Let \(\sum_{m=0}^{\infty }\sum_{n=0}^{\infty } g_{m,n}\) be double series with the sequence of \((m,n)\)th partial sums
$$ s_{m,n}=\sum_{j=0}^{m}\sum _{k=0}^{n} g_{j,k}. $$
A double Hausdorff matrix has the entries
$$ h_{m,n}^{j,k}= \binom{m }{j} \binom{n }{k} \Delta ^{m-j}_{1} \Delta ^{n-k}_{2} \mu _{j,k}, $$
where \(\{ \mu _{j,k} \} \) is any real or complex sequence, and
$$ \Delta ^{m-j}_{1} \Delta ^{n-k}_{2} \mu _{j,k} = \sum_{w=0}^{m-j} \sum _{z=0}^{n-k} (-1)^{j+k} \binom{m-j }{w} \binom{n-k }{z} \mu _{j+w,k+z} . $$
If \(t_{m,n}^{H} = \sum_{j=0}^{m}\sum_{k=0}^{n} h_{m,n}^{j,k} s_{j,k} \rightarrow g \) as \(m \rightarrow \infty \) and \(n \rightarrow \infty \), then \(\sum_{m=0}^{\infty }\sum_{n=0}^{\infty } g_{m,n}\) is said to be summable to the sum g by the double Hausdorff matrix summability method [15].
A necessary and sufficient condition for double Hausdorff matrix summability method to be regular is there exists a function \(\chi (s,t) \in BV[0,1]\times [0,1]\) such that
$$ \int _{0}^{1} \int _{0}^{1} \bigl\vert d \chi (s,t) \bigr\vert < \infty $$
and
$$ \mu _{m,n} = \int _{0}^{1} \int _{0}^{1} s^{m} t^{n} \,d \chi (s,t), $$
where \(\chi (s,0)=\chi (s,0^{+})=\chi (0^{+},t)=\chi (0,t) = 0\), \(0\leq s \), \(t \leq 1 \), and \(\chi (1,1)-\chi (1,0)-\chi (0,1)+\chi (0,0) = 1\) [10].
It is easy to see that the absolute value of the measure \(d \chi (s,t)\) can me majorized by \(K_{1} K_{2} \,ds \,dt\) for some constants \(K_{1}\) and \(K_{2}\) (see [16]).
The important particular cases of double Hausdorff matrix summability means are as follows:
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1
Almost Euler summability means (\((E,q_{1},q_{2})\) means) if \(\mu _{m,n} = \frac{1}{(1+q_{1})^{m}}\frac{1}{(1+q_{2})^{n}}\).
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2
\((E,1,1)\) means if \(q_{1}=1\) and \(q_{2}=1\) in \((E,q_{1},q_{2})\) means.
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3
\((C, \gamma , \delta )\) means if \(\mu _{m,n} = \frac{1}{A^{\gamma }_{m}}\frac{1}{A^{\delta }_{n}}\), where \(\gamma , \delta \geq -1\) and \(A^{\gamma }_{m} = \binom{{\gamma +m} }{m }\), \(A^{\delta }_{n} = \binom{{\delta +n} }{n }\).
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4
\((C,1,1)\) means if \(\gamma =\delta =1\) in \((C, \gamma , \delta )\) means.
Let \(f(x,y)\) be a Lebesgue-integrable function of period 2π with respect to both variables x and y and summable in the fundamental square \(Q:(-\pi ,\pi ) \times (-\pi ,\pi )\). The double Fourier series of \(f(x,y)\) is given by
$$ \begin{aligned} f(x,y)&=\sum_{m=0}^{\infty } \sum_{n=0}^{\infty } \lambda _{m,n} [ a_{m,n} \cos mx \cos ny +b_{m,n} \sin mx \cos ny \\ &\quad{} + c_{m,n} \cos mx \sin ny + d_{m,n} \sin mx \cos ny ] \end{aligned} $$
(1)
with \((m,n)\)th partial sums \(s_{m,n}(f;(x,y))\), where
$$\begin{aligned}& \lambda _{m,n}= \textstyle\begin{cases} 1/4 & \text{for } m=n=0, \\ 1/2 & \text{for } m>0, n=0 \mbox{ and } m=0, n>0, \\ 1 & \text{for } m>0, n>0 , \end{cases}\displaystyle \\& a_{m,n}=\pi ^{-2} \iint _{Q} f(x,y) \cos mx \cos ny \,dx \,dy, \end{aligned}$$
and similar expressions for \(b_{m,n}\), \(c_{m,n}\), and \(d_{m,n}\) [3].
We define the \(L^{r} \) norm by
$$ \Vert f \Vert _{r}= \textstyle\begin{cases} \{ \frac{1}{4\pi } \int _{0}^{2\pi } \int _{0}^{2\pi } \vert f(x,y) \vert ^{r} \,dx \,dy \} ^{1/r}, & r\geq 1, \\ \operatorname*{ess\,sup}_{0\leq x,y \leq 2\pi } \vert f(x,y) \vert , & r=\infty . \end{cases} $$
The degree of approximation of a function \(f :\mathbb{R}^{2} \rightarrow \mathbb{R}\) by a trigonometric polynomial [17]
$$ \begin{aligned} t_{m,n}(x,y)&=\sum _{j=0}^{m}\sum_{k=0}^{n} \lambda _{m,n} [ a_{j,k} \cos mx \cos ny +b_{j,k} \sin mx \cos ny \\ &\quad {}+ c_{j,k} \cos mx \sin ny + d_{j,k} \sin mx \cos ny ] \end{aligned} $$
of order \((m+n)\) is defined by
$$E_{m,n}\bigl(f,L^{r}\bigr) =\min_{0\leq x,y \leq2\pi} \Vert t_{m,n}-f \Vert _{r} . $$
A function \(f :\mathbb{R}^{2} \rightarrow \mathbb{R}\) of two variables x and y is said to belong to the class \(Lip(\alpha ,\beta )\) [4] if
$$ \bigl\vert f(x+u,y+v)-f(x,y) \bigr\vert =O\bigl( \vert u \vert ^{\alpha } + \vert v \vert ^{\beta }\bigr), \quad 0< \alpha \leq 1, 0< \beta \leq 1, $$
to the class \(Lip ((\alpha ,\beta );r )\) if
$$ \biggl\{ \frac{1}{4\pi } \int _{0}^{2\pi } \int _{0}^{2\pi } \bigl\vert f(x+u,y+v)-f(x,y) \bigr\vert ^{r} \,dx \,dy \biggr\} ^{1/r}= O \bigl( \vert u \vert ^{\alpha } + \vert v \vert ^{\beta } \bigr),\quad r\geq 1, $$
and to the class \(Lip ((\xi _{1},\xi _{2});r )\) if
$$ \biggl\{ \frac{1}{4\pi } \int _{0}^{2\pi } \int _{0}^{2\pi } \bigl\vert f(x+u,y+v)-f(x,y) \bigr\vert ^{r} \,dx \,dy \biggr\} ^{1/r}= O \bigl(\xi _{1}(u) + \xi _{2}(v) \bigr),\quad r\geq 1, $$
where \(\xi _{1}\) and \(\xi _{2}\) are moduli of continuity, that is, nonnegative nondecreasing continuous functions such that \(\xi _{1}(0) =\xi _{2}(0) = 0\), \(\xi _{1}(u_{1} + u_{2}) \le \xi _{1}(u_{1}) + \xi _{1}(u_{2})\), and \(\xi _{2}(v_{1} + v_{2}) \le \xi _{2}(v_{1}) + \xi _{2}(v_{2})\).
If \(\xi _{1}(u)=u^{\alpha }\) and \(\xi _{2}(v)=v^{\beta }\), \(0<\alpha \leq 1\), \(0 < \beta \leq 1\), then the class \(Lip ((\xi _{1},\xi _{2});r )\) coincides with \(Lip ((\alpha ,\beta );r )\). As \(r \rightarrow \infty \), \(Lip ((\alpha ,\beta );r )\) reduces to \(Lip(\alpha ,\beta )\). Clearly, \(Lip(\alpha ,\beta ) \subseteq Lip ((\alpha ,\beta );r ) \subseteq Lip ((\xi _{1},\xi _{2});r ) \).
We define the forward difference operator Δ as \(\Delta \mu _{k} = \mu _{k} - \mu _{k+1} \); also, \(\Delta ^{n+1}\mu _{k}=\Delta (\Delta ^{n} \mu _{k} )\), \(k\geq 0\). We denote
$$\begin{aligned}& \begin{aligned} \phi (u,v)&=(1/4) \bigl[f(x+u,y+v)+f(x+u,y-v) +f(x-u,y+v)+ f(x-u,y-v) \\ &\quad{} -4f(x,y) \bigr], \end{aligned} \\& M_{m}^{H}(u)= \frac{K_{1}}{2\pi }\sum _{j=0}^{m} \int _{0}^{1} \binom{m }{j} s^{j} (1-s)^{m-j} \,ds \frac{\sin (j+\frac{1}{2} )u}{\sin \frac{u}{2}}, \\& K_{n}^{H}(v) = \frac{K_{2}}{2\pi }\sum _{k=0}^{n} \int _{0}^{1} \binom{N }{K} t^{k} (1-t)^{n-k} \,dt \frac{\sin (k+\frac{1}{2} )v}{\sin \frac{v}{2}}. \end{aligned}$$