1 Introduction
Several results on the error estimation of a function g in Lipschitz and Hölder classes by a trigonometric polynomial using different single and product means have been obtained by the researchers like [1‐11], and [12].
Our motivation for this work is to consider a more advanced class of functions that can provide best approximation by a trigonometric polynomial of degree not more than r. Therefore, in this work, we generalize the results of Kushwaha and Dhakal [3] and Dhakal [1, 2]. In fact, we obtain the results on the error estimation for the function \(f\in H_{z}^{(w)}\) (\(z\geq1\)) by \(T.C^{1}\) method by F. S. Thus, the results of Kushwaha and Dhakal [3] and Dhakal [1, 2] become the particulars cases of our Theorem 2.1.
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We also obtain the results on the error estimation of the function \(\tilde{g} \in H_{z}^{(w)}\) (\(z\geq1\)) by \(T.C^{1}\) method of C. F. S.
Let “\(T=(a_{r,m})\) be an infinite triangular matrix satisfying the conditions of regularity [13], i.e.,
The sequence-to-sequence transformation
defines the sequence \(t_{r}^{T}\) of triangular matrix means of the sequence \(\{s_{r}\}\) generated by the sequence of coefficients \((a_{r,m})\).
$$ \begin{aligned} &\sum_{m=0}^{r}a_{r,m} =1 \quad \mbox{as } r\to\infty, \\ &a_{r,m} =0 \quad \mbox{for } m>r, \\ &\sum_{m=0}^{r}|a_{r,m}| \leq M, \quad \mbox{a finite constant}. \end{aligned} $$
(1)
$$ t_{r}^{T}:=\sum_{m=0}^{r}a_{r,m}s_{m}= \sum_{m=0}^{r}a_{r,r-m}s_{r-m} $$
(2)
If \(t_{r}^{T} \to s\) as \(r\to\infty\), then the infinite series \(\sum_{r=0}^{\infty}h_{r}\) or the sequence \(\{s_{r}\}\) is summable to s by a triangular matrix (T-method) [14].”
“Let
If \(C_{r}^{1} \to s\) as \(r\to\infty\), then the infinite series \(\sum_{r=0}^{\infty}h_{r}\) is summable to s by \(C^{1}\) means [14].” The \(TC^{1}\) means (T-means of \(C^{1}\) means) is given by
If \(t_{r}^{T.C^{1}} \to s\) as \(r \to\infty\), then the series \(\sum_{r=0}^{\infty} h_{r}\) or the sequence \(\{s_{r}\}\) is summable to s by \(T.C^{1}\) means.
$$\begin{aligned} C_{r}^{1} & = \frac{s_{0}+s_{1}+\cdots+s_{r}}{r+1} \\ &=\frac{1}{r+1}\sum_{m=0}^{r}s_{m} \to s \quad \mbox{as }r\to\infty. \end{aligned}$$
(3)
$$\begin{aligned} t_{r}^{T.C^{1}}&:=\sum_{m=0}^{r} a_{r,m} C_{m}^{1} \\ & = \sum_{m=0}^{r}a_{r,m} \frac{1}{m+1} \sum_{v=0}^{m} s_{m}. \end{aligned}$$
(4)
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The regularity of T and \(C^{1}\) methods implies the regularity of \(T.C^{1}\) method.
Remark 1
(Example)
Consider an infinite series
The nth partial sum of (5) is given by
and so
Therefore, series (5) is not summable by \((C,1)\) means.
$$ 1+\sum_{n=1}^{\infty}(-1)^{n}.2n. $$
(5)
$$s_{n}= \textstyle\begin{cases} n+1,&n\text{ is even}, \\ 0,&n\text{ is odd} \end{cases} $$
$$C^{1}_{n}= \textstyle\begin{cases} 1, &n\text{ is even}, \\ 0, &n\text{ is odd}. \end{cases} $$
Remark 2
\(TC^{1}\) means reduces to Let \(L^{z}[0,2\pi]= \{ g: [0,2\pi]\to\mathbb{R}: \int_{0}^{2\pi} |g(x)|^{z} \,dx <\infty, z\geq1\}\) be the space of functions (2π-periodic and integrable). We define the norm \(\|\cdot\|_{(z)}\) by
As defined in “[14], \(w:[0,2\pi] \to\mathbb{R}\) is an arbitrary function with \(w(l)>0\) for \(0< l\leq2\pi\) and \(\lim_{l\to 0^{+}}w(l)=w(0)=0\).” Now we define
and
(i)
\((H,\frac{1}{r+1})C^{1}\) or \(H.C^{1}\) means if \(a_{r,m}= \frac{1}{(r-m+1)\log(r+1)}\);
(ii)
\((N,p_{r})C^{1}\) or \(N_{p}C^{1}\) means if \(a_{r,m}=\frac {p_{r-m}}{P_{r}}\), where \(P_{r}=\sum_{m=0}^{r}p_{m} \neq0\);
(iii)
\((N,p,q)(C,1)\) or \(N_{p,q}C^{1}\) means if \(a_{r,m}=\frac {p_{r-m} q_{m}}{R_{r}}\), where \(R_{r}=\sum_{m=0}^{r}p_{m}q_{r-m}\);
(iv)
\((\bar{N},p_{r})(C,1)\) or \(\bar{N}_{p}C^{1}\) means if \(a_{r,m}=\frac{p_{m}}{P_{r}}\).
$$\biggl\{ \frac{1}{2\pi} \int_{0}^{2\pi} \bigl\vert g(x) \bigr\vert ^{z}\,dx \biggr\} ^{\frac {1}{z}},\quad z\geq1. $$
$$H_{z}^{(w)}= \biggl\{ g\in L^{z}[0,2\pi]:\sup _{l\neq0} \frac {\|g(\cdot,+l)-g(\cdot)\|_{z}}{w(l)} < \infty,z\geq1 \biggr\} $$
$$\Vert\cdot\Vert_{z}^{(w)}= \Vert g \Vert_{z}^{(w)}= \Vert g \Vert_{z}+ \sup_{l\neq0} \frac{\Vert g(\cdot+l)-g(\cdot)\Vert_{z}}{w(l)} ; \quad z\geq1. $$
Note 1
If we consider \(\frac{w(l)}{v(l)}\) as positive and non-decreasing,
Thus,
$$\Vert g \Vert_{z}^{(v)}\leq\max \biggl(1, \frac{w(2\pi)}{v(2\pi)} \biggr) \Vert g\Vert_{z}^{(w)} < \infty. $$
$$H_{z}^{(w)}\subset H_{z}^{(v)}\subset L^{z} ;\quad z\geq1. $$
Remark 3
(i)
If \(w(l)=l^{\alpha}\) in \(H^{(w)}\), \(H^{(w)}\) implies \(H_{\alpha}\) class.
(ii)
If \(w(l)=l^{\alpha}\) in \(H_{z}^{(w)}\), \(H^{(w)}\) implies \(H_{\alpha ,z}\) class.
(iii)
If \(z \to\infty\) in \(H_{z}^{(w)}\), \(H_{z}^{(w)}\) implies \(H^{(w)}\) class and \(H_{\alpha,z}\) class implies \(H_{\alpha}\) class.
Remark 4
We are not representing here the F. S. and C. F. S. as these trigonometric series are well known and the detailed work on these series can be found in [14].
We denote the rth partial sum of the F. S. as
The rth partial sum of C. F. S. is defined as
where
“The error estimation of function g is given by
where \(t_{r}\) is a trigonometric polynomial of degree r [14].”
$$s_{r}(g;x)-g(x)=\frac{1}{2\pi} \int_{0}^{\pi} \phi_{x}(l) \frac{\sin (r+\frac{1}{2})l}{\sin{\frac{l}{2}}}\,dl. $$
$$s_{r}(\tilde{g};x)-\tilde{g}(x)=\frac{1}{2\pi} \int_{0}^{\pi} \psi _{x}(l) \frac{\cos(r+\frac{1}{2})l}{\sin(\frac{l}{2})}\,dl, $$
$$\tilde{g}=-\frac{1}{2\pi} \int_{0}^{\pi}\psi_{x}(l)\cot\biggl( \frac{l}{2}\biggr)\,dl. $$
$$E_{r}(g)=\min\Vert g-t_{r}\Vert_{z}, $$
We write
$$\begin{aligned}& \phi_{x}(l) =\phi(x,l)=g(x+l)+g(x-l)-2g(x), \\& \psi_{x}(l) =\psi(x,l)=g(x+l)-g(x-l), \\& \Delta p_{m} =p_{m}-p_{m+1},\quad m\geq{0}, \\& H_{r}(l) =\frac{1}{2\pi}\sum_{m=0}^{r}a_{r,m} \frac{1}{m+1}\sum_{v=0}^{m} \frac{\sin(v+\frac{1}{2})l}{\sin(\frac{l}{2})}, \\& \tilde{H}_{r}(l) =\frac{1}{2\pi}\sum_{m=0}^{r}a_{r,m} \frac{1}{m+1}\sum_{v=0}^{m} \frac{\cos(v+\frac{1}{2})l}{\sin(\frac{l}{2})}. \end{aligned}$$
2 Main theorems
Theorem 2.1
If
\(g\in H_{z}^{(w)}\)
class; \(z\geq1\)
and
\(\frac{w(l)}{v(l)}\)
are positive and non-decreasing, then the error estimation of
g
by
\(TC^{1}\)
means of F. S. is
where
\(T=(a_{r,m})\)
is an infinite triangular matrix satisfying (1) and
w, v
are defined as in Note
1
provided
$$\bigl\Vert t_{r}^{T.C^{1}}-g \bigr\Vert _{z}^{(v)} =O \biggl( \frac{1}{r+1} \int _{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}v(l)}\,dl \biggr), $$
$$ \sum_{m =0}^{r-1}|\Delta a_{r,m}|=O \biggl(\frac{1}{r+1} \biggr) \quad \textit {and}\quad (r+1)a_{r,r}=O(1). $$
(6)
Theorem 2.2
If
\(\tilde{g}\in H_{z}^{(w)}\)
class; \(z\geq1\)
and
\(\frac{w(l)}{v(l)}\)
are positive and non-decreasing, then the error estimation of
g̃
by
\(TC^{1}\)
means of C. F. S. is
where
\(T=(a_{r,m})\)
is an infinite triangular matrix satisfying (1), (6) and
w, v
are defined as in Note
1.
$$\bigl\Vert \tilde{t_{r}}^{T.C^{1}} -\tilde{g} \bigr\Vert _{z}^{(v)}=O \biggl( \frac{(\log (r+1)+1)}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}v(l)}\,dl \biggr), $$
3 Lemmas
Lemma 3.1
Proof
For \(0< l<\frac{1}{r+1}\), \(\sin(\frac{l}{2}) \geq\frac{l}{\pi}\), \(\sin (r l) \leq r l\).
□
$$\begin{aligned}& H_{r}(l)=\frac{1}{2\pi} \sum_{m=0}^{r}a_{r,m} \frac{1}{m+1} \sum_{v=0}^{m} \frac{\sin(v+\frac{1}{2})l}{\sin(\frac{l}{2})}, \\& \begin{aligned} \bigl\vert H_{r}(l) \bigr\vert & \leq \frac{1}{2\pi} \times\frac{\pi}{l} \Biggl\vert \sum _{m=0}^{r}a_{r,m}\frac{1}{m+1}\sum _{v=0}^{m}\sin\biggl(v+\frac{1}{2} \biggr)l \Biggr\vert \\ & = \frac{1}{2l} \Biggl\vert \sum_{m=0}^{r}a_{r,m} \frac{1}{m+1} \sum_{v=0}^{m}\sin(2v+1) \frac{l}{2} \Biggr\vert \\ & \leq\frac{1}{2l} \Biggl\vert \sum_{m=0}^{r} a_{r,m} \frac{1}{m+1} \sum_{v=0}^{m}(2v+1) \frac{l}{2} \Biggr\vert \\ &=\frac{1}{4} \Biggl\vert \sum_{m=0}^{r}a_{r,m} \frac{1}{m+1}\sum_{v=0}^{m}(2v+1) \Biggr\vert \\ & = \frac{1}{4} \Biggl\vert \sum_{m=0}^{r} a_{r,m} \frac{1}{m+1} \times (m+1)^{2} \Biggr\vert \\ & = \frac{1}{4} \Biggl\vert \sum_{m=0}^{r}a_{r,m}(m+1) \Biggr\vert \\ & = \frac{1}{4}(m+1)\sum_{m=0}^{r}|a_{r,m}| \\ &=O(r+1). \end{aligned} \end{aligned}$$
Lemma 3.2
Proof
For \(\frac{1}{r+1} \leq l \leq\pi\), \(\sin(\frac{l}{2}) \geq\frac {l}{\pi}\), \(\sin^{2}{r}l \leq1\) and using Abel’s lemma, we have
□
$$\begin{aligned}& H_{r}(l) = \frac{1}{2\pi}\sum_{m=0}^{r}a_{r,m} \frac{1}{r+1} \sum_{v=0}^{r} \frac{\sin(v+\frac{1}{2})l}{\sin(\frac{l}{2})}, \\& \bigl\vert H_{r}(l) \bigr\vert \leq\frac{1}{2\pi} \times \frac{\pi}{l} \Biggl\vert \sum_{m=0}^{r}a_{r,m} \frac{1}{m+1} \sum_{v=0}^{m} \sin \biggl(v+\frac{1}{2}\biggr)l \Biggr\vert \\& \hphantom{ \vert H_{r}(l) \vert } = \frac{1}{2l} \Biggl\vert \sum _{m=0}^{r}a_{r,m} \frac{1}{m+1} \operatorname{Im} \Biggl\{ \sum_{v=0}^{m} e^{i(v+\frac{1}{2})l} \Biggr\} \Biggr\vert \\& \hphantom{ \vert H_{r}(l) \vert } = \frac{1}{2l} \Biggl\vert \sum _{m=0}^{r} a_{r,m} \frac{1}{m+1} \operatorname{Im} \Biggl\{ e^{i\frac{l}{2}} \sum_{v=0}^{m}e^{ivl} \Biggr\} \Biggr\vert \\& \hphantom{ \vert H_{r}(l) \vert } = \frac{1}{2l} \Biggl\vert \sum _{m=0}^{r} a_{r,m} \frac{1}{m+1} \operatorname{Im} \biggl\{ e^{\frac{il}{2}} \frac{1-e^{i(m+1)l}}{1-e^{il}} \biggr\} \Biggr\vert \\& \hphantom{ \vert H_{r}(l) \vert } = \frac{1}{2l} \Biggl\vert \sum _{m=0}^{r}a_{r,m} \frac{1}{m+1} \operatorname{Im} \biggl\{ \frac{e^{i(m+1)l}-1}{2i\sin(\frac{l}{2})} \biggr\} \Biggr\vert \\& \hphantom{ \vert H_{r}(l) \vert } \leq\frac{1}{2l} \times\frac{\pi}{l} \Biggl\vert \sum_{m=0}^{r} a_{r,m} \frac{1}{m+1} \sin^{2}(m+1) \frac{l}{2} \Biggr\vert \\& \hphantom{ \vert H_{r}(l) \vert } \leq\frac{\pi}{2l^{2}} \Biggl\vert \sum _{m=0}^{r} a_{r,m} \frac {1}{m+1} \Biggr\vert \\& \hphantom{ \vert H_{r}(l) \vert } = \frac{\pi}{2l^{2}} \Biggl\vert \sum _{m=0}^{r-1}(a_{r,m}-a_{r,m+1})\sum _{v=0}^{m}\frac{1}{v+1} +a_{r,r} \sum_{m=0}^{r} \frac{1}{m+1} \Biggr\vert \\& \hphantom{ \vert H_{r}(l) \vert } \leq \frac{\pi}{2l^{2}} \Biggl\vert \sum _{m=0}^{r-1} \Delta a_{r,m}\sum _{v=0}^{m}\frac{1}{v+1} \Biggr\vert + a_{r,r} \Biggl\vert \sum_{m=0}^{r} \frac {1}{m+1} \Biggr\vert \\& \hphantom{ \vert H_{r}(l) \vert } \leq\frac{\pi}{2l^{2}} \Biggl[ \sum _{m=0}^{r-1} |\Delta a_{r,m}| +a_{r,r} \Biggr] \max_{0\leq m \leq d} \Biggl\vert \sum _{m=0}^{d}\frac{1}{m+1} \Biggr\vert \\& \hphantom{ \vert H_{r}(l) \vert } =O \biggl( \frac{1}{(r+1)l^{2}} \biggr). \end{aligned}$$
Lemma 3.3
Proof
For \(0< l\leq\frac{1}{r+1}\), using \(\sin(\frac{l}{2}) \geq\frac{l}{\pi }\) and \(|\cos{rl}| \leq1\), we obtain
□
$$\begin{aligned}& \tilde{H_{r}}(l) =\frac{1}{2\pi} \sum _{m=0}^{r}a_{r,m} \frac{1}{m+1} \sum _{v=0}^{m} \frac{\cos(v+\frac{1}{2})l}{\sin(\frac{l}{2})}, \\& \begin{aligned} \bigl\vert \tilde{H_{r}}(l) \bigr\vert & \leq \frac{1}{2\pi} \times\frac{\pi}{l} \sum_{m=0}^{r} a_{r,m} \frac{1}{m+1} \sum_{v=0}^{m} \biggl|\cos\biggl(v+\frac{1}{2}\biggr)l\biggr| \\ & \leq\frac{1}{2l} \sum_{m=0}^{r} a_{r,m} \frac{1}{m+1} \sum_{v=0}^{m}1 \\ &\leq\frac{1}{2l} \sum_{m=0}^{r} a_{r,m}, \end{aligned} \\& \therefore\tilde{H_{r}}(l)= O \biggl( \frac{1}{l} \biggr). \end{aligned}$$
Lemma 3.4
Proof
For \(\frac{1}{r+1} \leq l \leq\pi\), using \(\sin(\frac{l}{2}) \geq \frac{l}{\pi}\), Abel’s lemma, and \(\vert \sum_{m=0}^{r}\frac{\sin (m+1)l}{m+1} \vert \leq1+\frac{\pi}{2}\) ∀r and l [15], we get
□
$$\begin{aligned} &\bigl\vert \tilde{H_{r}}(l) \bigr\vert \\ &\quad \leq\frac{1}{2\pi} \times\frac{\pi}{l} \Biggl\vert \sum_{m=0}^{r} a_{r,m} \frac{1}{m+1} \sum_{v=0}^{m} \cos\biggl(v+\frac {1}{2}\biggr)l \Biggr\vert \\ &\quad \leq\frac{1}{2l} \Biggl\vert \sum_{m=0}^{r} a_{r,m} \frac{1}{m+1} \biggl\{ \frac{2\sin(\frac{l}{2}) \cos\frac{l}{2}+ 2\sin(\frac{l}{2}) \cos \frac{3l}{2} +\cdots+ 2\sin(\frac{l}{2}) \cos(\frac{(2m+1)l}{2}) }{ 2\sin(\frac{l}{2})} \biggr\} \Biggr\vert \\ &\quad \leq\frac{1}{4l} \times\frac{\pi}{l} \Biggl\vert \sum _{m=0}^{r} a_{r,m} \frac{1}{m+1} \bigl\{ \sin{l} +\sin{2l} - \sin{l} + \sin{3l} - \sin{2l} + \cdots \\ &\qquad {}+ \sin{(m+1)l} -\sin{m l} \bigr\} \Biggr\vert \\ &\quad \leq\frac{\pi}{4l^{2}} \Biggl\vert \sum_{m=0}^{r} a_{r,m} \frac{\sin(m+1)l}{m+1} \Biggr\vert \\ &\quad \leq\frac{\pi}{4l^{2}} \Biggl\vert \sum_{m=0}^{r-1} (a_{r,m}-a_{r,m+1}) \sum_{v=0}^{m} \frac{\sin(v+1)l}{v+1} +a_{r,r} \sum_{m=0}^{r} \frac{\sin (m+1)l}{m+1} \Biggr\vert \\ &\quad \leq\frac{\pi}{4l^{2}} \Biggl[ \sum_{m=0}^{r-1}| \Delta a_{r,m}| \Biggl\vert \sum_{v=0}^{m} \frac{\sin(v+1)l}{v+1} \Biggr\vert +a_{r,r} \Biggl\vert \sum _{m=0}^{r} \frac{\sin(m+1)l}{m+1} \Biggr\vert \Biggr] \\ &\quad \leq \Biggl[ \frac{1}{l^{2}} \Biggl( \sum_{m=0}^{r-1}| \Delta a_{r,m}| +a_{r,r} \Biggr) \Biggr]. \\ &\quad = \biggl[\frac{1}{l^{2}} \biggl\{ O \biggl(\frac{1}{r+1} \biggr)+O \biggl(\frac{1}{r+1} \biggr) \biggr\} \biggr] \\ &\quad =O \biggl(\frac{1}{(r+1)l^{2}} \biggr). \end{aligned}$$
Lemma 3.5
(“([16], p. 93)”)
Let
\(g\in{H_{z}}^{(w)}\), then for
\(0< l\leq\pi\):
(i)
\(\Vert\phi(\cdot,l)\Vert_{z}=O(w(l))\);
(ii)
\(\Vert\phi(\cdot+y,l)-\phi(\cdot,l)\Vert_{z}= \scriptsize{\big\{\begin{array}{l} O(w(l)),\\ O(w(|y|)); \end{array}} \)
(iii)
If
\(w(l)\)
and
\(v(l)\)
are defined as in Note
1, then
\(\Vert\phi(\cdot+y,l)-\phi(\cdot,l)\Vert_{z}=O (v( \vert y \vert ) (\frac {w(l)}{v(l)} ) )\).
Lemma 3.6
Let
\(\tilde{g}\in{H_{z}}^{(w)}\), then for
\(0< l\leq\pi\):
(i)
\(\Vert\psi(\cdot,l)\Vert_{z}=O(w(l))\);
(ii)
\(\Vert\psi(\cdot+y,l)-\psi(\cdot,l)\Vert_{z}= \scriptsize{\bigl\{\begin{array}{l} O(w(l)),\\ O(w(|y|)); \end{array}\bigr.} \)
(iii)
If
\(w(l)\)
and
\(v(l)\)
are defined as in Note
1, then
\(\Vert \psi(\cdot+y,l)-\psi(\cdot,l) \Vert _{z}=O (v( \vert y \vert ) (\frac {w(l)}{v(l)} ) )\).
Proof
This lemma can be proved along the same lines as the proof of Lemma 3.5(iii). □
4 Proof of the main theorems
4.1 Proof of Theorem 2.1
Proof
Following Titchmarsh [17], \(s_{r}(g;x)\) of F. S. is given by
Now, denoting \(T.C^{1}\) transform of \(s_{r}(g;x)\) by \({t_{r}}^{T.C^{1}}\),
Let
Then
“Using generalized Minkowski’s inequality Chui [18],” we get
Using Lemmas 3.1 and 3.5(iii), we have
Also, using Lemmas 3.2 and 3.5(iii), we get
By (9), (10), and (11), we have
Again applying Minkowski’s inequality, Lemma 3.1, Lemma 3.2, and \(\Vert \phi(\cdot,l) \Vert_{z}=O(w(l))\), we obtain
Now, we have
Using (12) and (13), we get
By the monotonicity of \(v(l)\), \(w(l)=\frac{w(l)}{v(l)} v(l) \leq v(\pi ) \frac{w(l)}{v(l)}\) for \(0< l\leq\pi\), we get
Since w and v are moduli of continuity such that \(\frac {w(l)}{v(l)}\) is positive and non-decreasing, therefore
Then
From (16) and (17), we get
□
$$ s_{r}(g;x)- g(x) = \frac{1}{2\pi} \int_{0}^{\pi} \phi_{x}(l) \frac {\sin(m+\frac{1}{2})l}{\sin(\frac{l}{2})} \,dl. $$
$$\begin{aligned}& \begin{aligned} {t_{r}}^{T.C^{1}}(x)- g(x) & = \sum _{m=0}^{r} a_{r,m} \bigl(C_{m}^{1}(x)- g(x) \bigr) \\ & = \sum_{m=0}^{r} a_{r,m} \Biggl( \frac{1}{m+1} \sum_{v=0}^{m} s_{v}(g;x)-g(x) \Biggr) \\ &= \int_{0}^{\pi} \phi_{x}(l) \Biggl( \frac{1}{2\pi} \sum_{m=0}^{r} a_{r,m} \frac{1}{m+1} \sum_{v=0}^{m} \frac{\sin(v+\frac{1}{2})l}{\sin (\frac{l}{2})} \Biggr) \,dl, \end{aligned} \\& {t_{r}}^{T.C^{1}}(x)- g(x) = \int_{0}^{\pi} \phi_{x}(l) {H_{r}}(l)\,dl. \end{aligned}$$
(7)
$$ R_{r}(x)={t_{r}}^{T.C^{1}}(x)- g(x) = \int_{0}^{\pi} \phi_{x}(l) {H_{r}}(l)\,dl. $$
(8)
$$ R_{r}(x+y)-R_{r}(x) = \int_{0}^{\pi} \bigl( \phi(x+y,l)-\phi(x,l) \bigr) {H_{r}(l)}\,dl. $$
$$\begin{aligned} \bigl\Vert R_{r}(\cdot,+y)-R_{r}(\cdot)\bigr\Vert _{z} & \leq \int_{0}^{\pi} \bigl\Vert \phi (\cdot+y,l)-\phi(\cdot,l) \bigr\Vert _{z} H_{r}(l)\,dt \\ &= \biggl( \int_{0}^{\frac{1}{r+1}} + \int_{\frac{1}{r+1}}^{\pi} \biggr) \bigl\Vert \phi(\cdot+y,l)-\phi( \cdot,l) \bigr\Vert _{z} H_{r}(l)\,dl \\ &=I_{1}+I_{2}. \end{aligned}$$
(9)
$$\begin{aligned} I_{1} &= \int_{0}^{\frac{1}{r+1}} \bigl\Vert \phi(\cdot+y,l)-\phi( \cdot,l) \bigr\Vert _{z} H_{r}(l)\,dl \\ &=O(r+1) \biggl( v\bigl( \vert y \vert \bigr) \int_{0}^{\frac{1}{r+1}} \frac{w(l)}{v(l)} \,dl \biggr) \\ &=O \biggl( v\bigl( \vert y \vert \bigr) \frac{w(\frac{1}{r+1})}{v(\frac{1}{r+1})} \biggr). \end{aligned}$$
(10)
$$\begin{aligned} I_{2} &= \int_{\frac{1}{r+1}}^{\pi} \bigl\Vert \phi(\cdot+y,l)-\phi( \cdot,l) \bigr\Vert _{z} H_{r}(l)\,dl \\ &=O \biggl(\frac{1}{r+1} \int_{\frac{1}{r+1}}^{\pi} v\bigl( \vert y \vert \bigr) \frac {w(l)}{l^{2}v(l)} \,dl \biggr). \end{aligned}$$
(11)
$$ \sup_{y \neq0} \frac{\Vert R_{r}(\cdot,+y)-R_{r}(\cdot)\Vert_{z}}{v(|y|)} =O \biggl( \frac{w(\frac{1}{r+1})}{v(\frac{1}{r+1})} \biggr) + O \biggl( \frac {1}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}v(l)} \,dl \biggr). $$
(12)
$$\begin{aligned} \bigl\Vert R_{r}(\cdot) \bigr\Vert _{z} &= \bigl\Vert t_{r}^{T.C^{1}}-g \bigr\Vert _{z} \\ &\leq \biggl( \int_{0}^{\frac{1}{r+1}}+ \int_{\frac{1}{r+1}}^{\pi} \biggr) \bigl\Vert \phi(\cdot,l) \bigr\Vert _{z} H_{r}(l)\,dl \\ &=O \biggl( (r+1) \int_{0}^{\frac{1}{r+1}} w(l) \,dl \biggr)+O \biggl( \frac {1}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}} \,dl \biggr) \\ &= O \biggl( w \biggl(\frac{1}{r+1} \biggr) \biggr)+O \biggl( \frac{1}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}} \,dl \biggr). \end{aligned}$$
(13)
$$ \bigl\Vert R_{r}(\cdot) \bigr\Vert _{z}^{v} = \bigl\Vert R_{r}(\cdot) \bigr\Vert _{z}+ \sup_{y\neq0} \frac{\Vert R_{r}(\cdot,+y)-R_{r}(\cdot)\Vert_{z}}{v(|y|)}. $$
(14)
$$\begin{aligned} \bigl\Vert R_{r}(\cdot) \bigr\Vert _{z}^{v} &= O \biggl( w \biggl(\frac{1}{r+1} \biggr) \biggr)+ O \biggl( \frac{1}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac {w(l)}{l^{2}} \,dl \biggr) \\ &\quad {}+ O \biggl( \frac{w (\frac{1}{r+1} )}{v ( \frac {1}{r+1} )} \biggr)+ O \biggl( \frac{1}{r+1} \int_{\frac {1}{r+1}}^{\pi} \frac{w(l)}{l^{2}v(l)} \,dl \biggr). \end{aligned}$$
(15)
$$ \bigl\Vert R_{r}(\cdot) \bigr\Vert _{z}^{v} =O \biggl( \frac{w(\frac{1}{r+1})}{v(\frac {1}{r+1})} \biggr)+O \biggl( \frac{1}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}v(l)} \,dl \biggr). $$
(16)
$$ \frac{1}{r+1} \int_{\frac{1}{r+1}}^{\pi}\frac{w(l)}{l^{2}v(l)}\,dl \geq \frac{w(\frac{1}{r+1})}{v(\frac{1}{r+1})} \biggl(\frac{1}{r+1} \biggr) \int _{\frac{1}{r+1}}^{\pi} \frac{1}{l^{2}}\,dl \geq \frac{w ( \frac {1}{r+1} )}{2v ( \frac{1}{r+1} )}. $$
$$ \frac{w ( \frac{1}{r+1} )}{v ( \frac{1}{r+1} )} =O \biggl( \frac{1}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac {w(l)}{l^{2}v(l)} \,dl\biggr). $$
(17)
$$ \begin{aligned} & \bigl\Vert R_{r}(\cdot) \bigr\Vert _{z}^{(v)}=O \biggl(\frac{1}{r+1} \int_{\frac {1}{r+1}}^{\pi} \frac{w(l)}{l^{2}v(l)}\,dl \biggr), \\ & \bigl\Vert t_{r}^{T.C^{1}}-g \bigr\Vert _{z}^{(v)}=O \biggl(\frac{1}{r+1} \int _{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}v(l)} \,dl\biggr). \end{aligned} $$
(18)
4.2 Proof of Theorem 2.2
Proof
The integral representation of \(s_{r}(\tilde{g};x)\) is given by
Now, denoting \(T.C^{1}\) transform of \(s_{r}(\tilde{g};x)\) by \(\tilde {t_{r}}^{T.C^{1}}\), we get
Let
Then
Using “generalized Minkowski’s inequality Chui [18],” we get
$$ s_{r}(\tilde{g};x)- \tilde{g}(x) = \frac{1}{2\pi} \int_{0}^{\pi} \psi _{x}(l) \frac{\cos(r+\frac{1}{2})l}{\sin(\frac{l}{2})} \,dl. $$
$$\begin{aligned}& \begin{aligned} \tilde{t_{r}}^{T.C^{1}}(x)- \tilde{g}(x) & = \sum _{m=0}^{r} a_{r,m} \bigl(C_{m}^{1}(x)- \tilde{g}(x) \bigr) \\ & = \sum_{m=0}^{r} a_{r,m} \Biggl( \frac{1}{m+1} \sum_{v=0}^{m} s_{v}(\tilde{g};x)-\tilde{g}(x) \Biggr) \\ &= \int_{0}^{\pi} \psi_{x}(l) \Biggl( \frac{1}{2\pi} \sum_{m=0}^{r} a_{r,m} \frac{1}{m+1} \sum_{v=0}^{m} \frac{\cos(v+\frac{1}{2})}{\sin (\frac{l}{2})} \Biggr) \,dl, \end{aligned} \\& \tilde{t_{r}}^{T.C^{1}}(x)- \tilde{g}(x) = \int_{0}^{\pi} \psi_{x}(l) \tilde{H_{r}}(l)\,dl. \end{aligned}$$
$$ \tilde{R}_{r}(x) =\tilde{t}_{r}^{T.C^{1}}(x)- \tilde{g}(x)= \int_{0}^{\pi }\psi_{x}(l) \tilde{H}_{r} \,dl. $$
$$ \tilde{R}_{r}(x+y)-\tilde{R}_{r}(x)= \int_{0}^{\pi} \bigl\{ \psi _{x}(x+y,l)- \psi_{x}(x,l) \bigr\} \tilde{H}_{r}(l)\,dl. $$
$$\begin{aligned} \bigl\Vert \tilde{R}_{r}(\cdot+y)-\tilde{R}_{r}(\cdot) \bigr\Vert _{z} & \leq \int_{0}^{\pi } \bigl\Vert \psi_{x}( \cdot+y,l) \bigr\Vert _{z} \tilde{H}_{r}(l)\,dl \\ &= \biggl( \int_{0}^{\frac{1}{r+1}}+ \int_{\frac{1}{r+1}}^{\pi} \biggr) \bigl\Vert \psi(\cdot+y,l)- \psi(\cdot,l) \bigr\Vert _{z} \tilde{R}_{r}(l)\,dl \\ &= I_{1}+I_{2}. \end{aligned}$$
(19)
Using Lemmas 3.3 and 3.6(iii), we have
Again using Lemmas 3.4 and 3.6(iii), we have
Using (19), (20), and (21), we have
Again applying Minkowski’s inequality, Lemma 3.3, Lemma 3.4, and \(\Vert \psi(\cdot,l)\Vert_{z}= O(w(l))\), we have
Now, we have
Using (22) and (23), we get
By the monotonicity of \(v(l)\), we have \(w(l)=\frac{w(l)}{v(l)}v(l)\leq v(\pi) \frac{w(l)}{v(l)}\), \(0< l\leq\pi\), we get
Using the fact that \(\frac{w(l)}{v(l)}\) is positive and non-decreasing, we have
Then
From (24) and (25), we get
□
$$\begin{aligned} I_{1} & = \int_{0}^{\frac{1}{r+1}} \bigl\Vert \psi(\cdot+y,l)-\psi( \cdot,l) \bigr\Vert _{z} \tilde{H_{r}}(l)\,dl \\ &= O \biggl( v\bigl( \vert y \vert \bigr) \frac{w(\frac{1}{r+1})}{ v(\frac{1}{r+1})} \int _{0}^{\frac{1}{r+1}} \frac{1}{l}\,dl \biggr) \\ &=O \biggl( v\bigl( \vert y \vert \bigr) \frac{w(\frac{1}{r+1})}{v(\frac{1}{r+1})} \log(r+1) \biggr). \end{aligned}$$
(20)
$$\begin{aligned} I_{2} &= \int_{\frac{1}{r+1}}^{\pi} \bigl\Vert \psi(\cdot+y,l)-\psi( \cdot,l) \bigr\Vert _{z} \tilde{H_{r}}(l)\,dl \\ &=O \biggl( \frac{1}{r+1} \int_{\frac{1}{r+1}}^{\pi} v\bigl(|y|\bigr) \frac {w(l)}{l^{2} v(l)} \,dl \biggr). \end{aligned}$$
(21)
$$ \sup_{y \neq0} \frac{\Vert\tilde{R}_{r}(\cdot+y)-\tilde{R}_{r}(\cdot) \Vert _{z}}{v(|y|)}= O \biggl( \frac{w(\frac{1}{r+1})}{v(\frac{1}{r+1})}\log (r+1) \biggr)+O \biggl( \frac{1}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac {w(l)}{l^{2} v(l)} \,dl \biggr). $$
(22)
$$\begin{aligned} \bigl\Vert \tilde{R_{r}}(\cdot) \bigr\Vert _{z} &= \bigl\Vert \tilde{t_{r}}^{T.C^{1}}-\tilde {g}\bigr\Vert _{z} \leq \biggl( \int_{0}^{\frac{1}{r+1}} + \int_{\frac{1}{r+1}}^{\pi } \biggr) \bigl\Vert \psi(\cdot,l)\bigr\Vert _{z} \tilde{H_{r}}(l)\,dl \\ &= O \biggl( \int_{0}^{\frac{1}{r+1}}\frac{w(l)}{l}\,dl \biggr)+ O \biggl( \frac{1}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}}\,dl \biggr) \\ & = O \biggl( w \biggl(\frac{1}{r+1} \biggr) \log(r+1) \biggr)+ O \biggl( \frac{1}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}}\,dl \biggr). \end{aligned}$$
(23)
$$ \bigl\Vert \tilde{R_{r}}(\cdot) \bigr\Vert _{z}^{(v)} = \bigl\Vert \tilde{R_{r}}(\cdot) \bigr\Vert _{z}+\sup _{y \neq0} \frac{ \Vert \tilde{R}_{r}(\cdot+y)-\bar{R}_{r}(\cdot) \Vert _{z}}{v(|y|)}. $$
$$\begin{aligned} \bigl\Vert \tilde{R_{r}}(\cdot) \bigr\Vert _{z}^{(v)}&= O \biggl( \bigl(\log(r+1)\bigr) w \biggl(\frac{1}{r+1} \biggr) \biggr)+O \biggl( \frac{1}{r+1} \int_{\frac {1}{r+1}}^{\pi} \frac{w(l)}{l^{2}}\,dl \biggr) \\ &\quad {} + O \biggl( \frac{w(\frac{1}{r+1})}{v(\frac{1}{r+1})} \log(r+1) \biggr)+O \biggl( \frac{1}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac {w(l)}{l^{2}v(l)}\,dl \biggr). \end{aligned}$$
$$ \bigl\Vert \tilde{R_{r}}(\cdot) \bigr\Vert _{z}^{(v)}=O \biggl( \frac{w(\frac {1}{r+1})}{v(\frac{1}{r+1})} \log(r+1) \biggr)+ O \biggl( \frac{1}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}v(l)}\,dl \biggr). $$
(24)
$$\begin{aligned} \frac{1}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}v(l)}\,dl & \geq \frac{w(\frac{1}{r+1})}{v(\frac{1}{r+1})} \frac{1}{r+1} \int_{\frac {1}{r+1}}^{\pi} \frac{1}{l^{2}}\,dl \\ & \geq\frac{w(\frac{1}{r+1})}{2v(\frac{1}{r+1})}. \end{aligned}$$
$$ \frac{w(\frac{1}{r+1})}{v(\frac{1}{r+1})} = O \biggl( \frac{1}{r+1} \int _{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}v(l)}\,dl \biggr). $$
(25)
$$ \begin{aligned} & \bigl\Vert \tilde{R_{r}}(\cdot) \bigr\Vert _{z}^{(v)}=O \biggl( \frac{\log(r+1)}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}v(l)} \,dl \biggr)+O \biggl( \frac{1}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}v(l)} \,dl \biggr), \\ &\therefore \bigl\Vert \tilde{t_{r}}^{T.C^{1}} -\tilde{g} \bigr\Vert _{z}^{(v)}= O \biggl( \frac{\log(r+1)+1}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac {w(l)}{l^{2}v(l)} \,dl \biggr). \end{aligned} $$
(26)
5 Corollary
Corollary 5.1
Let
\(0 \leq\beta<\alpha\leq{1}\)
and
\(\tilde{g} \in H_{(\alpha),z}\); \(z\geq{1}\). Then
$$\bigl\Vert \tilde{t_{r}}^{T.C^{1}} -\tilde{g} \bigr\Vert _{(\beta),z}= \textstyle\begin{cases} O [ (\log(r+1)e)(r+1)^{\beta-\alpha} ]& \textit{if }0\leq {\beta}< \alpha< 1, \\ O [\frac{(\log(r+1)e)(\log(r+1) \pi)}{r+1} ] & \textit{if } \beta=0, \alpha=1. \end{cases} $$
Proof
Putting \(w(l)=l^{\alpha}\), \(v(l)=l^{\beta}\), \(0\leq\beta< \alpha\leq 1\) in (26)
□
$$\begin{aligned}& \bigl\Vert \tilde{t_{r}}^{T.C^{1}} -\tilde{g} \bigr\Vert _{(\beta),z} =O \biggl[ \frac{\log(r+1)e}{r+1} \int_{\frac{1}{r+1}}^{\pi} t^{\alpha-\beta-2} \,dl \biggr] \\& \quad \implies\quad \bigl\Vert \tilde{t_{r}}^{T.C^{1}} - \tilde{g} \bigr\Vert _{(\beta),z}= \textstyle\begin{cases} O ( \frac{(\log(r+1)e)}{(r+1)} \int_{\frac{1}{r+1}}^{\pi}l^{\alpha -\beta-2}\,dl )& \textit{if }0\leq{\beta}< \alpha< 1, \\ O ( \frac{\log(r+1)e}{r+1} \int_{\frac{1}{r+1}}^{\pi}l^{-1}\,dl ) & \textit{if }\beta=0, \alpha=1, \end{cases}\displaystyle \\& \therefore \bigl\Vert \tilde{t_{r}}^{T.C^{1}} -\tilde{g} \bigr\Vert _{(\beta),z}= \textstyle\begin{cases} O [ (\log(r+1)e)(r+1)^{\beta-\alpha} ]& \textit{if }0\leq {\beta}< \alpha< 1, \\ O [\frac{(\log(r+1)e)}{r+1} \times log(r+1)\pi ] & \textit{if } \beta=0, \alpha=1. \end{cases}\displaystyle \end{aligned}$$
Corollary 5.2
Let
\(0 \leq\beta<\alpha\leq{1}\), \(a,b \in\mathbb{R}\)
and suppose
\(w(l)=\frac{l^{\alpha}}{(\log\frac{1}{l})^{a}}\), \(w(l)=\frac{l^{\beta }}{(\log\frac{1}{l})^{b}}\), \(0< l\leq\pi\), \(\tilde{g} \in H_{z}^{(w)}\), \(z\geq{1}\). Then
$$\bigl\Vert \tilde{t_{r}}^{T.C^{1}} -\tilde{g} \bigr\Vert _{z}^{(v)}= \textstyle\begin{cases} O [ \frac{\log(r+1)e}{\{\log(r+1) \}^{b-a}} ]& \textit{if } \alpha=\beta \textit{ and } a-b \geq{-1}, \\ O [\frac{(\log(r+1)e)}{\log(r+1)} ] & \textit{if }\alpha =\beta \textit{ and } a-b=-1. \end{cases} $$
Proof
We have
□
$$\begin{aligned}& \begin{aligned} \bigl\Vert \tilde{t_{r}}^{T.C^{1}} - \tilde{f} \bigr\Vert _{z}^{(v)}&=O \biggl( \frac {\log(r+1)e}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac{l^{\alpha }}{l^{2}(\log\frac{1}{l})^{a} \times\frac{l^{\beta}}{(\log\frac {1}{l})^{b}}}\,dl \biggr) \\ &= O \biggl( \frac{\log(r+1)e}{r+1} \int_{\frac{1}{r+1}}^{\pi} l^{\alpha -\beta-2} \biggl(\log \frac{1}{l}\biggr)^{b-a}\,dl \biggr) \end{aligned} \\& \therefore \bigl\Vert \tilde{t_{\eta}}^{T.C^{1}} -\tilde{g} \bigr\Vert _{z}^{(v)}= \textstyle\begin{cases} O [ \frac{\log(r+1)e}{\{\log(r+1) \}^{b-a}} ]& \text{if $\alpha=\beta\mbox{ and } a-b \geq{-1} $}.\\ O [\frac{(\log(r+1)e)}{\log(r+1)} ] & \text{if $\alpha =\beta\mbox{ and }a-b=-1$}. \end{cases}\displaystyle \end{aligned}$$
Corollary 5.3
If
\(a_{r,m}= \frac{1}{(r-m+1)\log(r+1)}\), then
\(T.C^{1}\)
means reduces to
\((H, \frac{1}{r+1} )(C,1)\)
means and error estimation of a function
\(g\in H_{z}^{(w)}\)
by
\((H, \frac{1}{r+1})(C,1)\)
means of F. S. is
$$\bigl\Vert t_{r}^{H.C^{1}}-g \bigr\Vert _{z}^{(v)}=O \biggl( \frac{1}{r+1} \int _{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}v(l)}\,dl \biggr). $$
Corollary 5.4
If
\(a_{r,m}=\frac{p_{r-m}}{P_{r}}\), then
\(T.C^{1}\)
means reduces to
\(N_{p}.C^{1}\)
and the error estimation of
\(g \in H_{v}^{(w)}\)
by
\(N_{p}.C^{1}\)
means of F. S. is
$$\bigl\Vert t_{r}^{N_{p}.C^{1}}-g \bigr\Vert _{z}^{(v)}=O \biggl( \frac{1}{r+1} \int _{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}v(l)}\,dl \biggr). $$
Corollary 5.5
If
\(a_{r,m}=\frac{p_{r-m}q_{m}}{R_{r}}\), then
\(T.C^{1}\)
means reduces to
\(N_{p,q}.C^{1}\)
and the error estimation of
\(g \in H_{v}^{(w)}\)
by
\(N_{p,q}.C^{1}\)
means of F. S. is
$$\bigl\Vert t_{r}^{N_{p,q}.C^{1}}-g \bigr\Vert _{z}^{(v)}=O \biggl( \frac{1}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac{w(l)}{l^{2}v(l)}\,dl \biggr). $$
Corollary 5.6
If
\(a_{r,m}= \frac{1}{(r-m+1)\log(r+1)}\), then
\(T.C^{1}\)
means reduces to
\((H, \frac{1}{r+1} )(C,1)\)
means and the error estimation of a function
\(\tilde{g}\in H_{z}^{(w)}\)
by
\((H, \frac{1}{r+1})(C,1)\)
means of C. F. S. is
$$\bigl\Vert \tilde{t_{r}}^{H.C^{1}}-\tilde{g} \bigr\Vert _{z}^{(v)}=O \biggl( \frac {(\log(r+1)+1)}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac {w(l)}{l^{2}v(l)}\,dl \biggr). $$
Corollary 5.7
If
\(a_{r,m}=\frac{p_{r-m}}{P_{r}}\), then
\(T.C^{1}\)
means reduces to
\(N_{p}.C^{1}\)
and the error estimation of
\(\tilde{g} \in H_{v}^{(w)}\)
by
\(N_{p}.C^{1}\)
means of C. F. S. is
$$\bigl\Vert \tilde{t_{r}}^{N_{p}.C^{1}}-\tilde{f} \bigr\Vert _{z}^{(v)}=O \biggl( \frac{(\log(r+1)+1)}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac {w(l)}{l^{2}v(l)}\,dl \biggr). $$
Corollary 5.8
If
\(a_{r,m}=\frac{p_{r-m}q_{m}}{R_{r}}\), then
\(T.C^{1}\)
means reduces to
\(N_{p,q}.C^{1}\)
and the error estimation of
\(\tilde{f} \in H_{v}^{(w)}\)
by
\(N_{p,q}.C^{1}\)
means of C. F. S. is
$$\bigl\Vert \tilde{t_{r}}^{N_{p,q}.C^{1}}-\tilde{g} \bigr\Vert _{z}^{(v)}=O \biggl( \frac{(\log(r+1)+1)}{r+1} \int_{\frac{1}{r+1}}^{\pi} \frac {w(l)}{l^{2}v(l)}\,dl \biggr). $$
Remark 5
(i)
If \(z\to\infty\) in \(H_{z}^{(w)}\) class, then \(H_{z}^{(w)}\) class reduces to \(H^{(w)}\) class. Also putting \(w(l)=l^{\alpha}\) and \(v(l)=l^{\beta}\) in our Theorem 2.1, \(H^{(w)}\) class reduces to \(H_{\alpha}\) class; then, by putting \(\beta=0\) in \(H_{\alpha}\) class, \(H_{\alpha}\) class reduces to Lipα class.
(ii)
In our Theorem 2.1, by putting \(w(l)=l^{\alpha}\), \(v(l)=l^{\beta}\) in \(H_{z}^{(w)}\) class, \(H_{z}^{(w)}\) class reduces to \(H_{\alpha,z}\); then, by putting \(\beta=0\) in \(H_{\alpha,z}\) class, \(H_{\alpha,z}\) class reduces to \(\operatorname{Lip}(\alpha,z)\) class.
6 Particular cases
6.2.
7 Conclusion
Approximation by trigonometric polynomials is at the heart of approximation theory. Much of the advances in the theory of trigonometric approximation are due to the periodicity of the functions. The study of error approximation of periodic functions in Lipschitz and Hölder classes has been of great interest among the researchers [1‐11], and [12] in recent past. The trigonometric Fourier approximation (TFA) is of great importance due to its wide applications in different branches of engineering such as electronics and communication engineering, electrical and electronics engineering, computer science engineering, etc. Several elegant results on TFA can be found in a monograph [14].
In this paper, we, for the first time, obtain the best approximation of the functions g and g̃ in a generalized Hölder class \(H_{r}^{(w)}\) (\(r\geq1\)) using Matrix-\(C^{1}\)
\((T.C^{1})\) method of F. S. and C. F. S. respectively. Since, in view of Remark 2, the product summability means \(H.C^{1}\), \(N_{p}C^{1}\), \(N_{p,q}C^{1}\), and \(\bar{N}_{p}C^{1}\) are the particular cases of Matrix-\(C^{1}\) method, so our results also hold for these methods, which are represented in a form of corollaries. In view of Remark 1, it has been shown that \((TC^{1})\) method is more powerful than the individual T method and \(C^{1}\) method. Moreover, in view of Remark 5, some previous results (see Sect. 6) become the particular cases of our Theorem 2.1. We also deduce a corollary for the \(H_{\alpha ,r}\) class (\(r\geq1\)).
Some other studies regarding the modulus of continuity (smoothness) of functions using more generalized functional spaces may be addressed as a future work.
Acknowledgements
The first author expresses his gratitude towards his mother for her blessings. The first author also expresses his gratitude towards his father in heaven, whose soul is always guiding and encouraging him. The second author is thankful to the University Grants Commission (India) for providing Junior Research Fellowship (JRF) to carry out the present work as a part of PhD degree. The second author also expresses his gratitude towards his parents for blessings and is very grateful to his guide Dr. H. K. Nigam without whose help he couldn’t complete his work. Both the authors are also grateful to the Hon’ble vice-chancellor, Central University of South Bihar, for motivation to carry out this work.
Competing interests
The authors declare that they have no competing interests.
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