A study on degree of approximation by Karamata summability method

Abstract

Vuĉkoviĉ [Maths. Zeitchr. 89, 192 (1965)] and Kathal [Riv. Math. Univ. Parma, Italy 10, 33-38 (1969)] have studied summability of Fourier series by Karamata (K^λ) summability method. In present paper, for the first time, we study the degree of approximation of function f ∈ Lip (α,r) and f ∈ W(L _r ,ξ(t)) by K^λ-summability means of its Fourier series and conjugate of function $\tilde{f}\in \text{Lip}\left(\alpha ,r\right)$ and $\tilde{f}\in W\left(L_{r,}\xi \left(t\right)\right)$ by K^λ-summability means of its conjugate Fourier series and establish four quite new theorems.

MSC: primary 42B05; 42B08; 42A42; 42A30; 42A50.

1 Introduction

The method K^λwas first introduced by Karamata [1] and Lotosky [2] reintroduced the special case λ = 1. Only after the study of Agnew [3], an intensive study of these and similar cases took place. Vuĉkoviĉ [4] applied this method for summability of Fourier series. Kathal [5] extended the result of Vuĉkoviĉ [4]. Working in the same direction, Ojha [6], Tripathi and Lal [7] have studied K^λ-summability of Fourier series under different conditions. The degree of approximation of a function f ∈ Lip α by Cesàro and Nörlund means of the Fourier series has been studied by Alexits [8], Sahney and Goel [9], Chandra [10], Qureshi [11], Qureshi and Neha [12], Rhoades [13], etc. But nothing seems to have been done so far in the direction of present work. Therefore, in present paper, we establish two new theorems on degree of approximation of function f belonging to Lip (α,r) (r ≥ 1) and to weighted class W(L _r , ξ (t))(r ≥ 1) by K^λ-means on its Fourier series and two other new theorems on degree of approximation of function $\tilde{f}$, conjugate of a 2π-periodic function f belonging to Lip (α,r) (r > 1) and to weighted class W(L _r ,ξ (t)) (r ≥ 1) by K^λ-means on its conjugate Fourier series.

2 Definitions and notations

Let us define, for n = 0, 1, 2,..., the numbers $\left[\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right]$, for 0 ≤ m ≤ n, by

$$\begin{array}{ll}\hfill \prod _{v-0}^{n-1}\left(x+\nu \right)& =\sum _{m=0}^n\left[\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right]x^m=\frac{\mathrm{\Gamma }\left(x+n\right)}{\mathrm{\Gamma }\left(x\right)}\\ =x\left(x+1\right)\left(x+2\right)...\left(x+n-1\right).\end{array}$$

(2.1)

The numbers $\left[\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right]$ are known as the absolute value of stirling number of first kind

Let {s _n } be the sequence of partial sums of an infinite series ∑u _n , and let us write

$$s_n^{\lambda }=\frac{\Gamma (\lambda )}{\Gamma (\lambda +n)}{\displaystyle \sum _{m=0}^n\left[\begin{array}{c}n\\ m\end{array}\right]{\lambda }^m{s}_m}$$

(2.2)

to denote the n th K^λ-mean of order λ > 0. If $s_n^{\lambda }\to s$ as n → ∞, where s is a fixed finite number, then the sequence {s _n } or the series ∑u _n is said to be summable by Karamata method (K^λ) of order λ > 0 to the sum s, and we can write

$$s_n^{\lambda }\to s\left(K^{\lambda }\right)\mathsf{\text{as}}n\to \infty .$$

(2.3)

Let f be a 2π-periodic function and integrable in the sense of Lebesgue. The Fourier series associated with f at a point x is defined by

$$f\left(x\right)~\frac{a_0}{2}+\sum _{n=1}^{\infty }\left(a_{n}cosnx+b_{n}sinnx\right)\equiv \sum _{n=1}^{\infty }{A}_n\left(x\right)$$

(2.4)

with n th partial sums s _n (f;x).

The conjugate series of Fourier series (2.4) is given by

$$\sum _{n=1}^{\infty }\left(a_{n}sinnx-b_{n}cosnx\right)\equiv \sum _{n=1}^{\infty }{B}_n\left(x\right)$$

(2.5)

with n th partial sums ${\tilde{s}}_n\left(f;x\right)$.

Throughout this paper, we will call (2.5) as conjugate Fourier series of function f.

L_∞-norm of a function f: R → R is defined by

$$\parallel f{\parallel }_{\infty }=sup\left\{\mid f\left(x\right)\mid :x\in R\right\}$$

(2.6)

L _r -norm is defined by

$$\parallel f{\parallel }_r={\left({\int }_0^{2\pi }\mid f\left(x\right){\mid }^{r}dx\right)}^{\frac{1}{r}},r\ge 1.$$

(2.7)

The degree of approximation of a function f: R → R by a trigonometric polynomial t _n of degree n under sup norm || ||_∞ is defined by

(Zygmund [14])

$$\parallel t_n-f{\parallel }_{\infty }=sup\left\{\mid t_n\left(x\right)-f\left(x\right)\mid :x\in R\right\}$$

(2.8)

and E _n (f) of a function f ∈ L _r is given by

$$E_n\left(f\right)=\underset{t_n}{min}\parallel t_n-f{\parallel }_r.$$

(2.9)

This method of approximation is called trigonometric Fourier approximation. A function f ∈ Lip α if

$$\mid f\left(x+t\right)-f\left(x\right)\mid =O\left(\mid t{\mid }^{\alpha }\right)\mathsf{\text{for}}0<\alpha \le 1$$

(2.10)

and

f ∈ Lip (α, r) for 0 ≤ x ≤ 2π, if

$${\left({\int }_0^{2\pi }\mid f\left(x+t\right)-f\left(x\right){\mid }^{r}dx\right)}^{\frac{1}{r}}=O\left(\mid t{\mid }^{\alpha }\right),0<\alpha \le 1\mathsf{\text{ and }}r\ge 1$$

(2.11)

(definition 5.38 of McFadden [15]).

Given a positive increasing function ξ (t) and an integer r ≥ 1, f ∈ Lip (ξ(t), r), if

$${\left({\int }_0^{2\pi }\mid f\left(x+t\right)-f\left(x\right){\mid }^{r}dx\right)}^{\frac{1}{r}}=O\left(\xi \left(t\right)\right)$$

(2.12)

and that

f ∈ W (L _r , ξ (t)) if

$${\left({\int }_0^{2\pi }\mid \left\{f\left(x+t\right)-f\left(x\right)\right\}{sin}^{\beta }x{\mid }^{r}dx\right)}^{\frac{1}{r}}=O\left(\mid \xi \left(t\right)\right),\beta \ge 0$$

(2.13)

If β = 0, our newly defined weighted i.e. W (L _r , ξ (t)) reduces to Lip (ξ (t), r), if ξ (t) = t^αthen Lip (ξ (t), r) coincides with Lip (α, r) and if r → ∞ then Lip (α, r) reduces to Lip α.

We observe that

$$\mathsf{\text{Lip }}\alpha \subseteq \mathsf{\text{Lip(}}\alpha \mathsf{\text{,}}r\mathsf{\text{)}}\subseteq \mathsf{\text{Lip}}\left(\xi \left(t\right),r\right)\subseteq W\left(L_r,\xi \left(t\right)\right)\mathsf{\text{for}}0<\alpha \le 1,r\ge 1.$$

We write

$$\begin{array}{c}\varphi \left(t\right)=f\left(x+t\right)+f\left(x-t\right)-2f\left(x\right)\\ K_n\left(t\right)=\frac{{\sum }_{m=0}^n\left[\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right]{\lambda }^{m}sin\left(m+\frac{1}{2}\right)t}{\mathrm{\Gamma }\left(\lambda +n\right)sin\left(\frac{t}{2}\right)}\\ \psi \left(t\right)=f\left(x+t\right)-f\left(x-t\right)\\ {\tilde{K}}_n\left(t\right)=\frac{{\sum }_{m=0}^n\left[\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right]{\lambda }^{m}cos\left(m+\frac{1}{2}\right)t}{\mathrm{\Gamma }\left(\lambda +n\right)sin\left(\frac{t}{2}\right)}\\ \tilde{f}\left(x\right)=-\frac{1}{2\pi }{\int }_0^{\pi }\psi \left(t\right)cot\left(\frac{t}{2}\right)dt\end{array}$$

3 The main results

3.1 Theorem 1

If a function f, 2π-periodic, belonging to Lip (α, r) then its degree of approximation by K^λ-summability means on its Fourier series is given by

$$\begin{array}{c}\parallel s_n-f{\parallel }_r=O\left[\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\left\{\frac{log\left(n+1\right)e}{\left(n+1\right)}\right\}+\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right],\\ 0<\alpha \le 1,n=0,1,2...,\end{array}$$

(3.1)

where s _n is K^λ-mean of Fourier series (2.4).

3.2 Theorem 2

If a function f, 2π-periodic, belonging to W (L _r , ξ (t)) then its degree of approximation by K^λ-summability means on its Fourier series is given by

$$\parallel s_n-f{\parallel }_r=O\left[\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\xi \left(\frac{1}{n+1}\right)\right\}\left\{\frac{log\left(n+1\right)}{\left(n+1\right)}+\frac{1}{{\left(n+1\right)}^2}+\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right\}\right]$$

(3.2)

provided that ξ (t) satisfies the following conditions:

$$\left\{\frac{\xi \left(t\right)}{t}\right\}\mathsf{\text{is non - increasing }}int,$$

(3.3)

$${\left\{{\int }_0^{\frac{1}{n+1}}{\left(\frac{t\mid \varphi \left(t\right)\mid }{\xi \left(t\right)}\right)}^r{sin}^{\beta r}tdt\right\}}^{\frac{1}{r}}=O\left(\frac{1}{n+1}\right),$$

(3.4)

and

$${\left\{{\int }_{\frac{1}{n+1}}^{\pi }{\left(\frac{t^{-\delta }\mid \varphi \left(t\right)\mid }{\xi \left(t\right)}\right)}^{r}dt\right\}}^{\frac{1}{r}}=O\left({\left(n+1\right)}^{\delta }\right),$$

(3.5)

where δ is an arbitrary positive number such that s (1 - δ) - 1 > 0, $\frac{1}{r}+\frac{1}{s}=1$, 1 ≤ r ≤ ∞, conditions (3.4) and (3.5) hold uniformly in x, s _n is K^λ-mean of Fourier series (2.4).

3.3 Theorem 3

If a function $\tilde{f}$, conjugate to a 2π-periodic function f, belonging to Lip(α,r) then its degree of approximation by K^λ-summability means on its conjugate Fourier series is given by

$$\begin{array}{c}\parallel {\tilde{s}}_n-\tilde{f}{\parallel }_r=O\left[\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\left\{\frac{log\left(n+1\right)e}{{\left(n+1\right)}^2}\right\}+\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}+1\right],\\ 0<\alpha \le 1,n=0,1,2,...,\end{array}$$

(3.6)

where ${\tilde{s}}_n$ is K^λ-mean of conjugate Fourier series (2.5) and

$$\tilde{f}\left(x\right)=-\frac{1}{2\pi }{\int }_0^{\pi }\psi \left(t\right)cot\left(\frac{t}{2}\right)dt.$$

3.4 Theorem 4

If a function $\tilde{f}$, conjugate to a 2π-periodic function f, belonging to W (L _r ,ξ (t)) then its degree of approximation by K^λ-summability means on its conjugate Fourier series is given by

$$\parallel {\tilde{s}}_n-\tilde{f}{\parallel }_r=O\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\xi \left(\frac{1}{n+1}\right)\right\}\left[\frac{2}{{\left(n+1\right)}^2}+\frac{log\left(n+1\right)}{{\left(n+1\right)}^2}+\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right]$$

(3.7)

provided that ξ (t) satisfies the conditions (3.3)-(3.5) in which δ is an arbitrary positive number such that s (1 - δ) - 1 > 0, $\frac{1}{r}+\frac{1}{s}=1$,1≤r≤ ∞. Conditions (3.4) and (3.5) hold uniformly in x, ${\tilde{s}}_n$ is K^λ-mean of conjugate Fourier series (2.5) and

$$\tilde{f}\left(x\right)=-\frac{1}{2\pi }{\int }_0^{\pi }\psi \left(t\right)cot\left(\frac{t}{2}\right)dt.$$

(3.8)

4 Lemmas

For the proof of our theorems, following lemmas are required.

4.1 Lemma 1

(Vuĉkoviĉ [14]). Let λ > 0 and $0<t<\frac{\pi }{2}$, then

$$\frac{\mathrm{\Gamma }\left(\lambda e^{it}+n\right)}{\mathrm{\Gamma }\left(\lambda cost+n\right)sin\left(\frac{t}{2}\right)}=\frac{\mid sin\left(\lambda log\left(n+1\right).sint\right)\mid }{sin\left(\frac{t}{2}\right)}+O\left(1\right)n\to \infty \mathsf{\text{uniformly}}\mathsf{\text{in}}t.$$

4.2 Lemma 2

$$K_n\left(t\right)=O\left\{\lambda log\left(n+1\right)\right\}+O\left(1\right).$$

Proof. For $0<t<\frac{1}{n+1},1-cost<\frac{t^2}{2}$, sin nt ≤ nt and $sin\frac{t}{2}\ge \frac{t}{\pi }$

$$\begin{array}{ll}\hfill \left|K_n\left(t\right)\right|& \le \left|\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\sum _{m=0}^n\left[\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right]\cdot {\lambda }^m\frac{sin\left(m+\frac{1}{2}\right)t}{sin\frac{t}{2}}\right|\\ =O\left[\frac{Im\left\{e^{\frac{it}{2}}\frac{\mathrm{\Gamma }\left(\lambda e^{it}+n\right)}{\mathrm{\Gamma }\left(\lambda e^{it}\right)}\right\}}{\mathrm{\Gamma }\left(\lambda +n\right)sin\frac{t}{2}}\right]\mathsf{\text{by (2.1)}}\\ =O\left[\frac{Im\mathrm{\Gamma }\left(\lambda e^{it}+n\right)}{\mathrm{\Gamma }\left(\lambda +n\right)sin\frac{t}{2}}\right]+O\left[\frac{Re\mathrm{\Gamma }\left(\lambda e^{it}+n\right)}{\mathrm{\Gamma }\left(\lambda +n\right)}\right]\\ =O\left[\frac{\mathrm{\Gamma }\left(\lambda cost+n\right)}{\mathrm{\Gamma }\left(\lambda +n\right)}\cdot \frac{Im\mathrm{\Gamma }\left(\lambda e^{it}+n\right)}{\mathrm{\Gamma }\left(\lambda cost+n\right)sin\frac{t}{2}}\right]+O\left[\frac{\mathrm{\Gamma }\left(\lambda cost+n\right)}{\mathrm{\Gamma }\left(\lambda +n\right)}\right]\\ =O\left[n^{-\lambda \left(1-cost\right)}\cdot \frac{Im\mathrm{\Gamma }\left(\lambda e^{it}+n\right)}{\mathrm{\Gamma }\left(\lambda cost+n\right)sin\frac{t}{2}}\right]+O\left[n^{-\lambda \left(1-cost\right)}\right]\\ =O\left[e^{-\lambda \left(1-cost\right)logn}\cdot \frac{Im\mathrm{\Gamma }\left(\lambda e^{it}+n\right)}{\mathrm{\Gamma }\left(\lambda cost+n\right)sin\frac{t}{2}}\right]+O\left[e^{-\lambda \left(1-cost\right)logn}\right]\\ =O\left[e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\cdot \frac{Im\mathrm{\Gamma }\left(\lambda e^{it}+n\right)}{\mathrm{\Gamma }\left(\lambda cost+n\right)sin\frac{t}{2}}\right]+O\left[e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\right].\end{array}$$

(4.1)

Considering first part of (4.1) and using Lemma 1,

$$\begin{array}{ll}\hfill K_n\left(t\right)& =O\left[e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\cdot \frac{\mid sin\left(\lambda log\left(n+1\right)\cdot sint\right)\mid }{sin\left(\frac{t}{2}\right)}\right]+O\left[e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\right]\\ +O\left[e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\right]\\ =O\left[e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\cdot \frac{\mid sin\left(\lambda log\left(n+1\right)\cdot sint\right)\mid }{sin\left(\frac{t}{2}\right)}\right]+O\left[e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\right]\\ =O\left\{\lambda log\left(n+1\right)\right\}\left[\frac{\mid sin\left(\lambda log\left(n+1\right)\cdot sint\right)\mid }{sin\left(\frac{t}{2}\right)}\right]+O\left(1\right)\\ =O\left\{\lambda log\left(n+1\right)\right\}+O\left(1\right).\end{array}$$

4.3 Lemma 3

$$\begin{array}{ll}\hfill {\tilde{K}}_n\left(t\right)=& O\left[\frac{e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}}{t}\right]+O\left\{\lambda log\left(n+1\right)\right\}\mid sin\left(\lambda log\left(n+1\right)\cdot sint\right)\mid \\ +O\left[e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\cdot \mid sin\left(t∕2\right)\mid \right]\cdot \end{array}$$

Proof. For $0<t<\frac{1}{n+1},1-cost<\frac{t^2}{2}$, sin nt ≤ nt and $sin\frac{t}{2}\ge \frac{t}{\pi }$

$$\begin{array}{ll}\hfill \left|K_n\left(t\right)\right|& \le \left|\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\sum _{m=0}^n\left[\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right]\cdot {\lambda }^m\frac{cos\left(m+\frac{1}{2}\right)t}{sin\frac{t}{2}}\right|\\ =O\left[\frac{Re\left\{e^{\frac{it}{2}}\frac{\mathrm{\Gamma }\left(\lambda e^{it}+n\right)}{\mathrm{\Gamma }\left(\lambda e^{it}\right)}\right\}}{\mathrm{\Gamma }\left(\lambda +n\right)sin\frac{t}{2}}\right]\mathsf{\text{by (2.1)}}\\ =O\left[\frac{Re\mathrm{\Gamma }\left(\lambda e^{it}+n\right)}{\mathrm{\Gamma }\left(\lambda +n\right)sin\frac{t}{2}}\right]+O\left[\frac{Im\mathrm{\Gamma }\left(\lambda e^{it}+n\right)}{\mathrm{\Gamma }\left(\lambda +n\right)}\right]\\ =O\left[\frac{\mathrm{\Gamma }\left(\lambda cost+n\right)}{\mathrm{\Gamma }\left(\lambda +n\right)sin\frac{t}{2}}\right]+O\left[\frac{\mathrm{\Gamma }\left(\lambda cost+n\right)}{\mathrm{\Gamma }\left(\lambda +n\right)}\cdot \frac{Im\mathrm{\Gamma }\left(\lambda e^{it}+n\right)}{\mathrm{\Gamma }\left(\lambda cost+n\right)}\right]\\ =O\left[\frac{n^{-\lambda \left(1-cost\right)}}{sin\frac{t}{2}}\right]+O\left[n^{-\lambda \left(1-cost\right)}\cdot \frac{Im\mathrm{\Gamma }\left(\lambda e^{it}+n\right)}{\mathrm{\Gamma }\left(\lambda cost+n\right)}\right]\\ =O\left[\frac{e^{-\lambda \left(1-cost\right)logn}}{sin\frac{t}{2}}\right]+O\left[e^{-\lambda \left(1-cost\right)logn}\cdot \frac{Im\mathrm{\Gamma }\left(\lambda e^{it}+n\right)}{\mathrm{\Gamma }\left(\lambda cost+n\right)}\right]\\ =O\left[\frac{e^{-\frac{\lambda }{2}{t}^{2}logn}}{sin\frac{t}{2}}\right]+O\left[e^{-\frac{\lambda }{2}{t}^{2}logn}\cdot \frac{Im\mathrm{\Gamma }\left(\lambda e^{it}+n\right)}{\mathrm{\Gamma }\left(\lambda cost+n\right)}\right]\\ =O\left[\frac{e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}}{t}\right]+O\left[e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\cdot \frac{Im\mathrm{\Gamma }\left(\lambda e^{it}+n\right)}{\mathrm{\Gamma }\left(\lambda cost+n\right)}\right].\end{array}$$

Using Lemma 1,

$$\begin{array}{ll}\hfill K_n\left(t\right)& =O\left[\frac{e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}}{t}\right]+O\left[e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\cdot \mid sin\left(\lambda log\left(n+1\right)\cdot sint\right)\mid \right]\\ +O\left[e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\cdot \mid sin\left(t∕2\right)\mid \right]\\ =O\left[\frac{e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}}{t}\right]+O\left\{\lambda log\left(n+1\right)\right\}\mid sin\left(\lambda log\left(n+1\right)\cdot sint\right)\mid \\ +O\left[e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\cdot \mid sin\left(t∕2\right)\mid \right].\end{array}$$

□

4.4 Lemma 4

(McFadden [15]), Lemma 5.40) If f(x) belongs to Lip(α,r) on [0,π], then φ(t) belongs to Lip(α,r) on [0,π].

5 Proof of the theorems

5.1 Proof of Theorem 1

Following Titchmarsh [16] and using Riemann-Lebesgue theorem, the m th partial sum s _m (x) of series (2.4) at t = x is given by

$$s_m\left(x\right)-f\left(x\right)=\frac{1}{2\pi }{\int }_0^{\pi }\varphi \left(t\right)\frac{sin\left(m+\frac{1}{2}\right)t}{sin\frac{t}{2}}dt$$

Therefore,

$$\begin{array}{c}\frac{\Gamma \left(\lambda \right)}{\Gamma \left(\lambda +n\right)}{\displaystyle \sum _{m=0}^n\left[\begin{array}{c}n\\ m\end{array}\right]{\lambda }^m\left\{s_m\left(x\right)-f\left(x\right)\right\}}=\frac{1}{2\pi }{\displaystyle {\int }_0^{\pi }\varphi \left(t\right)\frac{\Gamma \left(\lambda \right)}{\Gamma \left(\lambda +n\right)}}{\displaystyle \sum _{m+0}^n\left[\begin{array}{c}n\\ m\end{array}\right]}\\ \cdot {\lambda }^m\frac{\sin \left(m+\frac{1}{2}\right)t}{\sin \frac{t}{2}}dt\\ s_m\left(x\right)-f\left(x\right)=\frac{\Gamma \left(\lambda \right)}{2\pi }{\displaystyle {\int }_0^{\pi }\varphi \left(t\right)}{K}_n\left(t\right)dt\\ =\frac{\Gamma \left(\lambda \right)}{2\pi }\left[{{\displaystyle \int }}_0^{\frac{1}{n+1}}+{{\displaystyle \int }}_{\frac{1}{n+1}}^{\pi }\right]\varphi \left(t\right)K_n\left(t\right)dt\\ =O\left(I_{1.1}\right)+O\left(I_{1.2}\right)\left(\text{say}\right).\end{array}$$

(5.1)

Now we consider,

$$I_{1.1}={\int }_0^{\frac{1}{n+1}}\mid \varphi \left(t\right)\parallel K_n\left(t\right)\mid dt.$$

Using Lemma 2,

$$I_{1.1}=O\left\{\lambda log\left(n+1\right)\right\}{\int }_0^{\frac{1}{n+1}}\mid \varphi \left(t\right)\mid dt+O\left[{\int }_0^{\frac{1}{n+1}}\mid \varphi \left(t\right)\mid dt\right].$$

Using Hölder's inequality and Lemma 4,

$$\begin{array}{ll}\hfill I_{1.1}& =O\left[\left\{\lambda log\left(n+1\right)\right\}+1\right]{\left[{{\int }_0^{\frac{1}{n+1}}\left\{\frac{t\varphi \left(t\right)}{t^{\alpha }}\right\}}^{r}dt\right]}^{\frac{1}{r}}{\left[{\int }_0^{\frac{1}{n+1}}{\left(t^{\alpha -1}\right)}^{s}dt\right]}^{\frac{1}{s}}\\ =O\left[\left\{\lambda log\left(n+1\right)\right\}+1\right]\left(\frac{1}{n+1}\right){\left[{\left\{\frac{t^{\alpha s-s+1}}{\alpha s-s+1}\right\}}_0^{\frac{1}{n+1}}\right]}^{\frac{1}{s}}\\ =O\left\{\frac{log\left(n+1\right)e}{\left(n+1\right)}\right\}{\left[\frac{1}{{\left(n+1\right)}^{\alpha s-s+1}}\right]}^{\frac{1}{s}}\\ =O\left[\left\{\frac{log\left(n+1\right)e}{\left(n+1\right)}\right\}\frac{1}{{\left(n+1\right)}^{\alpha -1+\frac{1}{s}}}\right]\\ =O\left[\left\{\frac{log\left(n+1\right)e}{\left(n+1\right)}\right\}\left\{\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\right\}\right]\mathsf{\text{since}}\frac{1}{r}+\frac{1}{s}=1.\end{array}$$

(5.2)

Since, for $\frac{1}{n+1}\le t\le \pi ,sin\frac{t}{2}\ge \frac{t}{\pi }$

$$\begin{array}{ll}\hfill K_n\left(t\right)& =O\left[\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)sin\frac{t}{2}}\right]\\ =O\left[\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)t}\right].\end{array}$$

(5.3)

Next we consider,

$$\mid I_{1.2}\mid \le \underset{\frac{1}{n+1}}{\overset{\pi }{\int }}\mid \varphi \left(t\right)\mid \mid K_n\left(t\right)\mid dt.$$

Using Hölder's inequality, (5.3) and Lemma 4,

$$\begin{array}{ll}\hfill I_{1.2}& =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right){\left[{\int }_{\frac{1}{n+1}}^{\pi }{\left\{\frac{t^{-\delta }\varphi \left(t\right)}{t^{\alpha }}\right\}}^{r}dt\right]}^{\frac{1}{r}}{\left[\underset{\frac{1}{n+1}}{\overset{\pi }{\int }}{\left(\frac{t^{\delta +\alpha }}{t}\right)}^{s}dt\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right)\frac{1}{{\left(n+1\right)}^{-\delta }}{\left[{\int }_{\frac{1}{n+1}}^{\pi }{t}^{\left(\delta +\alpha -1\right)s}dt\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right)\frac{1}{{\left(n+1\right)}^{-\delta }}{\left[{\left\{\frac{t^{\left(\delta +\alpha -1\right)s+1}}{\left(\delta +\alpha -1\right)s+1}\right\}}_{\frac{1}{n+1}}^{\pi }\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right)\frac{1}{{\left(n+1\right)}^{-\delta }}{\left[\frac{1}{{\left(n+1\right)}^{\left(\delta +\alpha -1\right)s+1}}\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right)\frac{1}{{\left(n+1\right)}^{-\delta }}\left[\frac{1}{{\left(n+1\right)}^{\left(\delta +\alpha -1\right)+\frac{1}{s}}}\right]\\ =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right)\left[\frac{1}{{\left(n+1\right)}^{\alpha -1+\frac{1}{s}}}\right]\\ =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right)\left[\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\right].\end{array}$$

(5.4)

Combining (5.1), (5.2) and (5.4),

$$\begin{array}{ll}\hfill S_m-f\left(x\right)& =O\left[\left(\frac{log\left(n+1\right)e}{\left(n+1\right)}\right)\left(\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\right)\right]+O\left[\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right)\left(\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\right)\right]\\ =O\left[\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\left\{\frac{log\left(n+1\right)e}{\left(n+1\right)}+\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right\}\right].\end{array}$$

This completes the proof of Theorem 1.

5.2 Proof of Theorem 2

Following the proof of Theorem 1,

$$\begin{array}{ll}\hfill S_m\left(x\right)-f\left(x\right)& =\frac{\mathrm{\Gamma }\left(\lambda \right)}{2\pi }\left[{\int }_0^{\frac{1}{n+1}}+{\int }_{\frac{1}{n+1}}^{\pi }\right]\varphi \left(t\right)K_n\left(t\right)dt\\ =O\left(I_{2.1}\right)+O\left(I_{2.2}\right)\left(\mathsf{\text{say}}\right).\end{array}$$

(5.5)

We have

$$\mid \varphi \left(x+t\right)-\varphi \left(x\right)\mid \le \mid f\left(u+x+t\right)-f\left(u+x\right)\mid +\mid f\left(u-x-t\right)-f\left(u-x\right)\mid .$$

Hence, by Minkowiski's inequality,

$$\begin{array}{ll}\hfill {\left[{\int }_0^{2\pi }\left\{\mid \varphi \left(x+t\right)-\varphi \left(x\right)\right\}{sin}^{\beta }x{\mid }^{r}dx\right]}^{\frac{1}{r}}& \le {\left[{\int }_0^{2\pi }\mid \left\{f\left(u+x+t\right)-f\left(u+x\right)\right\}si{n}^{\beta }x{\mid }^{r}dx\right]}^{\frac{1}{r}}\\ +{\left[{\int }_0^{2\pi }\mid \left\{f\left(u-x-t\right)-f\left(u-x\right)\right\}{sin}^{\beta }x{\mid }^{r}dx\right]}^{\frac{1}{r}}\\ =O\left\{\xi \left(t\right)\right\}.\end{array}$$

Then f ∈ W (L _r ,ξ(t))⇒ φ ∈ W(L _r , ξ (t)).

Now we consider,

$$\mid I_{2.1}\mid \le {\int }_0^{\frac{1}{n+1}}\mid \varphi \left(t\right)\mid \mid K_n\left(t\right)\mid dt.$$

Using Lemma 2,

$$I_{2.1}=\left[O\left\{\lambda log\left(n+1\right)\right\}+O\left(1\right)\right]{\int }_0^{\frac{1}{n+1}}\mid \varphi \left(t\right)\mid dt.$$

Using Hölder's inequality and the fact that φ (t) ∈ W (L _r , ξ (t)),

$$\begin{array}{c}{I}_{2.1}=O\left[\left\{\lambda \log \left(n+1\right)\right\}+1\right]{\left[{\displaystyle {\int }_0^{\frac{1}{n+1}}{\left\{\frac{t\left|\varphi \left(t\right)\right|{\sin }^{\beta }\left(t\right)}{\xi \left(t\right)}\right\}}^{r}dt}\right]}^{\frac{1}{r}}\\ \cdot {\left[{\displaystyle {\int }_0^{\frac{1}{n+1}}{\left\{\frac{\xi \left(t\right)}{t{\sin }^{\beta }t}\right\}}^{s}dt}\right]}^{\frac{1}{s}}\\ =O\left\{\lambda \log \left(n+1\right)e\right\}\left(\frac{1}{n+1}\right){\left[{\displaystyle {\int }_0^{\frac{1}{n+1}}{\left\{\frac{\xi \left(t\right)}{t{\sin }^{\beta }t}\right\}}^{s}dt}\right]}^{\frac{1}{s}}\text{by}(3.4)\end{array}$$

Since sin t ≥ 2t/π,

$$I_{2.1}=O\left(\frac{log\left(n+1\right)e}{n+1}\right){\left[{\int }_0^{\frac{1}{n+1}}{\left\{\frac{\xi \left(t\right)}{t^{1+\beta }}\right\}}^{s}dt\right]}^{\frac{1}{s}}.$$

Since ξ (t) is a positive increasing function and using second mean value theorem for integrals,

$$\begin{array}{ll}\hfill I_{2.1}& =O\left\{\left(\frac{log\left(n+1\right)e}{n+1}\right)\xi \left(\frac{1}{n+1}\right)\right\}{\left[{\int }_{\in }^{\frac{1}{n+1}}\left\{\frac{1}{t^{\left(1+\beta \right)s}}\right\}dt\right]}^{\frac{1}{s}}\mathsf{\text{for}}\mathsf{\text{some}}0<\in <\frac{1}{n+1}\\ =O\left\{\left(\frac{log\left(n+1\right)e}{n+1}\right)\xi \left(\frac{1}{n+1}\right)\right\}{\left[{\left\{\frac{t^{-\left(1+\beta \right)s+1}}{-\left(1+\beta \right)s+1}\right\}}_{\in }^{\frac{1}{n+1}}\right]}^{\frac{1}{s}}\\ =O\left[\left\{\left(\frac{log\left(n+1\right)e}{n+1}\right)\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\left(1+\beta \right)-\frac{1}{s}}\right\}\right]\\ =O\left[\left\{\left(\frac{log\left(n+1\right)e}{n+1}\right)\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\right\}\right]\mathsf{\text{since}}\frac{1}{r}+\frac{1}{s}=1.\end{array}$$

(5.6)

Next we consider,

$$\mid I_{2.2}\mid \le {\int }_{\frac{1}{n+1}}^{\pi }\mid \varphi \left(t\right)\mid \mid K_n\left(t\right)\mid dt.$$

Using Hölder's inequality, |sin t| ≤ 1,sin t ≥ 2t/π, (5.3), conditions (3.3), (3.5) and second mean value theorem for integrals,

$$\begin{array}{ll}\hfill I_{2.2}& =O\left[{\int }_{\frac{1}{n+1}}^{\pi }\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)t}\mid \varphi \left(t\right)\mid dt\right]\\ =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right){\left[{\int }_{\frac{1}{n+1}}^{\pi }{\left\{\frac{t^{-\delta }\mid \varphi \left(t\right)\mid {sin}^{\beta }\left(t\right)}{\xi \left(t\right)}\right\}}^{r}dt\right]}^{\frac{1}{r}}{\left[{\int }_{\frac{1}{n+1}}^{\pi }{\left\{\frac{\xi \left(t\right)}{t^{-\delta }{sin}^{\beta }tt}\right\}}^{s}dt\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right){\left[{\int }_{\frac{1}{n+1}}^{\pi }{\left\{\frac{t^{-\delta }\mid \varphi \left(t\right)\mid }{\xi \left(t\right)}\right\}}^{r}dt\right]}^{\frac{1}{r}}{\left[{\int }_{\frac{1}{n+1}}^{\pi }{\left\{\frac{\xi \left(t\right)}{t^{-\delta +\beta +1}}\right\}}^{s}dt\right]}^{\frac{1}{s}}\\ =O\left[\left\{\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right\}\left\{{\left(n+1\right)}^{\delta }\right\}\right]{\left[{\int }_{\frac{1}{n+1}}^{\pi }{\left\{\frac{\xi \left(t\right)}{t^{-\delta +\beta +1}}\right\}}^{s}dt\right]}^{\frac{1}{s}}.\end{array}$$

Putting $t=\frac{1}{y}$

$$\begin{array}{ll}\hfill I_{2.2}& =O\left\{\frac{{\left(n+1\right)}^{\delta }}{\mathrm{\Gamma }\left(\lambda +n\right)}\right\}{\left[{\int }_{\pi }^{n+1}{\left\{\frac{\xi \left(\frac{1}{y}\right)}{y^{\delta -\beta -1^{\prime }}}\right\}}^s\frac{dy}{y^2}\right]}^{\frac{1}{s}}\\ =O\left\{\frac{{\left(n+1\right)}^{\delta }}{\mathrm{\Gamma }\left(\lambda +n\right)}\xi \left(\frac{1}{n+1}\right)\right\}{\left[{\int }_{\eta }^{n+1}\left\{\frac{dy}{d^{s\left(\delta -\beta -1\right)+2}}\right\}dt\right]}^{\frac{1}{s}}\mathsf{\text{for}}\mathsf{\text{some}}\frac{1}{\pi }\le \eta \le n+1\\ =O\left\{\frac{{\left(n+1\right)}^{\delta }}{\mathrm{\Gamma }\left(\lambda +n\right)}\xi \left(\frac{1}{n+1}\right)\right\}{\left[{\int }_1^{n+1}\left\{\frac{dy}{y^{s\left(\delta -\beta -1\right)+2}}\right\}dt\right]}^{\frac{1}{s}}\mathsf{\text{for}}\mathsf{\text{some}}\frac{1}{\pi }\le 1\le n+1\\ =O\left\{\frac{{\left(n+1\right)}^{\delta }}{\mathrm{\Gamma }\left(\lambda +n\right)}\xi \left(\frac{1}{n+1}\right)\right\}{\left[{\left\{\frac{y^{s\left(\beta +1-\delta \right)-1}}{s\left(\beta +1-\delta \right)-1}\right\}}_1^{n+1}\right]}^{\frac{1}{s}}\\ =O\left\{\frac{{\left(n+1\right)}^{\delta }}{\mathrm{\Gamma }\left(\lambda +n\right)}\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{1+\beta -\delta -\frac{1}{s}}\right\}\\ =O\left\{\frac{\xi \left(\frac{1}{n+1}\right)}{\mathrm{\Gamma }\left(\lambda +n\right)}\right\}\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\right\}\mathsf{\text{since}}\frac{1}{r}+\frac{1}{s}=1.\end{array}$$

(5.7)

Now combining (5.5)-(5.7),

$$\begin{array}{ll}\hfill \left|s_m\left(x\right)-f\left(x\right)\right|& =O\left[\left\{\left(\frac{log\left(n+1\right)e}{\left(n+1\right)}\right)\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\right\}\right]\\ +O\left[\left\{\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right)\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\right\}\right]\\ =O\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\xi \left(\frac{1}{n+1}\right)\right\}\left[\frac{log\left(n+1\right)e}{\left(n+1\right)}+\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right].\end{array}$$

Now using L _r -norm, we get

$$\begin{array}{c}\parallel S_m\left(x\right)-f\left(x\right)\parallel ={\left\{{\displaystyle {\int }_0^{2\pi }|S_m\left(x\right)-f\left(x\right)}{|}^{r}dx\right\}}^{\frac{1}{r}}\\ =O[{\displaystyle {\int }_0^{2\pi }\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\xi \left(\frac{1}{n+1}\right)\right\}}\\ \cdot {\left\{\frac{\log \left(n+1\right)e}{\left(n+1\right)}+\frac{1}{\Gamma \left(\lambda +n\right)}\right\}dx]}^{\frac{1}{r}}\\ =[\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\xi \left(\frac{1}{n+1}\right)\right\}\\ \cdot \left\{\frac{\log \left(n+1\right)e}{\left(n+1\right)}+\frac{1}{\Gamma \left(\lambda +n\right)}\right\}]\left[{\left\{{\displaystyle {\int }_0^{2\pi }dx}\right\}}^{\frac{1}{r}}\right]\\ =O\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\xi \left(\frac{1}{n+1}\right)\right\}\left[\frac{\log \left(n+1\right)e}{\left(n+1\right)}+\frac{1}{\Gamma \left(\lambda +n\right)}\right].\end{array}$$

This completes the proof of Theorem 2.

5.3 Proof of Theorem 3

Following Lal [7], the m th partial sum ${\tilde{S}}_m\left(x\right)$ of series (2.5) at t = x

$${\tilde{S}}_m\left(x\right)-\left[-\frac{1}{2\pi }{\int }_0^{\pi }\psi \left(t\right)cot\left(\frac{t}{2}\right)dt\right]=\frac{1}{2\pi }{\int }_0^{\pi }\psi \left(t\right)\frac{cos\left(m+\frac{1}{2}\right)t}{sin\frac{t}{2}}dt.$$

Therefore,

$$\begin{array}{c}\frac{\Gamma \left(\lambda \right)}{\Gamma \left(\lambda +n\right)}{\displaystyle \sum _{m=0}^n\left[\begin{array}{c}n\\ m\end{array}\right]}{\lambda }^m\left\{\tilde{S}\left(x\right)-\left(-\frac{1}{2\pi }{\displaystyle {\int }_0^{\pi }\psi \left(t\right)\cot \left(\frac{t}{2}\right)dt}\right)\right\}\\ =\frac{1}{2\pi }{\displaystyle {\int }_0^{\pi }\psi \left(t\right)\frac{\Gamma \left(\lambda \right)}{\Gamma \left(\lambda +n\right)}}{\displaystyle \sum _{m=0}^n\left[\begin{array}{c}n\\ m\end{array}\right]}{\lambda }^m\frac{\cos \left(m+\frac{1}{2}\right)t}{\sin \frac{t}{2}}dt,\\ {\tilde{S}}_m\left(x\right)-\tilde{f}\left(x\right)=\frac{\Gamma \left(\lambda \right)}{2\pi }{\displaystyle {\int }_0^{\pi }\psi }\left(t\right){\tilde{K}}_n\left(t\right)dt\\ =\frac{\Gamma \left(\lambda \right)}{2\pi }\left[{\displaystyle {\int }_0^{\frac{1}{n+1}}+}{\displaystyle {\int }_{\frac{1}{n+1}}^{\pi }}\right]\left|\psi \left(t\right)\right|\left|{\tilde{K}}_n\left(t\right)\right|dt\\ =O\left(I_{3.1}\right)+O\left(I_{3.2}\right).\end{array}$$

(5.8)

We consider,

$$\mid I_{3.1}\mid ={\int }_0^{\frac{1}{n+1}}\mid \psi \left(t\right)\mid \left|{\tilde{K}}_n\left(t\right)\right|dt.$$

Using Lemma 3,

$$\begin{array}{ll}\hfill I_{3.1}& =O\left[{\int }_0^{\frac{1}{n+1}}\frac{e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}}{t}\mid \psi \left(t\right)\mid dt\right]\\ +O\left\{\lambda log\left(n+1\right)\right\}{\int }_0^{\frac{1}{n+1}}\mid sin\left(\lambda log\left(n+1\right).sint\right)\mid \mid \psi \left(t\right)\mid dt\\ +O\left[{\int }_0^{\frac{1}{n+1}}{e}^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\mid sin\left(t∕2\right)\mid \mid \psi \left(t\right)\mid dt\right]\\ =I_{3.1.1}+I_{3.1.2}+I_{3.1.3}\left(\mathsf{\text{say}}\right).\end{array}$$

(5.9)

Now consider,

$$\begin{array}{ll}\hfill I_{3.1.1}& =O\left({\int }_0^{\frac{1}{n+1}}\frac{e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}}{t}\mid \psi \left(t\right)\mid dt\right)\\ =O{\left[{\int }_0^{\frac{1}{n+1}}{\left\{\frac{t\psi \left(t\right)}{t^{\alpha }}\right\}}^{r}dt\right]}^{\frac{1}{r}}{\left[{\int }_0^{\frac{1}{n+1}}{\left\{t^{\alpha -2}{e}^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\right\}}^{s}dt\right]}^{\frac{1}{s}}.\end{array}$$

Using second mean value theorem for integrals,

$$\begin{array}{ll}\hfill I_{3.1.1}& =O\left\{\frac{e^{-\frac{\lambda }{2}\frac{log\left(n+1\right)}{{\left(n+1\right)}^2}}}{\left(n+1\right)}\right\}{\left[{\int }_{\in }^{\frac{1}{n+1}}{\left(t^{\alpha -2}\right)}^{s}dt\right]}^{\frac{1}{s}}\mathsf{\text{for}}0<\in <\frac{1}{n+1}\\ =O\left\{\frac{e^{-\frac{\lambda }{2}\frac{log\left(n+1\right)}{{\left(n+1\right)}^2}}}{\left(n+1\right)}\right\}{\left[{\left\{\frac{t^{s\alpha -2s+1}}{s\alpha -2s+1}\right\}}_{\in }^{\frac{1}{n+1}}\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{n+1}\right){\left[\frac{1}{{\left(n+1\right)}^{\alpha s-2s+1}}\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{n+1}\right)\left[\frac{1}{{\left(n+1\right)}^{\alpha -2+\frac{1}{s}}}\right]\\ =O\left(\frac{1}{n+1}\right)\left[\frac{1}{{\left(n+1\right)}^{\alpha -1-\frac{1}{r}}}\right]\mathsf{\text{since}}\frac{1}{r}+\frac{1}{s}=1\\ =O\left[\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\right].\end{array}$$

(5.10)

Now we consider,

$$I_{3.1.2}=O\left\{\lambda log\left(n+1\right)\right\}{\int }_0^{\frac{1}{n+1}}\mid sin\left(\lambda log\left(n+1\right)sint\right)\mid \mid \psi \left(t\right)\mid dt.$$

Since, for $0<t<\frac{1}{n+1},sinnt\le nt$,

$$I_{3.1.2}=O\left\{\lambda log\left(n+1\right)\right\}{\int }_0^{\frac{1}{n+1}}t\mid \psi \left(t\right)\mid dt.$$

Using Hölder's inequality and Lemma 4,

$$\begin{array}{ll}\hfill I_{3.1.2}& =O\left\{\lambda log\left(n+1\right)\right\}{\left[{\int }_0^{\frac{1}{n+1}}{\left\{\frac{t\psi \left(t\right)}{t^{\alpha }}\right\}}^{r}dt\right]}^{\frac{1}{r}}{\left[{\int }_0^{\frac{1}{n+1}}{\left(t^{\alpha }\right)}^{s}dt\right]}^{\frac{1}{s}}\\ =O\left\{\lambda log\left(n+1\right)\right\}\left(\frac{1}{n+1}\right){\left[{\left\{\frac{t^{\alpha s+1}}{\alpha s+1}\right\}}_0^{\frac{1}{n+1}}\right]}^{\frac{1}{s}}\\ =O\left\{\lambda log\left(n+1\right)\right\}\left(\frac{1}{n+1}\right){\left[\frac{1}{{\left(n+1\right)}^{\alpha s+1}}\right]}^{\frac{1}{s}}\\ =O\left\{\lambda log\left(n+1\right)\right\}\left(\frac{1}{n+1}\right)\left[\frac{1}{{\left(n+1\right)}^{\alpha +\frac{1}{s}}}\right]\\ =O\left\{log\left(n+1\right)\right\}\left(\frac{1}{n+1}\right)\left[\frac{1}{{\left(n+1\right)}^{\alpha +1-\left(1-\frac{1}{s}\right)}}\right]\\ =O\left\{log\left(n+1\right)\right\}\left(\frac{1}{n+1}\right)\left[\frac{1}{{\left(n+1\right)}^{\alpha +1-\frac{1}{r}}}\right]\mathsf{\text{since}}\frac{1}{r}+\frac{1}{s}=1\\ =O\left\{\frac{log\left(n+1\right)}{{\left(n+1\right)}^2}\right\}\left[\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\right].\end{array}$$

(5.11)

Next we consider,

$$\begin{array}{ll}\hfill I_{3.1.3}& =O{\int }_0^{\frac{1}{n+1}}{e}^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\mid sin\left(t∕2\right)\mid \mid \psi \left(t\right)\mid dt\\ =O\left\{{\int }_0^{\frac{1}{n+1}}t\mid \psi \left(t\right)\mid dt\right\}\\ =O{\left[{\int }_0^{\frac{1}{n+1}}{\left\{\frac{t\psi \left(t\right)}{t^{\alpha }}\right\}}^{r}dt\right]}^{\frac{1}{r}}{\left[{\int }_0^{\frac{1}{n+1}}{\left(t^{\alpha }\right)}^{s}dt\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{n+1}\right){\left[{\left\{\frac{t^{\alpha s+1}}{\alpha s+1}\right\}}_0^{\frac{1}{n+1}}\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{n+1}\right){\left[\frac{1}{{\left(n+1\right)}^{\alpha s+1}}\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{n+1}\right)\left[\frac{1}{{\left(n+1\right)}^{\alpha +\frac{1}{s}}}\right]\\ =O\left(\frac{1}{n+1}\right)\left[\frac{1}{{\left(n+1\right)}^{\alpha +1-\left(1-\frac{1}{s}\right)}}\right]\\ =O\left\{\frac{1}{{\left(n+1\right)}^2}\right\}\left[\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\right]\mathsf{\text{since}}\frac{1}{r}+\frac{1}{s}=1.\end{array}$$

(5.12)

Combining (5.9)-(5.12),

$$\begin{array}{ll}\hfill I_{3.1}& =O\left[\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\right]+O\left\{\frac{log\left(n+1\right)}{{\left(n+1\right)}^2}\right\}\left[\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\right]\\ +O\left\{\frac{1}{{\left(n+1\right)}^2}\right\}\left[\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\right]\\ =O\left[\frac{log\left(n+1\right)e}{{\left(n+1\right)}^2{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\right]+O\left[\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\right].\end{array}$$

(5.13)

Since, for $\frac{1}{n+1}<t<\pi ,\mid sin\left(\frac{t}{2}\right)\mid \ge \frac{t}{\pi }$

$$\begin{array}{ll}\hfill {\tilde{K}}_n\left(t\right)& =O\left[\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)sin\left(\frac{t}{2}\right)}\right]\\ =O\left[\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)t}\right].\end{array}$$

(5.14)

Next we consider,

$$I_{3.2}\le {\int }_{\frac{1}{n+1}}^{\pi }\mid \psi \left(t\right)\mid \mid {\tilde{K}}_n\left(t\right)\mid dt.$$

Using Hölder's inequality, (5.14) and Lemma 4,

$$\begin{array}{ll}\hfill I_{3.2}& =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right){\left[{\int }_{\frac{1}{n+1}}^{\pi }{\left\{\frac{t^{-\delta }\psi \left(t\right)}{t^{\alpha }}\right\}}^{r}dt\right]}^{\frac{1}{r}}{\left[{{\int }_{\frac{1}{n+1}}^{\pi }\left(\frac{t^{\delta +\alpha }}{t}\right)}^{s}dt\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right)\frac{1}{{\left(n+1\right)}^{-\delta }}{\left[{\int }_{\frac{1}{n+1}}^{\pi }{t}^{\left(\delta +\alpha -1\right)s}dt\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right)\frac{1}{{\left(n+1\right)}^{-\delta }}{\left[{\left\{\frac{t^{\left(\delta +\alpha -1\right)s+1}}{\left(\delta +\alpha -1\right)s+1}\right\}}_{\frac{1}{n+1}}^{\pi }\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right)\frac{1}{{\left(n+1\right)}^{-\delta }}{\left[\frac{1}{{\left(n+1\right)}^{\left(\delta +\alpha -1\right)s+1}}\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right)\frac{1}{{\left(n+1\right)}^{-\delta }}\left[\frac{1}{{\left(n+1\right)}^{\left(\delta +\alpha -1\right)+\frac{1}{s}}}\right]\\ =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right)\left[\frac{1}{{\left(n+1\right)}^{\alpha -1+\frac{1}{s}}}\right]\\ =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right)\left[\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\right]\mathsf{\text{since}}\frac{\mathsf{\text{1}}}{\mathsf{\text{r}}}+\frac{1}{s}=1.\end{array}$$

(5.15)

Collecting (5.8), (5.13) and (5.15),

$$\begin{array}{ll}\hfill {\tilde{S}}_m-\tilde{f}\left(x\right)& =O\left[\frac{log\left(n+1\right)e}{{\left(n+1\right)}^2{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\right]+\left[\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\right]\\ +O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right)\left\{\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\right\}\\ =O\left[\frac{1}{{\left(n+1\right)}^{\alpha -\frac{1}{r}}}\left\{\frac{log\left(n+1\right)e}{\left(n+1\right)}+\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}+1\right\}\right].\end{array}$$

This completes the proof of Theorem 3.

5.4 Proof of Theorem 4

Following the calculations of Theorem 3,

$$\begin{array}{l}\hfill {\tilde{S}}_m\left(x\right)-\tilde{f}\left(x\right)=\frac{\mathrm{\Gamma }\left(\lambda \right)}{2\pi }\left[{\int }_0^{\frac{1}{n+1}}+{\int }_{\frac{1}{n+1}}^{\pi }\right]\psi \left(t\right){\tilde{K}}_n\left(t\right)dt\\ \hfill =O\left(I_{4.1}\right)+O\left(I_{4.2}\right).\end{array}$$

(5.16)

Now,

$$I_{4.1}=O\left[{\int }_0^{\frac{1}{n+1}}\mid \psi \left(t\right)\mid \mid K_n\left(t\right)\mid dt\right].$$

Using Lemma 3,

$$\begin{array}{ll}\hfill I_{4.1}& =O\left({\int }_0^{\frac{1}{n+1}}\frac{e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}}{t}\mid \psi \left(t\right)\mid dt\right)\\ +O\left\{\lambda log\left(n+1\right)\right\}{\int }_0^{\frac{1}{n+1}}\mid sin\left(\lambda log\left(n+1\right)sint\right)\mid \mid \psi \left(t\right)\mid dt\\ +O\left[{\int }_0^{\frac{1}{n+1}}{e}^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\mid sin\left(t∕2\right)\mid \mid \psi \left(t\right)\mid dt\right]\\ =I_{4.1.1}+I_{4.1.2}+I_{4.1.3}\left(\mathsf{\text{say}}\right).\end{array}$$

(5.17)

Using Minkowiski's inequality, we have a fact that f ∈ W (L _r , ξ (t))⇒ ψ ∈ W (L _r , ξ (t)). Now we consider,

$$\begin{array}{ll}\hfill I_{4.1.1}& =O\left({\int }_0^{\frac{1}{n+1}}\frac{e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}}{t}\mid \psi \left(t\right)\mid dt\right)\\ =O{\left[{\int }_0^{\frac{1}{n+1}}{\left\{\frac{t\psi \left(t\right){sin}^{\beta }\left(t\right)}{\xi \left(t\right)}\right\}}^{r}dt\right]}^{\frac{1}{r}}{\left[{\int }_0^{\frac{1}{n+1}}{\left\{\frac{\xi \left(t\right)e^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}}{{sin}^{\beta }\left(t\right)}\right\}}^{s}dt\right]}^{\frac{1}{s}}\\ =O\left\{e^{-\frac{\lambda }{2}\frac{log\left(n+1\right)}{{\left(n+1\right)}^2}}\right\}O\left(\frac{1}{n+1}\right){\left[{\int }_0^{\frac{1}{n+1}}{\left(\frac{\xi \left(t\right)}{{sin}^{\beta }\left(t\right)}\right)}^{s}dt\right]}^{\frac{1}{s}}\text{by}(3.4)\\ =O\left(\frac{1}{n+1}\right){\left[{\int }_0^{\frac{1}{n+1}}{\left(\frac{\xi \left(t\right)}{{sin}^{\beta }\left(t\right)}\right)}^{s}dt\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{n+1}\right){\left[{\int }_0^{\frac{1}{n+1}}{\left(\frac{\xi \left(t\right)}{t^{\beta }}\right)}^{s}dt\right]}^{\frac{1}{s}}\mathsf{\text{since}}\mathsf{\text{sin}}t\ge \frac{2t}{\pi }.\end{array}$$

Since ξ (t) is a positive increasing function and using second mean value theorem for integrals,

$$\begin{array}{ll}\hfill I_{4.1.1}& =O\left\{\left(\frac{1}{n+1}\right)\xi \left(\frac{1}{n+1}\right)\right\}{\left[{\int }_{\in }^{\frac{1}{n+1}}\left\{\frac{1}{t^{\beta s}}\right\}dt\right]}^{\frac{1}{s}}\mathsf{\text{for}}\mathsf{\text{some}}0<\in <\frac{1}{n+1}\\ =O\left\{\left(\frac{1}{n+1}\right)\xi \left(\frac{1}{n+1}\right)\right\}{\left[{\left\{\frac{t^{-\beta s+1}}{-\beta s+1}\right\}}_{\in }^{\frac{1}{n+1}}\right]}^{\frac{1}{s}}\\ =O\left[\left\{\left(\frac{1}{n+1}\right)\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\beta -1+\left(1-\frac{1}{s}\right)}\right\}\right]\\ =O\left[\left\{\frac{1}{{\left(n+1\right)}^2}\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\right\}\right]\mathsf{\text{since}}\frac{\mathsf{\text{1}}}{\mathsf{\text{r}}}+\frac{1}{s}=1.\end{array}$$

(5.18)

Now,

$$I_{4.1.2}=O\left\{\lambda log\left(n+1\right)\right\}{\int }_0^{\frac{1}{n+1}}\mid \mathsf{\text{sin}}\left(\lambda log\left(n+1\right)sint\right)\mid \mid \psi \left(t\right)\mid dt.$$

Since for $0<t<\frac{1}{n+1},sinnt\le nt$,

$$I_{4.1.2}=O\left\{\lambda log\left(n+1\right)\right\}{\int }_0^{\frac{1}{n+1}}t\mid \psi \left(t\right)\mid dt.$$

Hölder's inequality and the fact that ψ (t) ∈ W (L _r , ξ (t)),

$$\begin{array}{ll}\hfill I_{4.1.2}& =O\left\{\lambda log\left(n+1\right)\right\}{\left[{\int }_0^{\frac{1}{n+1}}{\left\{\frac{t\mid \psi \left(t\right)\mid {sin}^{\beta }\left(t\right)}{\xi \left(t\right)}\right\}}^{r}dt\right]}^{\frac{1}{r}}{\left[{\int }_0^{\frac{1}{n+1}}{\left\{\frac{\xi \left(t\right)}{{sin}^{\beta }t}\right\}}^{s}dt\right]}^{\frac{1}{s}}\\ =O\left\{log\left(n+1\right)\right\}O\left(\frac{1}{n+1}\right){\left[{\int }_0^{\frac{1}{n+1}}{\left\{\frac{\xi \left(t\right)}{{sin}^{\beta }t}\right\}}^{s}dt\right]}^{\frac{1}{s}}\text{by}(3.4)\\ =O\left(\frac{log\left(n+1\right)}{n+1}\right){\left[{\int }_0^{\frac{1}{n+1}}{\left\{\frac{\xi \left(t\right)}{t^{\beta }}\right\}}^{s}dt\right]}^{\frac{1}{s}}\mathsf{\text{since}}\mathsf{\text{sin}}t\ge 2t∕\pi .\end{array}$$

Since ξ (t) is a positive increasing function and using second mean value theorem for integrals,

$$\begin{array}{ll}\hfill I_{4.1.2}& =O\left\{\left(\frac{log\left(n+1\right)}{n+1}\right)\xi \left(\frac{1}{n+1}\right)\right\}{\left[{\int }_{\in }^{\frac{1}{n+1}}\left\{\frac{1}{t^{\beta s}}\right\}dt\right]}^{\frac{1}{s}}\mathsf{\text{for}}\mathsf{\text{some}}<\in <\frac{1}{n+1}\\ =O\left\{\left(\frac{log\left(n+1\right)}{n+1}\right)\xi \left(\frac{1}{n+1}\right)\right\}{\left[{\left\{\frac{t^{-\beta s+1}}{-\beta s+1}\right\}}_{\in }^{\frac{1}{n+1}}\right]}^{\frac{1}{s}}\\ =O\left[\left\{\left(\frac{log\left(n+1\right)}{n+1}\right)\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\beta -1+\left(1-\frac{1}{s}\right)}\right\}\right]\\ =O\left[\left\{\frac{log\left(n+1\right)}{{\left(n+1\right)}^2}\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\right\}\right]\mathsf{\text{since}}\frac{1}{r}+\frac{1}{s}=1.\end{array}$$

(5.19)

Next we consider,

$$\begin{array}{ll}\hfill I_{4.1.3}& =O\left[{\int }_0^{\frac{1}{n+1}}{e}^{-\frac{\lambda }{2}{t}^{2}log\left(n+1\right)}\mid sin\frac{t}{2}\mid \mid \varphi \left(t\right)\mid dt\right]\\ =O\left[{\int }_0^{\frac{1}{n+1}}t\mid \psi \left(t\right)\mid dt\right].\end{array}$$

Using Hölder's inequality and the fact that ψ (t) ∈ W (L _r ,ξ (t)),

$$\begin{array}{ll}\hfill I_{4.1.3}& =O{\left[{\int }_0^{\frac{1}{n+1}}{\left\{\frac{t\mid \psi \left(t\right)\mid {sin}^{\beta }\left(t\right)}{\xi \left(t\right)}\right\}}^{r}dt\right]}^{\frac{1}{r}}{\left[{\int }_0^{\frac{1}{n+1}}{\left\{\frac{\xi \left(t\right)}{{sin}^{\beta }t}\right\}}^{s}dt\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{n+1}\right){\left[{\int }_0^{\frac{1}{n+1}}{\left\{\frac{\xi \left(t\right)}{{sin}^{\beta }t}\right\}}^{s}dt\right]}^{\frac{1}{s}}\text{by}(3.4)\end{array}$$

Since, sin t ≥ 2t/π,

$$I_{4.1.3}=O\left(\frac{1}{n+1}\right){\left[{\int }_0^{\frac{1}{n+1}}{\left\{\frac{\xi \left(t\right)}{t^{\beta }}\right\}}^{s}dt\right]}^{\frac{1}{s}}.$$

Since ξ (t) is a positive increasing function and using second mean value theorem for integrals,

$$\begin{array}{ll}\hfill I_{4.1.3}& =O\left\{\left(\frac{1}{n+1}\right)\xi \left(\frac{1}{n+1}\right)\right\}{\left[{\int }_{\in }^{\frac{1}{n+1}}\left\{\frac{1}{t^{\beta s}}\right\}dt\right]}^{\frac{1}{s}}\\ =O\left\{\left(\frac{1}{n+1}\right)\xi \left(\frac{1}{n+1}\right)\right\}{\left[{\left\{\frac{t^{-\beta s+1}}{-\beta s+1}\right\}}_{\in }^{\frac{1}{n+1}}\right]}^{\frac{1}{s}}\\ =O\left[\left\{\left(\frac{1}{n+1}\right)\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\beta -\frac{1}{s}}\right\}\right]\\ =O\left[\left\{\left(\frac{1}{n+1}\right)\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\beta -1+\left(1-\frac{1}{s}\right)}\right\}\right]\\ =O\left[\left\{\frac{1}{{\left(n+1\right)}^2}\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\right\}\right]\mathsf{\text{since}}\frac{1}{r}+\frac{1}{s}=1.\end{array}$$

(5.20)

Combining from (5.17) to (5.20),

$$\begin{array}{ll}\hfill I_{4.1}& =O\left[\left\{\frac{1}{{\left(n+1\right)}^2}\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\right\}\right]\\ +O\left[\left\{\frac{log\left(n+1\right)}{{\left(n+1\right)}^2}\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\right\}\right]\\ +O\left[\left\{\frac{1}{{\left(n+1\right)}^2}\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\right\}\right].\end{array}$$

(5.21)

Using Hölder's inequality, |sin t| ≤ 1, sin t ≥ 2t/π, conditions (3.3), (3.5) and second mean value theorem for integrals and the fact ψ (t) ∈ W (L_r,ξ (t)),

$$\begin{array}{ll}\hfill I_{4.2}& =O\left({\int }_{\frac{1}{n+1}}^{\pi }\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)t}\mid \psi \left(t\right)\mid dt\right)\\ =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right){\left[{\int }_{\frac{1}{n+1}}^{\pi }{\left\{\frac{t^{-\delta }\mid \psi \left(t\right)\mid {sin}^{\beta }\left(t\right)}{\xi \left(t\right)}\right\}}^{r}dt\right]}^{\frac{1}{r}}{\left[{\int }_{\frac{1}{n+1}}^{\pi }{\left\{\frac{\xi \left(t\right)}{t^{1-\delta }{sin}^{\beta }t}\right\}}^{s}dt\right]}^{\frac{1}{s}}\\ =O\left(\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right){\left[{{\int }_{\frac{1}{n+1}}^{\pi }\left\{\frac{t^{-\delta }\mid \psi \left(t\right)\mid }{\xi \left(t\right)}\right\}}^{r}dt\right]}^{\frac{1}{r}}{\left[{\int }_{\frac{1}{n+1}}^{\pi }{\left\{\frac{\xi \left(t\right)}{t^{1-\delta }{sin}^{\beta }t}\right\}}^{s}dt\right]}^{\frac{1}{s}}\\ =O\left[\left\{\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right\}\left\{{\left(n+1\right)}^{\delta }\right\}\right]{\left[{\int }_{\frac{1}{n+1}}^{\pi }{\left\{\frac{\xi \left(t\right)}{t^{1-\delta }{sin}^{\beta }t}\right\}}^{s}dt\right]}^{\frac{1}{s}}\\ =O\left[\left\{\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right\}\left\{{\left(n+1\right)}^{\delta }\right\}\right]{\left[{\int }_{\frac{1}{n+1}}^{\pi }{\left\{\frac{\xi \left(t\right)}{t^{-\delta +\beta +1}}\right\}}^{s}dt\right]}^{\frac{1}{s}}.\end{array}$$

Putting $t=\frac{1}{y}$

$$\begin{array}{ll}\hfill I_{4.2}& =O\left\{\frac{{\left(n+1\right)}^{\delta }}{\mathrm{\Gamma }\left(\lambda +n\right)}\right\}{\left[{\int }_{\pi }^{n+1}{\left\{\frac{\xi \left(\frac{1}{y}\right)}{y^{\delta -\beta -1^{\prime }}}\right\}}^s\frac{dy}{y^2}\right]}^{\frac{1}{s}}\\ =O\left\{\frac{{\left(n+1\right)}^{\delta }}{\mathrm{\Gamma }\left(\lambda +n\right)}\xi \left(\frac{1}{n+1}\right)\right\}{\left[{\int }_{\eta }^{n+1}\left\{\frac{dy}{y^{s\left(\delta -\beta -1\right)+2}}\right\}dt\right]}^{\frac{1}{s}}\mathsf{\text{for}}\mathsf{\text{some}}\frac{\mathsf{\text{1}}}{\pi }\le \eta \le n+1\\ =O\left\{\frac{{\left(n+1\right)}^{\delta }}{\mathrm{\Gamma }\left(\lambda +n\right)}\xi \left(\frac{1}{n+1}\right)\right\}{\left[{\int }_1^{n+1}\left\{\frac{dy}{y^{s\left(\delta -\beta -1\right)+2}}\right\}dt\right]}^{\frac{1}{s}}\mathsf{\text{for}}\mathsf{\text{some}}\frac{\mathsf{\text{1}}}{\pi }\le 1\le n+1\\ =O\left\{\frac{{\left(n+1\right)}^{\delta }}{\mathrm{\Gamma }\left(\lambda +n\right)}\xi \left(\frac{1}{n+1}\right)\right\}{\left[{\left\{\frac{y^{s\left(\beta +1-\delta \right)-1}}{s\left(\beta +1-\delta \right)-1}\right\}}_1^{n+1}\right]}^{\frac{1}{s}}\\ =O\left\{\frac{{\left(n+1\right)}^{\delta }}{\mathrm{\Gamma }\left(\lambda +n\right)}\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{1+\beta -\delta -\frac{1}{s}}\right\}\\ =O\left\{\frac{\xi \left(\frac{1}{n+1}\right)}{\mathrm{\Gamma }\left(\lambda +n\right)}\right\}\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\right\}\mathsf{\text{since}}\frac{\mathsf{\text{1}}}{\mathsf{\text{r}}}+\frac{1}{s}=1.\end{array}$$

(5.22)

Combining from (5.16), (5.21) and (5.22)

$$\begin{array}{ll}\hfill \mid S_m\left(x\right)-f\left(x\right)\mid & =O\left[\left\{\frac{1}{{\left(n+1\right)}^2}\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\right\}\right]\\ +O\left[\left\{\frac{log\left(n+1\right)}{{\left(n+1\right)}^2}\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\right\}\right]\\ +O\left[\left\{\frac{1}{{\left(n+1\right)}^2}\xi \left(\frac{1}{n+1}\right)\right\}\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\right\}\right]\\ +O\left\{\frac{\xi \left(\frac{1}{n+1}\right)}{\mathrm{\Gamma }\left(\lambda +n\right)}\right\}\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\right\}\\ =O\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\xi \left(\frac{1}{n+1}\right)\right\}\left[\frac{2}{{\left(n+1\right)}^2}+\frac{log\left(n+1\right)}{{\left(n+1\right)}^2}+\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right].\end{array}$$

Now using L_r-norm, we get

$$\begin{array}{ll}\hfill \parallel S_m\left(x\right)-f\left(x\right)\parallel & ={\left\{{\int }_0^{2\pi }\mid S_m\left(x\right)-f\left(x\right){\mid }^{r}dx\right\}}^{\frac{1}{r}}\\ =O\left[{\int }_0^{2\pi }{\left(n+1\right)}^{\beta +\frac{1}{r}}\xi \left(\frac{1}{n+1}\right)\right.\\ \cdot {\left.\left\{\frac{2}{{\left(n+1\right)}^2}+\frac{log\left(n+1\right)}{{\left(n+1\right)}^2}+\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right\}dx\right]}^{\frac{1}{r}}\\ =\left[\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\xi \left(\frac{1}{n+1}\right)\right\}\right.\\ \cdot \left.\left\{\frac{2}{{\left(n+1\right)}^2}+\frac{log\left(n+1\right)}{{\left(n+1\right)}^2}+\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right\}\right]\left[{\left\{{\int }_0^{2\pi }dx\right\}}^{\frac{1}{r}}\right]\\ =O\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\xi \left(\frac{1}{n+1}\right)\right\}\left[\frac{2}{{\left(n+1\right)}^2}+\frac{log\left(n+1\right)}{{\left(n+1\right)}^2}+\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right]\\ =O\left\{{\left(n+1\right)}^{\beta +\frac{1}{r}}\xi \left(\frac{1}{n+1}\right)\right\}\left[1+\frac{log\left(n+1\right)e}{{\left(n+1\right)}^2}+\frac{1}{\mathrm{\Gamma }\left(\lambda +n\right)}\right].\end{array}$$

This completes the proof of Theorem 4.