4.1 Proof of Theorem 1
It is well known [26] that
$$ s_{k}(f,x)- f(x)= \frac{1}{2 \pi} \int_{0}^{\pi} \phi(x, t) \frac{\sin (k+2^{-1})t}{\sin(t/2)} \,dt. $$
(4.1)
Since \(\sum_{k=0}^{n} h_{n, k} = 1\) for each n in any regular Hausdorff matrix H ([13], p.397), so we have
$$\begin{aligned} H_{n}(x)-f(x) &= \frac{1}{2 \pi} \int_{0}^{\pi} \phi(x, t) \sum _{k=0}^{n} h_{n, k} \frac{\sin(k+2^{-1})t}{\sin(t/2)} \,dt \\ &= \frac{1}{2 \pi} \int_{0}^{\pi} \frac{\phi(x, t)}{\sin(t/2)} \sum _{k=0}^{n} \int_{0}^{1} {n \choose k} u^{k} (1-u)^{n-k}\, d \gamma(u) \operatorname{Im} \bigl(e^{i(k+2^{-1})t}\bigr) \,dt \\ &= \frac{1}{2 \pi} \int_{0}^{\pi} \frac{\phi(x, t)}{\sin(t/2)} \int _{0}^{1} \operatorname{Im} \Biggl[\sum _{k=0}^{n} {n \choose k} u^{k} (1-u)^{n-k} e^{i(k+2^{-1})t} \Biggr]\, d \gamma(u) \,dt \\ &= \frac{1}{2 \pi} \int_{0}^{\pi} \frac{\phi(x, t)}{\sin(t/2)} \int _{0}^{1} g(u, t)\, d \gamma(u) \,dt. \end{aligned}$$
Let
$$ l_{n}(x) := H_{n}(x)-f(x)= \int_{0}^{\pi} \phi(x, t) K_{n}^{H}(t) \,dt. $$
(4.2)
Then
$$l_{n}(x+y)+ l_{n}(x-y)-2 l_{n}(x) = \int_{0}^{\pi} \bigl[\phi(x+y, t)+\phi (x-y, t)- 2 \phi(x,t)\bigr] K_{n}^{H}(t) \,dt. $$
Using the generalized Minkowski inequality ([21], p.37), we get
$$\begin{aligned} & \bigl\Vert l_{n}(\cdot+ y)+ l_{n}(\cdot- y)-2 l_{n}(\cdot) \bigr\Vert _{p} \\ &\quad = \biggl\{ \frac{1}{2 \pi} \int_{0}^{2 \pi} \bigl\vert l_{n}(x+y)+ l_{n}(x-y)-2 l_{n}(x) \bigr\vert ^{p} \,dx \biggr\} ^{1/p} \\ &\quad = \biggl\{ \frac{1}{2 \pi} \int_{0}^{2 \pi} \biggl\vert \int_{0}^{\pi} \bigl[\phi(x+y, t)+\phi(x-y, t)- 2 \phi(x, t)\bigr] K_{n}^{H}(t) \,dt \biggr\vert ^{p} \,dx \biggr\} ^{1/p} \\ &\quad \le \int_{0}^{\pi} \biggl\{ \frac{1}{2 \pi} \int_{0}^{2 \pi} \bigl\vert \bigl[\phi(x+y, t)+ \phi(x-y, t)- 2 \phi(x, t)\bigr] K_{n}^{H}(t) \bigr\vert ^{p} \,dx \biggr\} ^{1/p} \,dt \\ &\quad = \int_{0}^{\pi} \bigl( \bigl\vert K_{n}^{H}(t) \bigr\vert ^{p} \bigr)^{1/p} \biggl\{ \frac{1}{2 \pi} \int_{0}^{2 \pi} \bigl\vert \phi(x+y, t)+\phi(x-y, t)- 2 \phi(x, t) \bigr\vert ^{p} \,dx \biggr\} ^{1/p} \,dt \\ &\quad = \int_{0}^{\pi} \bigl\Vert \phi(\cdot+y, t)+\phi( \cdot-y, t)-2\phi (\cdot, t) \bigr\Vert _{p} \bigl\vert K_{n}^{H}(t) \bigr\vert \,dt \\ &\quad = \int_{0}^{1/(n+1)} \bigl\Vert \phi(\cdot+y, t)+\phi( \cdot-y, t)-2\phi (\cdot, t) \bigr\Vert _{p} \bigl\vert K_{n}^{H}(t) \bigr\vert \,dt \\ &\qquad{} + \int_{1/(n+1)}^{\pi} \bigl\Vert \phi(\cdot+y, t)+\phi( \cdot -y, t)-2\phi(\cdot, t) \bigr\Vert _{p} \bigl\vert K_{n}^{H}(t) \bigr\vert \,dt \\ &\quad := I_{1} + I_{2}, \quad\text{say}. \end{aligned}$$
(4.3)
Using Lemma 2, Lemma 3 {part(iii)} and the monotonicity of \(\omega(t)/v(t)\) with respect to t, we have
$$ \begin{aligned}[b] I_{1} &= \int_{0}^{(n+1)^{-1}} \bigl\Vert \phi(\cdot+y, t)+\phi( \cdot-y, t)-2\phi(\cdot, t) \bigr\Vert _{p} \bigl\vert K_{n}^{H}(t) \bigr\vert \,dt \\ &= O \biggl( \int_{0}^{(n+1)^{-1}} v(y) \frac{\omega(t)}{v(t)} (n+1) \,dt \biggr) \\ &= O \biggl((n+1) v(y) \int_{0}^{(n+1)^{-1}} \frac{\omega(t)}{v(t)} \,dt \biggr) \\ &= O \biggl((n+1) v(y) \frac{\omega((n+1)^{-1})}{v((n+1)^{-1})} \int _{0}^{(n+1)^{-1}} \,dt \biggr) \\ &= O \biggl( v(y) \frac{\omega((n+1)^{-1})}{v((n+1)^{-1})} \biggr). \end{aligned} $$
(4.4)
Using Lemma 2 and Lemma 3 {part(iii)}, we get
$$\begin{aligned} I_{2} &= \int_{(n+1)^{-1}}^{\pi} \bigl\Vert \phi(\cdot+ y, t)+\phi( \cdot - y, t)-2\phi(\cdot, t) \bigr\Vert _{p} \bigl\vert K_{n}^{H}(t) \bigr\vert \,dt \\ &= O \biggl( \int_{(n+1)^{-1}}^{\pi} v(y) \frac{\omega(t)}{v(t)} (n+1)^{-1} t^{-2} \,dt \biggr) \\ &= O \biggl((n+1)^{-1} v(y) \int_{(n+1)^{-1}}^{\pi} \frac{t^{-2} \omega (t)}{v(t)} \,dt \biggr). \end{aligned}$$
(4.5)
Thus, from (4.3), (4.4) and (4.5),
$$ \begin{gathered} \bigl\Vert l_{n}(\cdot+ y)+l_{n}(\cdot- y)-2l_{n}(\cdot) \bigr\Vert _{p}\\ \quad = O \biggl( v(y) \frac{\omega((n+1)^{-1})}{v((n+1)^{-1})} \biggr) + O \biggl((n+1)^{-1} v(y) \int_{(n+1)^{-1}}^{\pi} \frac{t^{-2} \omega (t)}{v(t)} \,dt \biggr) , \\ \sup_{y \neq0} \frac{ \Vert l_{n}(\cdot+ y)+l_{n}(\cdot- y)-2l_{n}(\cdot) \Vert _{p}}{v(y)}\\ \quad = O \biggl( \frac{\omega ((n+1)^{-1})}{v((n+1)^{-1})} \biggr) + O \biggl((n+1)^{-1} \int _{(n+1)^{-1}}^{\pi} \frac{t^{-2} \omega(t)}{v(t)} \,dt \biggr). \end{gathered} $$
(4.6)
Again using Lemmas 2 and 3, we have
$$\begin{aligned} \bigl\Vert l_{n}(\cdot) \bigr\Vert _{p} &\le \biggl( \int _{0}^{(n+1)^{-1}} + \int_{(n+1)^{-1}}^{\pi} \biggr) \bigl\Vert \phi(\cdot , t) \bigr\Vert _{p} \bigl\vert K_{n}^{H}(t) \bigr\vert \,dt \\ &= O \biggl((n+1) \int_{0}^{(n+1)^{-1}} \omega(t) \,dt \biggr)+ O \biggl( (n+1)^{-1} \int_{(n+1)^{-1}}^{\pi} t^{-2} \omega(t) \,dt \biggr) \\ &= O \bigl(\omega\bigl((n+1)^{-1}\bigr) \bigr) + O \biggl((n+1)^{-1} \int _{(n+1)^{-1}}^{\pi} t^{-2} \omega(t) \,dt \biggr). \end{aligned}$$
(4.7)
Now, from (4.6) and (4.7), we obtain
$$ \begin{aligned}[b] \bigl\Vert l_{n}(\cdot) \bigr\Vert _{p}^{(v)} &= \bigl\Vert l_{n}(\cdot ) \bigr\Vert _{p} + \sup_{y \neq0} \frac{ \Vert l_{n}(\cdot+ y)+l_{n}(\cdot- y)-2l_{n}(\cdot) \Vert _{p}}{v(y)} \\ &= O \bigl(\omega\bigl((n+1)^{-1}\bigr) \bigr) + O \biggl((n+1)^{-1} \int _{(n+1)^{-1}}^{\pi} t^{-2} \omega(t) \,dt \biggr) \\ &\quad{}+ O \biggl( \frac{\omega((n+1)^{-1})}{v((n+1)^{-1})} \biggr) + O \biggl((n+1)^{-1} \int_{(n+1)^{-1}}^{\pi} \frac{t^{-2} \omega(t)}{v(t)} \,dt \biggr) \\ &:= \sum_{i=1}^{4} O(J_{i}),\quad \text{say}. \end{aligned} $$
(4.8)
Now we write \(J_{1}\) in terms of \(J_{3}\) and further \(J_{2}\), \(J_{3}\) in terms of \(J_{4}\).
In view of the monotonicity of \(v(t)\), we have
$$\omega(t)= \frac{\omega(t)}{v(t)} v(t) \le v(\pi) \frac{\omega (t)}{v(t)}= O \biggl( \frac{\omega(t)}{v(t)} \biggr)\quad \text{for } 0 < t \le\pi. $$
Hence, for \(t=(n+1)^{-1}\),
$$ J_{1}=O ( J_{3} ). $$
(4.9)
Again by the monotonicity of \(v(t)\),
$$\begin{aligned} J_{2}&= (n+1)^{-1} \int_{(n+1)^{-1}}^{\pi} \frac{t^{-2} \omega (t)}{v(t)} v(t) \,dt \\ &\le(n+1)^{-1} v(\pi) \int_{(n+1)^{-1}}^{\pi} \frac{t^{-2} \omega (t)}{v(t)} \,dt = O (J_{4} ). \end{aligned}$$
(4.10)
Using the fact that \({\omega(t)}/{v(t)}\) is positive and non-decreasing, we have
$$\begin{aligned} J_{4}&=(n+1)^{-1} \int_{(n+1)^{-1}}^{\pi} \frac{t^{-2} \omega(t)}{v(t)} \,dt \\ &\ge\frac{\omega((n+1)^{-1})}{v((n+1)^{-1})} (n+1)^{-1} \int _{(n+1)^{-1}}^{\pi} t^{-2} \,dt = \frac{\omega ((n+1)^{-1})}{v((n+1)^{-1})} (n+1)^{-1} \bigl( n+1 -{\pi}^{-1} \bigr) \\ &\ge\frac{\omega((n+1)^{-1})}{2 v((n+1)^{-1})}, \end{aligned}$$
as \((n+1)^{-1} (n+1-{\pi}^{-1}) > (n+1)^{-1} {n} \ge1/2\). Therefore
$$ J_{3}= O (J_{4} ). $$
(4.11)
Combining (4.8) with (4.11), we get
$$\bigl\Vert l_{n}(\cdot) \bigr\Vert _{p}^{(v)} = O(J_{4}) = O \biggl((n+1)^{-1} \int_{(n+1)^{-1}}^{\pi} \frac{t^{-2} \omega(t)}{v(t)} \,dt \biggr). $$
Hence
$$E_{n}(f)= \inf_{n} \bigl\Vert l_{n}( \cdot) \bigr\Vert _{p}^{(v)} = O \biggl((n+1)^{-1} \int_{(n+1)^{-1}}^{\pi} \frac{t^{-2} \omega(t)}{v(t)} \,dt \biggr). $$
This completes the proof of Theorem 1.
4.2 Proof of Theorem 2
Following the proof of Theorem 1, we have
$$E_{n}(f)= O \biggl((n+1)^{-1} \int_{(n+1)^{-1}}^{\pi} \frac{t^{-2} \omega (t)}{v(t)} \,dt \biggr). $$
From the assumption that \({t^{-1} \omega(t)}/{v(t)}\) is positive and non-increasing with t, we have
$$\begin{aligned} E_{n}(f) &= O \biggl( (n+1)^{-1} (n+1) \frac{\omega ((n+1)^{-1})}{v((n+1)^{-1})} \int_{(n+1)^{-1}}^{\pi} t^{-1} \,dt \biggr) \\ &= O \biggl( \frac{\omega((n+1)^{-1})}{v((n+1)^{-1})} \log(n+1) \biggr). \end{aligned}$$
This completes the proof of Theorem 2.