\documentclass[10pt]{amsart}
\usepackage{cma}

\begin{document}
\title{Applications of the deferred generalized de la Vall\'ee Poussin means in approximation of continuous functions}
\author{Xhevat Z. Krasniqi}
\address{
 University of Prishtina "Hasan Prishtina" \newline
\indent Faculty of Education \newline
\indent Avenue "Mother Theresa " no. 5, Prishtin\"{e} 10000 \newline
\indent Republic of Kosovo}
\email{xhevat.krasniqi@uni-pr.edu}

\setcounter{page}{1}% to be added afterwards
\coordinates{xx}{201x}{1}{00-00}% to be added afterwards
\date{date}% to be added afterwards % submission date 
\undate{date}% to be added afterwards % revised date 
\subjclass[2010]{}% AMS Subject Classification (2010)
\keywords{\em }

\begin{cmaAbstract}
In this paper we have proved a theorem which show the degree of approximation of periodic functions by some generalized means of their Fourier series. In addition, our result is extended to two-dimensional setting as well. 

\vspace{5pt}
\noindent 
\textbf{Keywords: }{Fourier series, Generalized de la Vall\'ee Poussin means, Degree of approximation, Modulus of continuity, Fourier series.}

\vspace{5pt}
\noindent
\textbf{2010 Mathematics Subject Classification:} 42A10, 42A16.
\end{cmaAbstract}

\maketitle
\markboth{Xhevat Z. Krasniqi}{Applications of the deferred generalized de la Vall\'ee Poussin means}


\section{Introduction}

Let $f$ be a $2\pi$-periodic function, $f\in L[0,2\pi]$, and 
\begin{equation}\label{eq11}
\frac{a_0}{2}+\sum_{k=1}^{\infty}(a_k\cos kx+b_k\sin kx),    
\end{equation}
its Fourier series at the point $x$,
where
$$a_k=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos kxdx, \,\, (k=0,1,\dots);\,\,\, b_k=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin kxdx, \,\,(k=1,2,\dots).$$

By 
$$\|f\|=\sup_{0\leq x\leq 2\pi}|f(x)|$$
we denote the sup-norm of $f$ over $[0,2\pi]$, and by $C[0,2\pi]$ the class 
of all $2\pi$-periodic continuous functions defined in $[0,2\pi]$.

In 1928, was G. Alexits \cite{GA} who studied the degree of approximation of function a $f\in \text{Lip}\alpha$ 
by Ces\`aro means $(C,\delta )$ of its Fourier series. This study may be considered as a starting point for other 
studies of this nature, and another type of similar studies can be found in \cite{PC}-\cite{MR}. Recent studies of other researchers can be found in \cite{AM}, \cite{FA}, and \cite{PeC}. 

For our purpose, we are going to recall a result proved in \cite{PC}. To do this we need first to 
present the generalized Vall\'ee Poussin mean given in \cite{LL}.  

Let $\sum_{n=1}^{\infty}w_n$ be a given infinite series and let $s_n$ be its $n$-th partial sum. 
Let $\lambda:=(\lambda_n)$ be a monotone non-decreasing sequence of integers such that $\lambda_1=1$ 
and $\lambda_{n+1}-\lambda_n\leq 1$. 

The mean
\begin{equation}\label{eq12}
    V_n(\lambda)=\frac{1}{\lambda_n}\sum_{m=n-\lambda_n}^{n-1}s_m,\quad (n\geq 1),
\end{equation}
is called the $n$-th generalized de la Vall\'ee Poussin mean of the sequence $(s_n)$ generated by sequence $(\lambda_n)$.

For $n$-th partial sum
$$s_n(f;x)=\frac{a_0}{2}+\sum_{k=1}^{n}(a_k\cos kx+b_k\sin kx)$$
of the series (\ref{eq11}), its $n$-th generalized de la Vall\'ee Poussin mean is defined by
\begin{equation}\label{eq13}
    V_n(\lambda;f;x)=\frac{1}{\lambda_n}\sum_{m=n-\lambda_n}^{n-1}s_m(f;x),\quad (n\geq 1),
\end{equation}
and the modulus of continuity of $f(x)$, for a given real number $\delta>0$, is defined as follows 
$$\omega (f;\delta):=\sup_{|x-y|\leq \delta}|f(x)-f(y)|,$$
where $x,y\in [0,2\pi]$. 



Throughout this paper we write $u=\mathcal{O}(v)$ if there exists a positive constant $K$, such that $u\leq Kv$. 
Now, we are ready to recall the result mentioned above.

\begin{theorem}[\cite{PC}]\label{the11}
Let $f\in C[0,2\pi]$ and $\omega (f;t)$ be its modulus of continuity satisfying the 
following conditions as $t\to +0$:
\begin{equation}\label{eq14}
 \int_{t}^{\frac{\pi}{2}}u^{-2}\omega (f;u)du=\mathcal{O}(F(t)),
\end{equation}
where $F(t)\geq 0$, and
\begin{equation}\label{eq15}
 \int_{0}^{t}F(u)du=\mathcal{O}(tF(t)).
\end{equation}

Then 
\begin{equation}\label{eq16}
 \|f-V_n(\lambda;f)\|=\mathcal{O}\left(\frac{1}{\lambda_n}F\left(\frac{\pi}{2\lambda_n}\right)\right).
\end{equation}
\end{theorem}

For our further investigation let $a:=(a_n)$ and $b:=(b_n)$ be sequences of non-negative integers with condition
\begin{equation}\label{eq17}
1\leq b_n-a_n+\lambda_n,\quad (n=1,2,\dots). 
\end{equation}

Whence, we are in able to generalize the mean $V_n(\lambda)$ defined in (\ref{eq12}) as follows. 

The mean
\begin{equation}\label{eq19}
    V_n(\lambda,a,b)=\frac{1}{b_n-a_n+\lambda_n}\sum_{m=a_n-\lambda_n}^{b_n-1}s_m,\quad (n\geq 1),
\end{equation}
is called the $n$-th deferred generalized de la Vall\'ee Poussin mean of the sequence $(s_n)$ generated by sequences $\lambda$, $a$, and $b$.

It is the purpose of this paper to estimate the deviation $f-V_n(\lambda,a,b)$ in the sup-norm, which in fact generalize Theorem \ref{the11} (as well as we extend it in the two-dimensional setting, see subsec. \ref{sub32}). To do this we need some helpful lemmas given in next section.

\section{Auxiliary Lemma}

Next lemma has been proved implicitly in \cite{PC}. 

\begin{lemma}\label{le20}
Let (\ref{eq14}) hold. Then, $\omega (f;t)=\mathcal{O}(tF(t))$.
\end{lemma}

Now, we prove next helpful lemma.
\begin{lemma}\label{le21} Denote by
$$\mathbb{K}_n^{a,b}(t):=\sum_{m=a_n-\lambda_n}^{b_n-1}D_m(t)=\sum_{m=a_n-\lambda_n}^{b_n-1}\frac{\sin\left(2m+1\right)t}{\sin t}$$
the deferred de la Vall\'ee Poussin kernel, where $D_m(t):=\frac{\sin\left(2m+1\right)t}{\sin t}$.

Then, 
\begin{itemize}
    \item[(i)] $$\mathbb{K}_n^{a,b}(t)=\frac{\sin (b_n-a_n+\lambda_n)t\sin (b_n+a_n-\lambda_n)t}{\sin^2 t},$$
    \item[(ii)] $$|\mathbb{K}_n^{a,b}(t)|=\mathcal{O}\left(\frac{b_n-a_n+\lambda_n}{t}\right),\quad 0<t\leq \frac{\pi}{2(b_n-a_n+\lambda_n)},$$
    \item[(iii)] $$|\mathbb{K}_n^{a,b}(t)|=\mathcal{O}\left(\frac{1}{t^2}\right),\quad \frac{\pi}{2(b_n-a_n+\lambda_n)}<t\leq \frac{\pi}{2}.$$
\end{itemize}
\end{lemma}
\begin{proof}
(i) We have
\begin{eqnarray*}
\mathbb{K}_n^{a,b}(t)&=&\sum_{m=a_n-\lambda_n}^{b_n-1}\frac{\sin\left(2m+1\right)t}{\sin t}\\
&=&\sum_{m=0}^{b_n-1}\frac{2\sin\left(2m+1\right)t\sin t}{2\sin^2 t}-\sum_{m=0}^{a_n-\lambda_n-1}\frac{2\sin\left(2m+1\right)t\sin t}{2\sin^2 t}\\
&=&\frac{1-\cos (2b_nt)}{2\sin^2 t}-\frac{1-\cos (a_n-\lambda_n)t}{2\sin^2 t}\\
&=&\frac{\sin^2(b_nt)-\sin^2(a_n-\lambda_n)t}{\sin^2 t}\\
&=&\frac{\sin (b_n-a_n+\lambda_n)t\sin (b_n+a_n-\lambda_n)t}{\sin^2 t}.
\end{eqnarray*}

(ii) Using the inequalities $|\sin \beta |\leq 1$, $|\sin \beta |\leq \beta $, and $\sin \beta \geq \frac{2}{\pi}\beta$ for $0<\beta \leq \frac{\pi}{2}$, we have:
$$|\mathbb{K}_n^{a,b}(t)|\leq \frac{\pi^2(b_n-a_n+\lambda_n)t}{4t^2}=\mathcal{O}\left(\frac{b_n-a_n+\lambda_n}{t}\right).$$

(iii) Similarly, using the inequalities $|\sin \beta |\leq 1$ and $\sin \beta \geq \frac{2}{\pi}\beta$ for $0<\beta\leq \frac{\pi}{2}$, we also have:
$$|\mathbb{K}_n^{a,b}(t)|\leq \frac{\pi^2}{4t^2}=\mathcal{O}\left(\frac{1}{t^2}\right).$$

The proof is completed.
\end{proof}
 
In the sequel we pass to the main result.

\section{Main Result}
\subsection{Approximation by deferred  generalized de la Vall\'ee Poussin mean of single  Fourier series}
Here, we prove the following.

\begin{theorem}\label{the21}
Let $f\in C[0,2\pi]$ and $\omega (f;t)$ be its modulus of continuity satisfying conditions (\ref{eq14}) 
and (\ref{eq15}) as $t\to +0$, where $F(t)\geq 0$.

Then 
\begin{equation}\label{eq210}
 \|f-V_n(\lambda,a,b;f)\|=\mathcal{O}\left(\frac{1}{b_n-a_n+\lambda_n}F\left(\frac{\pi}{2(b_n-a_n+\lambda_n)}\right)\right).
\end{equation}
\end{theorem}
\begin{proof}
After some calculation we have: 
$$s_m(f;x)=\frac{1}{\pi}\int_{0}^{\frac{\pi}{2}}[f(x+2t)+f(x-2t)]D_m(t)dt,$$
where
$D_m(t)=\frac{\sin\left(2m+1\right)t}{\sin t}$.

Denoting by $V_n(\lambda,a,b;f;x)$ the deferred generalized de la Vall\'ee Poussin mean of $s_m(f;x)$, i.e.,
\begin{equation*}
V_n(\lambda,a,b;f;x):=\frac{1}{b_n-a_n+\lambda_n}\sum_{m=a_n-\lambda_n}^{b_n-1}s_m(f;x),    
\end{equation*}
we get:
\begin{equation*}
 V_n(\lambda,a,b;f;x)-f(x)=\frac{1}{(b_n-a_n+\lambda_n)\pi}\int_{0}^{\frac{\pi}{2}}\psi_x(t)\mathbb{K}_n^{a,b}(t)dt,   
\end{equation*}
where
$$\psi_x(t):=f(x+2t)+f(x-2t)-f(x).$$

Whence,
\begin{eqnarray}\label{eq211}
 &&\|V_n(\lambda,a,b;f)-f\|\leq \frac{1}{(b_n-a_n+\lambda_n)\pi}\int_{0}^{\frac{\pi}{2}}|\psi_x(t)||\mathbb{K}_n^{a,b}(t)|dt\nonumber \\
&&\hspace{1.3cm}\leq \frac{4}{(b_n-a_n+\lambda_n)\pi}\left(\int_{0}^{\frac{\pi}{2(b_n-a_n+\lambda_n)}}+\int_{\frac{\pi}{2(b_n-a_n+\lambda_n)}}^{\frac{\pi}{2}}\right)\omega(f;t)|\mathbb{K}_n^{a,b}(t)|dt\nonumber \\ 
&&\hspace{1.3cm}:=\mathbb{P}_1+\mathbb{P}_2.
\end{eqnarray}

Using Lemma \ref{le21}, part (ii), we obtain:
$$|\mathbb{P}_1|= \mathcal{O}\left(1\right)\int_{0}^{\frac{\pi}{2(b_n-a_n+\lambda_n)}}t^{-1}\omega(f;t)dt,$$
and applying Lemma \ref{le20}, (\ref{eq14}) 
and (\ref{eq15}), we get:
\begin{eqnarray}\label{eq212}
|\mathbb{P}_1|
&=& \mathcal{O}\left(1\right)\int_{0}^{\frac{\pi}{2(b_n-a_n+\lambda_n)}}\int_{t}^{\frac{\pi}{2}}u^{-2}\omega(f;u)dudt\nonumber \\
&=& \mathcal{O}\left(1\right)\int_{0}^{\frac{\pi}{2(b_n-a_n+\lambda_n)}}F(t)dt\nonumber \\
&=& \mathcal{O}\left(\frac{1}{b_n-a_n+\lambda_n}F\left(\frac{\pi}{2(b_n-a_n+\lambda_n)}\right)\right).
\end{eqnarray}

To estimate $\mathbb{P}_2$, we use Lemma \ref{le21}, part (iii). Namely, based on (\ref{eq14}), we have
\begin{eqnarray}\label{eq213}
|\mathbb{P}_2|
&=& \mathcal{O}\left(\frac{1}{(b_n-a_n+\lambda_n)\pi}\right)\int_{\frac{\pi}{2(b_n-a_n+\lambda_n)}}^{\frac{\pi}{2}}t^{-2}\omega(f;t)dt\nonumber \\
&=& \mathcal{O}\left(\frac{1}{b_n-a_n+\lambda_n}F\left(\frac{\pi}{2(b_n-a_n+\lambda_n)}\right)\right).
\end{eqnarray}

Finally, inserting (\ref{eq211}) and (\ref{eq212}) into (\ref{eq213}), we immediately obtain (\ref{eq210}) as required.

The proof is completed. 
\end{proof}

\begin{remark}
Since, in general, $\lambda_n\leq b_n-a_n+\lambda_n$, then we observe  that the degree of approximation obtained in Theorem \ref{the21} is not worse than that appears in Theorem \ref{the11}. 
\end{remark}

\begin{remark}
For $b_n=a_n=n$, we immediately obtain the result given in \cite{PC}. 
\end{remark}

Further, let the sequences $a:=(a_n)$ and $b:=(b_n)$ be of non-negative integers with conditions
\begin{equation}\label{eq214}
a_n<b_n,\quad n=1,2,\dots ,
\end{equation}
and
\begin{equation}\label{eq215}
\lim _{n\to \infty}b_n=+\infty . 
\end{equation}

If $\lambda_n=1$ for all $n\geq 1$, then the deferred de la Vall\'ee Poussin mean $V_n(1,a+2,b+1;f;x)$ reduces to
$$D_{a}^{b}(f;x):=\frac{1}{b_n-a_n}\sum_{m=a_n+1}^{b_n}s_m(f;x),$$
which is the deferred Ces\`aro mean of the sum $s_n(f;x)$ introduced in \cite{RPA}. 
In the same paper, it was shown that (\ref{eq214}) and (\ref{eq215}) are conditions of 
regularity for $D_{a}^{b}$. Consequently, if conditions (\ref{eq214}) and (\ref{eq215}) are satisfied, then from Theorem \ref{the21} we deduce the following. 
\begin{corollary}\label{cor21}
Let $f\in C[0,2\pi]$ and $\omega (f;t)$ be its modulus of continuity satisfying conditions (\ref{eq14}) 
and (\ref{eq15}) as $t\to +0$, where $F(t)\geq 0$.

Then 
\begin{equation*}
 \|f-D_{a}^{b}(f)\|=\mathcal{O}\left(\frac{1}{b_n-a_n}F\left(\frac{\pi}{2(b_n-a_n)}\right)\right).
\end{equation*}
\end{corollary}

Also, if we take $\lambda_n=n$, $a_n=n$, $b_n=n+1$, $\forall n\geq 1$, then the deferred generalized de la Vall\'ee Poussin mean reduces to ordinary Ces\`aro mean of the sum $s_n(f;x)$,
$$\sigma_n(f;x):=\frac{1}{n+1}\sum_{m=0}^{n}s_m(f;x).$$

Therefore, Theorem \ref{the21} also implies:
\begin{corollary}\label{cor22}
Let $f\in C[0,2\pi]$ and $\omega (f;t)$ be its modulus of continuity satisfying conditions (\ref{eq14}) 
and (\ref{eq15}) as $t\to +0$, where $F(t)\geq 0$.

Then 
\begin{equation*}
 \|f-\sigma_n(f)\|=\mathcal{O}\left(\frac{1}{n+1}F\left(\frac{\pi}{2(n+1)}\right)\right).
\end{equation*}
\end{corollary}

Let us specify the function $F(t)$ as follows:

\begin{equation*}
   F(t)= \begin{cases}
               t^{\gamma -1},               & 0< \gamma <1;\\
               \log \left(\frac{\pi}{t}\right),               & \gamma =1.
           \end{cases}
\end{equation*}

Using this function the following estimations from Theorem \ref{the21}, Corollary \ref{cor21}, and Corollary \ref{cor22}
can be deduced (of course all other conditions are maintaining): 
\begin{itemize}
    \item[(a)] From Theorem \ref{the21}:
    \begin{equation*}
   \|f-V_n(\lambda,a,b;f)\|= \begin{cases}
               \mathcal{O}_{\gamma}\left(\frac{1}{(b_n-a_n+\lambda_n)^{\gamma }}\right),               & 0< \gamma <1;\\
               \frac{\log \left(2(b_n-a_n+\lambda_n)\right)}{b_n-a_n+\lambda_n},               & \gamma =1.
           \end{cases}
\end{equation*}
\item[(b)] From Corollary \ref{cor21}:
    \begin{equation*}
   \|f-D_{a}^{b}(f)\|= \begin{cases}
               \mathcal{O}_{\gamma}\left(\frac{1}{(b_n-a_n)^{\gamma }}\right),               & 0< \gamma <1;\\
               \frac{\log \left(2(b_n-a_n)\right)}{b_n-a_n},               & \gamma =1.
           \end{cases}
\end{equation*}
\item[(c)] From Corollary \ref{cor22} (this is a particular case of a result given in \cite{GA}):
    \begin{equation*}
   \|f-\sigma_n(f)\|= \begin{cases}
               \mathcal{O}_{\gamma}\left(\frac{1}{(n+1)^{\gamma }}\right),               & 0< \gamma <1;\\
               \frac{\log \left(2(n+1)\right)}{n+1},               & \gamma =1.
           \end{cases}
\end{equation*}
\end{itemize}
\subsection{Approximation by deferred generalized de la Vall\'ee Poussin mean of double Fourier series}\label{sub32}

Let $C([-\pi, \pi ]^2)$ be the class of real-valued functions of two variables that are continuous
on $[-\pi, \pi ]\times[-\pi, \pi ]:=[-\pi, \pi ]^2$ and $2\pi $ periodic with respect to $x$ and $y$. We recall that the double Fourier series of the function  $f(x,y)\in C([-\pi, \pi ]^2)$ is defined by
\begin{equation*}
\begin{split}
f(x,y)\sim \sum_{m=1}^{\infty }\sum_{n=1}^{\infty }\lambda _{mn}\bigg[&a_{mn}\cos mx\cos ny+b_{mn}\sin mx\cos ny\\
&+c_{mn}\cos mx\sin ny+d_{mn}\sin mx\sin ny\bigg],
\end{split}
\end{equation*}
where
$$
\lambda _{mn} = \begin{cases} 1/4, & \mbox{if } m=n=0, \\ 
1/2, & \mbox{if } m>0, n=0 \vee m=0, n>0, \\
1, & \mbox{if } m>0, n>0 ,
\end{cases}
$$
and 
$$a_{mn}=\frac{1}{\pi ^2}\int_{-\pi}^{\pi }\int_{-\pi}^{\pi }f(u,v)\cos mu\cos nvdudv,$$
$$b_{mn}=\frac{1}{\pi ^2}\int_{-\pi}^{\pi }\int_{-\pi}^{\pi }f(u,v)\sin mu\cos nvdudv,$$
$$c_{mn}=\frac{1}{\pi ^2}\int_{-\pi}^{\pi }\int_{-\pi}^{\pi }f(u,v)\cos mu\sin nvdudv,$$
$$d_{mn}=\frac{1}{\pi ^2}\int_{-\pi}^{\pi }\int_{-\pi}^{\pi }f(u,v)\sin mu\sin nvdudv,$$
are the Fourier coefficients of the function $f(x,y)$. 

The sequence $\{s_{m,n}(f;x,y)\}$ represents the sequence of partial sums of the double Fourier series which can be rewritten in integral form by
$$s_{m,n}(x,y):=s_{m,n}(f;x,y):=\frac{1}{\pi ^2}\int_{-\pi}^{\pi }\int_{-\pi}^{\pi }f(x+u,y+v)D_{m}(u)D_{n}(v)dudv.$$

To my best knowledge the double de la  Vall\'ee Poussin mean of $s_{m,n}(x,y)$ is defined by (see \cite{RK})
\begin{equation}\label{eq315}
    V_{m,n}^{(p,q)}(f;x,y):=\frac{1}{(p+1)(q+1)}\sum_{k=n}^{n+p}\sum_{\ell=m}^{m+q}s_{k,\ell}(x,y),\quad p\geq 0,\, q\geq 0.
\end{equation}

The mean $V_{m,n}^{(p,q)}(f;x,y)$ is generalized in \cite{MA} as follows (for our purposes we modify it "a little bit").  
Let $\lambda:=(\lambda_m)$ and $\mu:=(\mu_n)$ be two monotone non-decreasing sequences of integers such that $\lambda_1=\mu_1=1$, $\lambda_{m+1}-\lambda_m\leq 1$, and $\mu_{n+1}-\mu_n\leq 1$. 

The mean
\begin{equation}\label{eq16}
    V_{m,n}^{\lambda,\mu}(f;x,y)=\frac{1}{\lambda_m\mu_n}\sum_{k=m-\lambda_m}^{m-1}\sum_{k=n-\mu_n}^{n-1}s_{k,\ell}(x,y),\quad (m,n\geq 1),
\end{equation}
is called the $(mn)$-th deferred  generalized de la Vallée-Poussin mean of the sequence $(s_{k,\ell}(x,y))$ generated by sequences $(\lambda_m)$ and $(\mu_n)$.

The (total) modulus of continuity of a continuous function $f(x,y)$, $2\pi$-periodic in each variable, in symbols $f\in C([-\pi, \pi ]^2)$, is defined by (see \cite{FB}, page 283)
$$\omega_1(f,\delta _1, \delta _2)=\sup_{x,y}\sup_{|u|\leq \delta_1,|v|\leq \delta_2}|f(x+u,y+v)-f(x,y)|,\quad \delta _1, \delta _2\geq 0.$$

To estimate the deviation
$$\max_{(x,y)\in Q}\left|V_{m,n}^{\lambda,\mu}(f;x,y)-f(x,y)\right|,$$
which is the main result of this subsection, first we denote
\begin{eqnarray*}\phi_{xy}(s,t):=f(x+s,y+t)\!\!\!\!&+&\!\!\!\!f(x-s,y+t)\\
\!\!\!\!&+&\!\!\!\!f(x+s,y-t)+f(x-s,y-t)-4f(x,y).
\end{eqnarray*}

Now, we are in able to prove the following.

\begin{theorem}\label{the33} Let $f\in C([-\pi,\pi]^2)$, 
$\omega_1(f,s,t)=\mathcal{O}\left(\omega^{(1)}(s) \omega^{(2)}(t)\right)$, where $\omega^{(1)}(s)$ and $\omega^{(2)}(t)$ are two non-negative functions of modulus type satisfying conditions  (\ref{eq14}) and (\ref{eq15}) as $s, t\to +0$, and $F_1(s), F_2(t)\geq 0$ two mediate functions. Then 
\begin{eqnarray}\label{eq317}
\max_{(x,y)\in Q}\left|V_{m,n}^{\lambda,\mu}(f;x,y)-f(x,y)\right|
=\mathcal{O}\left(\frac{1}{\lambda_m\lambda_n}F_1\left(\frac{\pi}{2\lambda_m}\right)F_2\left(\frac{\pi}{2\lambda_n}\right)\right).\nonumber      
\end{eqnarray}
\end{theorem}
\begin{proof}
After some transforms we get:
\begin{equation}\label{eq318}
V_{m,n}^{\lambda,\mu}(f;x,y)-f(x,y)=\frac{1}{\pi^2}\int_{0}^{\frac{\pi}{2}}\int_{0}^{\frac{\pi}{2}}\phi_{xy}(2s,2t)K_{mn}^{\lambda,\mu}(s,t)dsdt,  
\end{equation}
where
$$K_{mn}^{\lambda,\mu}(s,t):=\frac{1}{\lambda_m\mu_n}\sum_{k=m-\lambda_m}^{m-1}\sum_{\ell=n-\mu_n}^{n-1}\frac{\sin\left(2k+1\right)s}{\sin s}\frac{\sin\left(2\ell+1\right)t}{\sin t}.$$

Without difficulty the quantity $K_{mn}^{\lambda,\mu}(s,t)$ can be written as
$$K_{mn}^{\lambda,\mu}(s,t)=\frac{\sin (\lambda_ms)\sin[\left(2m-\lambda_m\right)s]\sin (\mu_nt)\sin[\left(2n-\mu_n\right)t]}{\lambda_m\mu_n\sin^2 s\sin^2 t}.$$

Therefore, we have:
\begin{eqnarray}\label{eq318}
&&|V_{m,n}^{\lambda,\mu}(f;x,y)-f(x,y)|\leq  \left(\frac{4}{\pi}\right)^2\int_{0}^{\frac{\pi}{2}}\int_{0}^{\frac{\pi}{2}}\omega_1(f,s,t)|K_{mn}^{\lambda,\mu}(s,t)|dsdt\nonumber \\
&&=\mathcal{O}\left(\int_{0}^{\frac{\pi}{2\lambda_m}}\int_{0}^{\frac{\pi}{2\mu_n}}+\int_{\frac{\pi}{2\lambda_m}}^{\frac{\pi}{2}}\int_{0}^{\frac{\pi}{2\mu_n}}+\int_{0}^{\frac{\pi}{2\lambda_m}}\int_{\frac{\pi}{2\mu_n}}^{\frac{\pi}{2}}+\int_{\frac{\pi}{2\lambda_m}}^{\frac{\pi}{2}}\int_{\frac{\pi}{2\mu_n}}^{\frac{\pi}{2}}\right)\nonumber \\
&&:=\mathcal{O}\left(\mathbb{S}_1+\mathbb{S}_2+\mathbb{S}_3+\mathbb{S}_4\right).
\end{eqnarray}

Using Jordan's inequality $\sin \nu \geq \frac{2}{\pi}\nu$ for $0<\nu\leq \frac{\pi}{2}$, given assumptions, and Lemma \ref{le20}, we obtain:
\begin{eqnarray}\label{eq319}
\mathbb{S}_1&=&\mathcal{O}\left(1\right)\int_{0}^{\frac{\pi}{2\lambda_m}}\int_{0}^{\frac{\pi}{2\mu_n}}s^{-1}t^{-1}\omega_1(f,s,t)dsdt\\
&=&\mathcal{O}\left(\frac{1}{\lambda_m\mu_n}F_1\left(\frac{\pi}{2\lambda_m}\right)F_2\left(\frac{\pi}{2\mu_n}\right)\right). \nonumber   
\end{eqnarray}

Using the same arguments and Lemma \ref{le21}, we also obtain:
\begin{eqnarray}\label{eq320}
\mathbb{S}_2&=&\mathcal{O}\left(1\right)\int_{\frac{\pi}{2\lambda_m}}^{\frac{\pi}{2}}\int_{0}^{\frac{\pi}{2\mu_n}}s^{-2}t^{-1}\omega_1(f,s,t)dsdt\\
&=&\mathcal{O}\left(\frac{1}{\lambda_m\mu_n}F_1\left(\frac{\pi}{2\lambda_m}\right)F_2\left(\frac{\pi}{2\mu_n}\right)\right).  \nonumber  
\end{eqnarray}

With very similar reasoning, we get:
\begin{eqnarray}\label{eq321}
\mathbb{S}_3&=&\mathcal{O}\left(1\right)\int_{0}^{\frac{\pi}{2\lambda_m}}\int_{\frac{\pi}{2\mu_n}}^{\frac{\pi}{2}}s^{-1}t^{-2}\omega_1(f,s,t)dsdt\\
&=&\mathcal{O}\left(\frac{1}{\lambda_m\mu_n}F_1\left(\frac{\pi}{2\lambda_m}\right)F_2\left(\frac{\pi}{2\mu_n}\right)\right). \nonumber  
\end{eqnarray}

Finally, based on given assumptions, and Lemma \ref{le21} twice, we have:
\begin{eqnarray}\label{eq322}
\mathbb{S}_4&=&\mathcal{O}\left(1\right)\int_{\frac{\pi}{2\lambda_m}}^{\frac{\pi}{2}}\int_{\frac{\pi}{2\mu_n}}^{\frac{\pi}{2}}s^{-2}t^{-2}\omega_1(f,s,t)dsdt\\
&=&\mathcal{O}\left(\frac{1}{\lambda_m\mu_n}F_1\left(\frac{\pi}{2\lambda_m}\right)F_2\left(\frac{\pi}{2\mu_n}\right)\right).\nonumber    
\end{eqnarray}

Subsequently, inserting (\ref{eq319}), (\ref{eq320}),(\ref{eq321}), and (\ref{eq322}) into (\ref{eq318}), the requested estimation follows.

The proof is completed. 
\end{proof}

Specifying functions $F_i(z)$, $(i=1,2)$, by:

\begin{equation*}
   F_i(z)= \begin{cases}
               z^{\gamma_i -1},               & 0< \gamma_i <1;\\
               \log \left(\frac{\pi}{z}\right),               & \gamma_i =1
           \end{cases}
\end{equation*}
then Theorem \ref{the33} implies:
\begin{corollary}\label{cor33}
Let $f\in C([-\pi,\pi]^2)$, 
$\omega_1(f,s,t)=\mathcal{O}\left(\omega^{(1)}(s) \omega^{(2)}(t)\right)$, where $\omega^{(1)}(s)$ and $\omega^{(2)}(t)$ are two non-negative functions of modulus type satisfying conditions  (\ref{eq14}) and (\ref{eq15}) as $s, t\to +0$. Then 

\begin{equation*}
   \max_{(x,y)\in Q}\left|V_{m,n}^{\lambda,\mu}(f;x,y)-f(x,y)\right|= \begin{cases}
               \mathcal{O}\left(\frac{1}{\lambda_m^{\gamma_1}\mu_n^{\gamma_2}}\right),               & 0< \gamma_1,\gamma_2 <1;\\
               \mathcal{O}\left(\frac{\log \left(2\mu_n\right)}{\lambda_m^{\gamma_1}\mu_n}\right),               & 0< \gamma_1 <1,\gamma_2=1;\\
               \mathcal{O}\left(\frac{\log \left(2\lambda_m\right)}{\lambda_m\mu_n^{\gamma_2}}\right),               & \gamma_1=1, 0< \gamma_2 <1;\\
               \mathcal{O}\left(\frac{\log \left(2\lambda_m\right)\log \left(2\mu_n\right)}{\lambda_m\mu_n}\right),               & \gamma_1= \gamma_2=1
           \end{cases}
\end{equation*}
\end{corollary}

In particular case, it is clear that $V_{m+1,n+1}^{m,n}(f;x,y)\equiv \sigma_{m,n}(f;x,y)$, which is the double Fej\`er mean of the sequence $(s_{k,\ell}(x,y))$. Thus, Theorem \ref{the33} also implies:
\begin{corollary}\label{cor34}
Let $f\in C([-\pi,\pi]^2)$, 
$\omega_1(f,s,t)=\mathcal{O}\left(\omega^{(1)}(s) \omega^{(2)}(t)\right)$, where $\omega^{(1)}(s)$ and $\omega^{(2)}(t)$ are two non-negative functions of modulus type satisfying conditions  (\ref{eq14}) and (\ref{eq15}) as $s, t\to +0$. Then 

\begin{equation*}
   \max_{(x,y)\in Q}\left|\sigma_{m,n}(f;x,y)-f(x,y)\right|= \begin{cases}
               \mathcal{O}\left(\frac{1}{(m+1)^{\gamma_1}(n+1)^{\gamma_2}}\right),               & 0< \gamma_1,\gamma_2 <1;\\
               \mathcal{O}\left(\frac{\log \left(2(n+1)\right)}{(m+1)^{\gamma_1}(n+1)}\right),               & 0< \gamma_1 <1,\gamma_2=1;\\
               \mathcal{O}\left(\frac{\log \left(2(m+1)\right)}{(m+1)(n+1)^{\gamma_2}}\right),               & \gamma_1=1, 0< \gamma_2 <1;\\
               \mathcal{O}\left(\frac{\log \left(2(m+1)\right)\log \left(2(n+1)\right)}{(m+1)(n+1)}\right),               & \gamma_1= \gamma_2=1
           \end{cases}
\end{equation*}
\end{corollary}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $a:=(a_n)$, $b:=(b_n)$, $c:=(c_n)$, and  $d:=(d_n)$ be sequences of non-negative integers with conditions
\begin{equation}\label{eq323}
1\leq b_m-a_m+\lambda_m,\quad 1\leq d_n-c_n+\mu_n,\quad (m, n=1,2,\dots). 
\end{equation}

The mean $V_{m,n}^{\lambda,\mu}(f;x,y)$ can be generalized further by
\begin{equation}\label{eq324}
    V_{m,n}^{\lambda,\mu}(a,b,c,d;f;x,y)=\frac{1}{\lambda_m\mu_n}\sum_{k=a_m-\lambda_m}^{b_m-1}\sum_{k=c_n-\mu_n}^{d_n-1}s_{k,\ell}(x,y),\quad (m,n\geq 1),
\end{equation}
is called the $(mn)$-th double deferred  generalized de la Vall\'ee Poussin mean of the sequence $(s_{k,\ell}(x,y))$ generated by sequences $(\lambda_m)$ and $(\mu_n)$.

\begin{remark}
Note that for $a_m=b_m=m$ and  $c_n=d_n=n$, for all $m,n\geq 1$, we obtain $$V_{m,n}^{\lambda,\mu}(a,b,c,d;f;x,y)\equiv V_{m,n}^{\lambda,\mu}(f;x,y),$$
and 
$$V_{m+1,n+1}^{m,n}(a,b,c,d;f;x,y)\equiv \sigma_{m,n}(f;x,y).$$
\end{remark}
 

The mean $V_{m,n}^{\lambda,\mu}(a,b,c,d;f;x,y)$ given by (\ref{eq324}) can be used to prove the following general theorem.


\begin{theorem}\label{the34} Let $f\in C([-\pi,\pi]^2)$, 
$\omega_1(f,s,t)=\mathcal{O}\left(\omega^{(1)}(s) \omega^{(2)}(t)\right)$, where $\omega^{(1)}(s)$ and $\omega^{(2)}(t)$ are two non-negative functions of modulus type satisfying conditions  (\ref{eq14}) and (\ref{eq15}) as $s, t\to +0$, and $F_1(s), F_2(t)\geq 0$ two mediate functions. Then 
\begin{eqnarray*}
&&\max_{(x,y)\in Q}\left|V_{m,n}^{\lambda,\mu}(a,b,c,d;f;x,y)-f(x,y)\right|\\
&&\hspace{2.5cm}=\mathcal{O}\bigg(\frac{1}{(b_m-a_m+\lambda_m)(d_n-c_n+\mu_n)}\\
&&\hspace{3cm}\times F_1\bigg(\frac{\pi}{2(b_m-a_m+\lambda_m)}\bigg)F_2\bigg(\frac{\pi}{2(d_n-c_n+\mu_n)}\bigg)\bigg).\nonumber      
\end{eqnarray*}
\end{theorem}
\begin{proof}
Because of the similarity with the proof of Theorem \ref{the33} we omit the proof of this theorem.
\end{proof}

\begin{remark}
One should note that Theorem \ref{the33} is a particular case of Theorem \ref{the34} (when $a_m=b_m$ and $c_n=d_n$; $\forall m,n\geq 1$). Moreover, it covers Corollary \ref{cor33} and Corollary \ref{cor34} as well (when $a_m=b_m$, $c_n=d_n$, $\lambda_m=m$, and $\mu_n=n$; $\forall m,n\geq 1$).
\end{remark}

Further, let $a:=(a_m)$, $b:=(b_m)$, $c:=(c_n)$, and  $d:=(d_n)$ be sequences of non-negative integers with conditions

\begin{equation}\label{eq325}
a_m<b_m,\quad c_n<d_n,\quad (m,n=1,2,\dots ),
\end{equation}
and
\begin{equation}\label{eq326}
\lim _{m\to \infty}b_m=+\infty,\quad  \lim _{n\to \infty}d_n=+\infty. 
\end{equation}

If $\lambda_m=1$ and $\mu_n=1$ for all $m,n\geq 1$, then the double deferred de la Vall\'ee Poussin mean $V_{m,n}^{\lambda,\mu}(a+2,b+1,c+2,d+1;f;x,y)$ reduces to
$$D_{a,c}^{b,d}(f;x,y):=\frac{1}{(b_m-a_m)(d_n-c_n)}\sum_{k=a_m+1}^{b_m}\sum_{\ell =c_n+1}^{d_n}s_{k,\ell}(f;x,y),$$
which is the double deferred Ces\`aro mean of the sum $s_{k,\ell}(f;x,y)$ introduced implicitly in \cite{SI}. 
It was shown there, that (\ref{eq325}) and (\ref{eq326}) are conditions of 
regularity for $D_{a,c}^{b,d}$. Therefore, if conditions (\ref{eq325}) and (\ref{eq326})) are satisfied, then Theorem \ref{the34} implies the following. 
\begin{corollary}\label{cor35}
Let $f\in C([-\pi,\pi]^2)$, 
$\omega_1(f,s,t)=\mathcal{O}\left(\omega^{(1)}(s) \omega^{(2)}(t)\right)$, where $\omega^{(1)}(s)$ and $\omega^{(2)}(t)$ are two non-negative functions of modulus type satisfying conditions  (\ref{eq14}) and (\ref{eq15}) as $s, t\to +0$, and $F_1(s), F_2(t)\geq 0$ two mediate functions. Then 
\begin{eqnarray*}
&&\max_{(x,y)\in Q}\left|D_{a,c}^{b,d}(f;x,y)-f(x,y)\right|\\
&&\hspace{1cm}=\mathcal{O}\bigg(\frac{1}{(b_m-a_m)(d_n-c_n)}
F_1\bigg(\frac{\pi}{2(b_m-a_m)}\bigg)F_2\bigg(\frac{\pi}{2(d_n-c_n)}\bigg)\bigg).    
\end{eqnarray*}
\end{corollary}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{document}