{\rm O}\left(t^{-2}{p}_1/P_n\right).\end{array}$$

Again by using Abel’s transformation and condition (3.1), we have

$$\begin{array}{l}{\mathrm{Q}}_2\left(n,t\right)={\displaystyle {\sum }_{k=0}^{n-1}{p}_{n-k}/\pi P_n\frac{\mathrm{\text{cos}}\left(n-k\right)t}{t}}\\ ={\rm O}\left(t^{-1}\right)\left[\mathrm{P}\left(\tau \right)+{\displaystyle {\sum }_{k=0}^{n-2}\left({\Delta }_k{p}_{n-k}\right)}{\displaystyle {\sum }_{r=0}^k\text{cos}\left(n-r\right)t}-p_{n-\tau }{\displaystyle {\sum }_{r=0}^{\tau -1}\text{cos}\left(n-r\right)t}\right]P_n^{-1}\\ +{\rm O}\left(t^{-1}\right)P_n^{-1}{p}_1{\displaystyle {\sum }_{r=0}^n\text{cos}\left(n-r\right)t}\end{array}$$

$$\begin{array}{l}\left|{\mathrm{Q}}_2\left(n,t\right)\right|={\rm O}\left(t^{-1}\right)\left[\mathrm{P}\left(\tau \right)+t^{-1}{\displaystyle {\sum }_{k=\tau }^{n-2}\left|{\Delta }_k{p}_{n-k}\right|}+t^{-1}{p}_{n-\tau }+t^{-1}{p}_1\right]/{\mathrm{P}}_n\\ ={\rm O}\left(t^{-1}\mathrm{P}\left(\tau \right)/P_n\right)+{\rm O}\left(t^{-2}{p}_1/P_n\right).\end{array}$$

By collecting Q₁(n, t), Q₂(n, t) and Q(n, t), we get

$$\mathrm{Q}\left(n,t\right)={\rm O}\left(t^{-1}\mathrm{P}\left(\tau \right)/P_n\right)+{\rm O}\left(t^{-2}{p}_1/P_n\right)\text{.}$$

This completes the proof of Lemma 4.3.

Proof of Theorem 3.1 The C₁ transform of the sequence {n B _n (x)} denoted by C _n (x) is defined by

$$C_n\left(x\right)=\frac{1}{n}{\displaystyle {\sum }_{k=1}^{n}k{\mathrm{B}}_k\left(x\right)}.$$

The N _p . C₁ transform of the sequence {n B _n (x)}, which is denoted by t^N C _n (x), is given by

$$\begin{array}{l}{t^{NC}}_n\left(x\right)=P_n^{-1}{\displaystyle {\sum }_{k=1}^n{p}_{n-k}{C}_k\left(x\right)}\\ =P_n^{-1}{\displaystyle {\sum }_{k=1}^n{p}_{n-k}\left(\frac{1}{k}{\displaystyle {\sum }_{r=1}^{k}r{\mathrm{B}}_r\left(x\right)}\right)}.\hfill \end{array}$$

Therefore, following Mohanty and Nanda [2], we obtain

$$\begin{array}{l}{t^{NC}}_n\left(x\right)-l/\pi =P_n^{-1}{\displaystyle {\sum }_{k=1}^n\left\{p_{n-k}\left(\frac{1}{k}{\displaystyle {\sum }_{r=1}^{k}r{\mathrm{B}}_r\left(x\right)}-l/\pi \right)\right\}}\\ =P_n^{-1}{\displaystyle \sum _{k=1}^n{p}_{n-k}\left\{\frac{1}{\pi }{\displaystyle {\int }_0^{\pi }\psi \left(t\right)\left(\frac{\mathrm{\text{sin}}\mathit{kt}}{k{t}^2}-\frac{\mathrm{\text{cos}}\mathit{kt}}{t}\right)\mathit{\text{dt}}+o\left(1\right)}\right\}}\\ =\frac{1}{\pi }{\displaystyle {\int }_0^{\pi }\psi \left(t\right)P_n^{-1}{\displaystyle {\sum }_{k=1}^n{p}_{n-k}\left(\frac{\mathrm{\text{sin}}\mathit{kt}}{k{t}^2}-\frac{\mathrm{\text{cos}}\mathit{kt}}{t}\right)\mathit{\text{dt}}}}+o\left(1\right)\\ =\frac{1}{\pi }\left({\displaystyle {\int }_0^{1/n}+}{\displaystyle {\int }_{1/n}^{\delta }+}{\displaystyle {\int }_{\delta }^{\pi }}\right)\psi \left(t\right)\text{Q}\left(n,t\right)dt+o\left(1\right),\mathrm{where}0<\delta <\pi \end{array}$$

$$=\frac{1}{\pi }\left(I_1+I_2+I_3\right)+o\left(1\right),$$

(5.1)

where

$$\begin{array}{l}{I}_1={\displaystyle {\int }_0^{1/n}}\psi \left(t\right)\mathrm{Q}\left(n,t\right)\mathit{\text{dt}}={\rm O}\left(n\right){\displaystyle {\int }_0^{1/n}}\left|\psi \left(t\right)\right|\mathit{\text{dt}}\\ ={\rm O}\left(n\right)\Psi \left(1/n\right)={\rm O}\left(n\right)o\left(1/n\mathrm{\alpha }\left(n\right)\right)\end{array}$$

$$=o\left(1/\mathrm{\alpha }\left(n\right)\right)=o\left(1\right),\mathrm{\text{as}}n\to \infty ,$$

(5.2)

in view of Lemma 4.1, conditions (3.2), and (3.3).

Using Lemma 4.3, we have

$$\begin{array}{l}{I}_2={\displaystyle {\int }_{1/n}^{\delta }}\psi \left(t\right)\mathrm{Q}\left(n,t\right)\mathit{\text{dt}}\\ ={\displaystyle {\int }_{1/n}^{\delta }}\left|\psi \left(t\right)\right|P_n^{-1}\left\{{\rm O}\left(t^{-2}{p}_1\right)+{\rm O}\left(t^{-1}\mathrm{P}\left(\tau \right)\right)\right\}\mathit{\text{dt}}\hfill \\ =I_{2,1}+I_{2,2}\text{as we say.}\hfill \end{array}$$

(5.3)

Now, using conditions (3.1-ii), (3.2), (3.3), and second mean value theorem for integrals, we have

$$\begin{array}{l}{I}_{2,1}={\rm O}\left(1\right){\displaystyle {\int }_{1/n}^{\delta }}{t}^{-2}{P}_n^{-1}{p}_1\left|\psi \left(t\right)\right|\mathit{\text{dt}}={\rm O}\left(P_n^{-1}{p}_1\right){\displaystyle {\int }_{1/n}^{\delta }}{t}^{-2}\left|\psi \left(t\right)\right|\mathit{\text{dt}}\\ ={\rm O}\left(\frac{1}{n}\right)\left\{{\left(t^{-2}\Psi \left(t\right)\right)}_{1/n}^{\delta }+{\displaystyle {\int }_{1/n}^{\delta }{t}^{-3}\Psi \left(t\right)\mathit{\text{dt}}}\right\}\\ =o\left(\frac{1}{n}\right){\left(\frac{1}{t\mathrm{\alpha }\left(1/t\right)}\right)}_{1/n}^{\delta }+o\left(\frac{1}{n}\right){\displaystyle {\int }_{1/n}^{\delta }\frac{\mathit{\text{dt}}}{t^2\mathrm{\alpha }\left(1/t\right)}}\\ =o\left(\frac{1}{n}\right)+o\left(\frac{1}{\mathrm{\alpha }\left(n\right)}\right)+o\left(\frac{1}{n\alpha \left(1/\delta \right)}\right){\displaystyle {\int }_{1/n}^{\delta }\frac{\mathit{\text{dt}}}{t^2}}\\ =o\left(\frac{1}{n}\right)+o\left(\frac{1}{\mathrm{\alpha }\left(n\right)}\right)+o\left(\frac{1}{n\alpha \left(1/\delta \right)}\right)(\delta -1/n)\end{array}$$

$$=o\left(1\right),\mathrm{\text{as}}n\to \infty .$$

(5.4)

Using conditions (3.1-ii), (3.2), (3.3), and (3.4), we have

$$\begin{array}{l}{I}_{2,2}={\rm O}\left(1\right){\displaystyle {\int }_{1/n}^{\delta }\frac{\left|\psi \left(t\right)\right|}{t}{P}_n^{-1}P\left(\tau \right)\mathit{\text{dt}}}={\rm O}\left(1\right){\left(\Psi \left(t\right)P_n^{-1}\frac{\mathrm{P}\left(\tau \right)}{t}\right)}_{1/n}^{\delta }\\ +{\rm O}\left(1\right){\displaystyle {\int }_{1/n}^{\delta }\Psi \left(t\right)P_n^{-1}\frac{\mathrm{P}\left(\tau \right)}{t^2}\mathit{\text{dt}}+{\rm O}\left(1\right){\displaystyle {\int }_{1/n}^{\delta }\frac{\Psi \left(t\right)}{t}d\left(P_n^{-1}\mathrm{P}\left(\tau \right)\right)}}\\ =o\left(1\right)+o\left(\frac{P_n^{-1}{P}_n}{\mathrm{\alpha }\left(n\right)}\right)+o\left(1\right){\displaystyle {\int }_{1/n}^{\delta }\frac{P_n^{-1}\mathrm{P}\left(\tau \right)}{t\mathrm{\alpha }\left(1/t\right)}\mathit{\text{dt}}}\\ +{\rm O}\left(P_n^{-1}\right){\displaystyle {\int }_{1/n}^{\delta }\Psi \left(t\right)d\left(\frac{\mathrm{P}\left(1/t\right)\alpha \left(1/t\right)}{t\alpha \left(1/t\right)}\right)}\end{array}$$

$$\begin{array}{l}=o\left(1\right)+o\left(\frac{P_n^{-1}{P}_n}{\mathrm{\alpha }\left(n\right)}\right)+o\left(1\right){\displaystyle {\int }_{1/n}^{\delta }\frac{P_n^{-1}\mathrm{P}\left(\tau \right)}{t\mathrm{\alpha }\left(1/t\right)}\mathit{\text{dt}}}\\ +{\rm O}\left(P_n^{-1}\right){\displaystyle {\int }_{1/n}^{\delta }\mathrm{o}\left(\frac{t}{\alpha \left(1/t\right)}\right)d\left(\frac{\mathrm{P}\left(1/t\right)}{t\alpha \left(1/t\right)}\right)\alpha \left(1/t\right)}\\ +{\rm O}\left(P_n^{-1}\right){\displaystyle {\int }_{1/n}^{\delta }\mathrm{o}\left(\frac{t}{\alpha \left(1/t\right)}\right)\frac{\mathrm{P}\left(1/t\right)}{t\alpha \left(1/t\right)}d\left(\alpha \left(1/t\right)\right)}\end{array}$$

$$\begin{array}{l}=o\left(1\right)+o\left(\frac{1}{\mathrm{\alpha }\left(n\right)}\right)+o\left(1\right){\displaystyle {\int }_{1/n}^{\delta }\frac{P_n^{-1}P\left(\tau \right)}{t\mathrm{\alpha }\left(1/t\right)}\mathit{\text{dt}}}\\ +\mathrm{o}\left(P_n^{-1}\right){\displaystyle {\int }_{1/n}^{\delta }td\left(\frac{\mathrm{P}\left(1/t\right)}{t\alpha \left(1/t\right)}\right)}+\mathrm{o}\left(1\right){\displaystyle {\int }_{1/n}^{\delta }\frac{d\left(\alpha \left(1/t\right)\right)}{{\left\{\alpha \left(1/t\right)\right\}}^2}}\end{array}$$

$$=o\left(1\right)+o\left(\frac{1}{\mathrm{\alpha }\left(n\right)}\right)+o\left(1\right){\displaystyle {\int }_{1/n}^{\delta }\frac{P_n^{-1}P\left(\tau \right)}{t\mathrm{\alpha }\left(1/t\right)}\mathit{\text{dt}}}$$

$$\begin{array}{l}+\text{o}\left(P_n^{-1}\right)\left\{{\left[\frac{tP\left(1/t\right)}{t\alpha \left(1/t\right)}\right]}_{1/n}^{\delta }-{\displaystyle \underset{1/n}{\overset{\delta }{\int }}\frac{\mathrm{P}\left(1/t\right)}{t\alpha \left(1/t\right)}\mathit{\text{dt}}}\right\}\\ +\text{o}\left(1\right){\left[-\frac{1}{\alpha \left(1/t\right)}\right]}_{1/n}^{\delta }\end{array}$$

$$\begin{array}{l}=o\left(1\right)+o\left(\frac{1}{\mathrm{\alpha }\left(n\right)}\right)+o\left(1\right){\displaystyle {\int }_{1/n}^{\delta }\frac{P_n^{-1}P\left(\tau \right)}{t\mathrm{\alpha }\left(1/t\right)}\mathit{\text{dt}}}\\ =o\left(1\right)+o\left(1\right){\displaystyle {\int }_{1/\delta }^n\frac{P_n^{-1}P\left(x\right)}{x\mathrm{\alpha }\left(x\right)}\mathit{\text{dx}}}\end{array}$$

$$=o\left(1\right)+o\left(1\right){\rm O}\left(P_n^{-1}P\left(n\right)\right)=o\left(1\right),\mathrm{\text{as}}n\to \infty .$$

(5.5)

On combining (5.3), (5.4) and (5.5), we get

$$I_2=o\left(1\right),\mathrm{\text{as}}n\to \infty$$

(5.6)

Finally, by Riemann-Lebesgue Theorem, we have

$$I_3={\displaystyle {\int }_{\delta }^{\pi }\psi \left(t\right)\mathrm{Q}\left(n,t\right)\mathit{\text{dt}}}=o\left(1\right),\mathrm{\text{as}}n\to \infty$$

(5.7)

By collecting (5.2), (5.6), and (5.7), we get

$${t^{NC}}_n\left(x\right)-l/\pi =o\left(1\right),\mathrm{\text{as}}n\to \infty .$$

This completes the proof of Theorem 3.1.

Conclusions

Various results pertaining to the C₁ and H₁C₁ summabilities of the sequence {n B _n (x)} have been reviewed, and the condition of monotonicity on the means of generating the sequence {p_n−k} has been relaxed. Moreover, a proper set of conditions have been discussed to rectify the errors pointed out in Remark 3.2 (1) and (2).