1 Introduction

Bojanic and Waterman [4, 5, 27] examined the quantitative version of the Dirichlet-Jordan test, concentrating on a class of functions with generalized bounded variation. In [27], Waterman presented an estimate of the Fourier series convergence rate for functions that are closer to the class of harmonic bounded variation. Móricz contributed a quantitative version of the Dirichlet-Jordan test for double Fourier series [14], considering functions with bounded variation.

This study extends and broadens the findings to include double rational Fourier series applied to continuous two-variable functions, which are close to the harmonic bounded variation class, building on the foundation established in [27] and [14]. The study of (double) rational Fourier series is important from a variety of perspectives, including approximation theory and Fourier analysis [10, 12, 16, 21, 22, 24] as well as signal processing, system identification, and control theory [3, 15, 17,18,19, 25].

During the 1920 s, two independent definitions of the rational orthogonal system emerged, attributed to Malmquist and Takenaka. Consequently, this orthogonal system is alternatively known as the Takenaka-Malmquist (or Malmquist-Takenaka) system. M. M. Džrbažyan [8] conducted research on rational Fourier series, employing the rational orthogonal system as his foundation. The definition of the rational orthogonal system is outlined as follows:

$$\begin{aligned} \phi _0(e^{ix})=1, \phi _n(e^{ix})=\frac{\sqrt{1-{\mid \alpha _n\mid }^2} e^{ix}}{1-\overline{{\alpha _n}} e^{ix}}\prod _{k=1}^{n-1}\frac{e^{ix}-\alpha _k}{1-\overline{\alpha _k}e^{ix}}, \end{aligned}$$
(1)

and \(\phi _{-n}(e^{ix})=\overline{\phi _n(e^{ix})},\ \forall n\in {\mathbb {N}}\). Here, \(\{\alpha _n\}_{n\in {\mathbb {N}}}\) is complex sequence such that \(\alpha _k\)’s are in open unit disk \({\mathbb {D}}\). Achieser [2, p. 244] (or see [6, p. 151]) observed that the system in (1) is complete in \(L^2[0,2\pi ]\) if and only if \(\sum _{n=1}^{\infty }(1-\mid \alpha _k\mid )=\infty .\) The simplest way to satisfy the previous completeness condition is to assume

$$\begin{aligned} \sup \mid \alpha _k\mid := r<1. \end{aligned}$$
(2)

In the sequel, the above condition (2) is assumed to be satisfied.

In 2009, Fu and Li [9] worked on convergence of the Cesàro mean of rational Fourier series. In 2012, Tan and Zhou [24] studied convergence behaviour of rational Fourier series and proved Jordan Dirichlet test for it. Later on, in 2013, Tan and Qian [22] gave quantitative version of Jordan Dirichlet test and proved convergence of rational Fourier series for functions of harmonic bounded variation. Recently, the quantitative version of Jordan Dirichlet test of rational Fourier series was generalized for certain functions of \(\Lambda -\) bounded variation [12]. In 2013, Tan and Zhou [23] worked on order of magnitude of rational Fourier coefficients and recently those results were further extended for multiple rational Fourier coefficients [11]. In 2022, Mnatsakanyan [13] proved \(L^p\) bounds for the maximal partial sum operator of the rational Fourier series. Recently, Abdullayev and Savchuk [1] gave analogous version of Fejer theorem for rational Fourier series.

If an integrable function f is \(2\pi \) periodic in both the variables, then double rational Fourier series of f is defined as

$$\begin{aligned} f(x,y)\sim \sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty } \hat{f}(m,n)\phi _m(e^{ix})\phi _n(e^{iy}), \end{aligned}$$
(3)

where \(\hat{f}(m,n)\) is the \((m,n)^{th}\) double rational Fourier coefficient of f, given by

$$\begin{aligned} \hat{f}(m,n)=\frac{1}{4\pi ^2}\int \int _{\mathbb {\overline{T}}^2}f(x,y)\overline{\phi _m(e^{ix})\phi _n(e^{iy})} dxdy. \end{aligned}$$
(4)

Note that, if \(\alpha _k=0,\ \forall k\in {\mathbb {N}}\) in (1) then \(\{\phi _n(e^{ix})\}_{n=-\infty }^{\infty }\) reduces to \(\{e^{inx}\}_{n=-\infty }^{\infty }\) and therefore double rational Fourier series reduces to classical double Fourier series.

2 Preliminaries and Notations

The notations and definitions mentioned in this section will be used in the sequel.

  1. 1.

    Waterman [26] defined the concept of \(\Lambda -\) bounded variation as follows.

    Let \(\{\lambda _n\}\) be a non decreasing sequence of positive numbers such that \(\sum \frac{1}{\lambda _n}\) diverges then a real valued function f is said to be of \(\Lambda -\) bounded variation on [a, b] (i.e. \(f\in \Lambda BV[a,b])\) if

    $$\begin{aligned} V_1(f,[a,b])=\sup \sum _{k=1}^n\frac{\mid f(b_k)-f(a_k)\mid }{\lambda _k}<\infty , \end{aligned}$$

    for every sequence of non-overlapping intervals \([a_k,b_k],k=1,...,n,\) which is contained in [a, b].

    The notion of bounded variation is extended for two variables in various ways like in the sense of Arzelà, Vitali, Hardy and others as can be seen in [7]. Here, we will need the definition of \(\Lambda \) bounded variation in the sense of Hardy, as given by Sablin [20]. A measurable function f defined on a rectangle \(I\times J:=[a,b]\times [c,d]\) is said to be of \(\Lambda -\) bounded variation in the sense of Hardy (that is, \(f\in \Lambda BV(R^2)\)) if the marginal functions \(f(.,c)\in \Lambda _1 BV([a,b]),\ f(a,.)\in \Lambda _2 BV([c,d])\) and

    $$\begin{aligned}V_2(f,R^2)=\sup _{J_1,J_2}\left\{ \sum _i\sum _j \frac{\mid f(I_i\times K_j)\mid }{\lambda _i\mu _j}\right\} <\infty . \end{aligned}$$

    Here, \(\Lambda =(\Lambda _1,\Lambda _2)\) where \(\Lambda _1=\{\lambda _i\}_{i=1}^{\infty }\) and \(\Lambda _2=\{\mu _j\}_{j=1}^{\infty }\); \(J_1\) and \(J_2\) are finite collections of non-overlapping subintervals \(\{I_i\}\) and \(\{K_j\}\) in [a, b] and [c, d] respectively; and \(f(I\times J)=\Delta f_{(a,c)}^{(b,d)}=f(b,d)-f(a,d)-f(b,c)+f(a,c).\)

  2. 2.

    As mentioned in [27], we suppose that \(\frac{\lambda _{\mid j\mid }}{\mid j\mid }\) is non increasing and if m is fixed then H(t) is a continuously non increasing function on \([-\pi ,0)\) and \((0,\pi ]\) such that

    $$\begin{aligned} H(t)=\frac{\lambda _{\mid j\mid }}{t};\ t=\frac{j\pi }{m+1} \text{ and } j=\pm 1,\pm 2,...,\pm (m+1). \end{aligned}$$

    Similarly, we suppose that \(\frac{\mu _{\mid k\mid }}{\mid k\mid }\) is non increasing and if n is fixed then G(t) is a continuously non increasing function on \([-\pi ,0)\) and \((0,\pi ]\) such that

    $$\begin{aligned} G(t)=\frac{\mu _{\mid k\mid }}{t};\ t=\frac{k\pi }{n+1} \text{ and } k=\pm 1,\pm 2,...,\pm (n+1). \end{aligned}$$
  3. 3.

    The oscillation of a function \(g:[a,b]\rightarrow {\mathbb {C}}\) over a subinterval \([a_1,b_1]\) of [a, b] is defined as

    $$\begin{aligned}osc_1(g,[a_1,b_1]) =\sup _{t,y\in [a_1,b_1]}\mid g(t)-g(y)\mid .\end{aligned}$$
  4. 4.

    The oscillation of a function \(h:[a,b]\times [c,d]\rightarrow {\mathbb {C}}\) over a sub-rectangle \([a_1,b_1]\times [c_1,d_1]\) of \([a,b]\times [c,d]\) is defined as

    $$\begin{aligned} osc_2&(h,[a_1,b_1]\times [c_1,d_1])\\&=\sup _{\begin{array}{c} u_1,u_2\in [a_1,b_1];\\ v_1,v_2\in [c_1,d_1] \end{array}}\mid h(u_1,v_1) -h(u_2,v_1)-h(u_1,v_2)+h(u_2,v_2)\mid . \end{aligned}$$
  5. 5.

    For \(m\in {\mathbb {N}}\cup \{0\}\), we define

    $$\begin{aligned}\eta _{km}=\frac{k\pi }{m+1},\ \forall k=0,1,2,...,m;\\ I_{km}^{+}=[\eta _{km},\eta _{(k+1)m}]\end{aligned}$$

    and

    $$\begin{aligned}I_{km}^{-}=[-\eta _{(k+1)m},-\eta _{km}].\end{aligned}$$
  6. 6.

    For a function \(f\in \mathbb {\overline{T}}^2:=[-\pi ,\pi ]^2,\) we define

    $$\begin{aligned} S(f;x,y)&:=\frac{1}{4}\{f(x+0,y+0)+f(x-0,y+0)+f(x+0,y-0)\\&\quad +f(x-0,y-0)\} \end{aligned}$$

    and \( \psi (u,v):=\psi _{xy}(u,v)\)

    $$\begin{aligned}:= {\left\{ \begin{array}{ll} S(f;x,y)-f(x-u,y-v) &{}\quad \hbox {if}\ u,v\ne 0 \\ S(f;x,y)-\frac{f(x-0,y-v)+f(x+0,y-v)}{2} &{} \quad \text {if } u=0,\ v\ne 0 \\ S(f;x,y)-\frac{f(x-u,y+0)+f(x-u,y-0)}{2} &{} \quad \text {if } u\ne 0,\ v=0 \\ 0 &{} \quad \hbox {if}\ u=v=0. \end{array}\right. } \end{aligned}$$

    Note that, here \(f(x+0,y+0):=\lim \{f(x+u,y+v):u,v\rightarrow 0\text { and }u,v>0\}\) and similary other limits like \(f(x-0,y+0),\ f(x+0,y-0)\) and \(f(x-0,y-0)\) are defined.

  7. 7.

    In view of [24, Lemma 2.1] and [22, Lemma 2.1], for \(\alpha _k=\mid \alpha _k\mid e^{ia_k},\ n\in {\mathbb {N}}\) and \(x\in [-\pi ,\pi ]\), the partial sums of rational Fourier series of integrable function g(x) is given by

    $$\begin{aligned} S_ng(x)=\frac{1}{\pi }\int _{-\pi }^{\pi }g(x-t)D_n(x-t,x)dt,\end{aligned}$$

    where rational Dirchlet kernel is given by

    $$\begin{aligned} D_n(t,x)=\frac{1}{2}\sum _{k=-n}^{n}\overline{\phi _k(e^{it})} \phi _k(e^{ix})=\frac{\sin \left[ \frac{x-t}{2}+\theta _n(t,x) \right] }{2\sin \left( \frac{x-t}{2}\right) } \end{aligned}$$
    (5)

    and

    $$\begin{aligned}\theta _n(t,x)=\int _t^x\sum _{k=1}^n\frac{1-\mid \alpha _k\mid ^2}{1-2\mid \alpha _k\mid \cos (y-a_k)+\mid \alpha _k\mid ^2}dy.\end{aligned}$$

    Here,

    $$\begin{aligned} \int _{-\pi }^{\pi }D_n(t,x)du=\pi \end{aligned}$$
    (6)
  8. 8.

    For \(m,n\in {\mathbb {N}}\), the partial sum of double rational Fourier series of an integrable function f(x, y) is given by

    $$\begin{aligned}S_{mn}(f;x,y)=\sum _{j=-m}^{m}\sum _{k=-n}^{n}\hat{f}(j,k)\phi _j(e^{ix})\phi _k(e^{iy}).\end{aligned}$$

    Using (4), (5) and the fact that f is \(2\pi \) periodic in both variables, we have

    $$\begin{aligned} S_{mn}(&f;x,y)\nonumber \\&=\frac{1}{4\pi ^2}\sum _{j=-m}^{m}\sum _{k=-n}^{n}\left\{ \int \int _{\mathbb {\overline{T}}^2}f(u,v) \overline{\phi _j(e^{iu})\phi _k(e^{iv})} \phi _j(e^{ix})\phi _k(e^{iy})dudv\right\} \nonumber \\&=\frac{1}{\pi ^2}\int \int _{\mathbb {\overline{T}}^2}f(u,v)D_m(u,x)D_n(v,y)du dv\nonumber \\&=\frac{1}{\pi ^2}\int \int _{\mathbb {\overline{T}}^2}f(x-u,y-v)D_m(x-u,x)D_n(y-v,y)du dv. \end{aligned}$$
    (7)
  9. 9.

    A two variable function f(x, y) is said to be regular in \(\mathbb {\overline{T}}^2\) if \(f(x\pm 0, y\pm 0),\ f(x\pm 0,.)\) and \(f(.,y\pm 0)\) exist for \((x,y)\in \mathbb {\overline{T}}^2.\)

3 Results

Understanding the rate of convergence is essential because it provides insights into how many terms are needed for a rational Fourier series to approximate a given function accurately. Analyzing and characterizing the rate of convergence is a crucial step in utilizing rational Fourier series effectively in a wide range of applications. The upcoming result, Theorem 1, gives rate of convergence of double rational Fourier series for a bounded, measurable and regular two variable function in terms of osciallations. Later on, the convergence is given in terms for two variable functions of generalized bounded variation in Theorem 2 and Corollary 1.

Theorem 1

If f is bounded, measurable and regular function on \(\mathbb {\overline{T}}^2\) and \(2\pi \) periodic in each variable, then for any \(m,n\in {\mathbb {N}}\cup \{0\},\)

$$\begin{aligned}&\mid S_{mn}(f;x,y)-S(f;x,y)\mid \nonumber \\&\quad \le 2\left( \frac{1+r}{1-r}\right) \sum _{k=0}^m\frac{1}{k+1} \{osc_1(\psi (0,.),I_{km}^+)+osc_1(\psi (0,.),I_{km}^-)\}\nonumber \\&\qquad + 2\left( \frac{1+r}{1-r}\right) \sum _{k=0}^n\frac{1}{k+1} \{osc_1(\psi (.,0),I_{kn}^+)+osc_1(\psi (.,0),I_{kn}^-)\}\nonumber \\&\qquad + 4\left( \frac{1+r}{1-r}\right) ^2\ \sum _{j=0}^m\sum _{k=0}^n\frac{1}{(j+1)(k+1)}\left\{ osc_2(\psi ,I_{jm}^+\times I_{kn}^+)\right. \nonumber \\&\qquad \left. +osc_2(\psi ,I_{jm}^-\times I_{kn}^+)+osc_2(\psi ,I_{jm}^+\times I_{kn}^-)+osc_2(\psi ,I_{jm}^-\times I_{kn}^-)\right\} . \end{aligned}$$
(8)

Proof

By using (7) and (6), we have

$$\begin{aligned}&S(f;x,y)-S_{mn}(f;x,y)\\&\quad =\frac{1}{\pi ^2}\int \int _{\mathbb {\overline{T}}^2}\{S(f;x,y)-f(x-u,v-y)\}D_m(x-u,x)D_n(y-v,y)dudv\\&\quad =\frac{1}{\pi ^2}\int \int _{\mathbb {\overline{T}}^2}\Bigg [S(f;x,y)-f(x-u,v-y)\\&\qquad -\left\{ S(f;x,y)-\frac{f(x-0,y-v)+f(x+0,y-v)}{2}\right\} \\&\qquad -\left\{ S(f;x,y)-\frac{f(x-u,y+0)+f(x-u,y-0)}{2}\right\} \Bigg ]\\&\qquad \times D_m(x-u,x)D_n(y-v,y)dudv\\&\qquad +\frac{1}{\pi ^2}\int \int _{\mathbb {\overline{T}}^2}\Bigg [\left\{ S(f;x,y)-\frac{f(x-0,y-v)+f(x+0,y-v)}{2}\right\} \\&\qquad +\left\{ S(f;x,y)-\frac{f(x-u,y+0)+f(x-u,y-0)}{2}\right\} \Bigg ]\\&\qquad \times D_m(x-u,x)D_n(y-v,y)dudv\\&\quad =\frac{1}{\pi ^2}\int _{0}^{\pi }\int _{0}^{\pi }\{\psi (u,v)- \psi (u,0)-\psi (0,v)\}D_m(x-u,x)D_n(y-v,y)dudv\\&\qquad +\frac{1}{\pi ^2}\int _{0}^{\pi }\int _{-\pi }^{0}\{\psi (u,v)- \psi (u,0)-\psi (0,v)\}D_m(x-u,x)D_n(y-v,y)dudv\\&\qquad +\frac{1}{\pi ^2}\int _{-\pi }^{0}\int _{0}^{\pi }\{\psi (u,v)- \psi (u,0)-\psi (0,v)\}D_m(x-u,x)D_n(y-v,y)dudv\\&\qquad +\frac{1}{\pi ^2}\int _{-\pi }^{0}\int _{-\pi }^{0}\{\psi (u,v)- \psi (u,0)-\psi (0,v)\}D_m(x-u,x)D_n(y-v,y)dudv\\&\qquad +\frac{1}{\pi }\left\{ \int _{-\pi }^{0}\psi (u,0)D_m(x-u,x)du+ \int _{0}^{\pi }\psi (u,0)D_m(x-u,x)du\right\} \\&\qquad +\frac{1}{\pi }\left\{ \int _{-\pi }^{0}\psi (0,v)D_n(y-v,y)dv+ \int _{0}^{\pi }\psi (0,v)D_n(y-v,y)dv\right\} \\&\quad :=A_1+A_2+A_3+A_4+A_5+A_6. \end{aligned}$$

Let \(h(u,v)=\psi (u,v)-\psi (u,0)-\psi (0,v).\) Therefore,

$$\begin{aligned}&\pi ^2A_1\\&\quad =\int _{I_{0m}^+}\int _{I_{0n}^+} h(u,v)D_m(x-u,x)D_n(y-v,y)dudv\\&\qquad +\sum _{j=1}^m\int _{I_{jm}^+}\int _{I_{0n}^+}\{h(u,v)-h(\eta _{jm},v)\}D_m(x-u,x)D_n(y-v,y)dudv\\&\qquad +\sum _{j=1}^m\int _{I_{jm}^+}\int _{I_{0n}^+}h(\eta _{jm},v) D_m(x-u,x)D_n(y-v,y)dudv\\&\qquad +\sum _{k=1}^n\int _{I_{0m}^+}\int _{I_{kn}^+}\{h(u,v)-h(u,\eta _{kn})\}D_m(x-u,x)D_n(y-v,y)dudv\\&\qquad +\sum _{k=1}^n\int _{I_{0m}^+}\int _{I_{kn}^+}h(u,\eta _{kn},) D_m(x-u,x)D_n(y-v,y)dudv\\&\qquad +\sum _{j=1}^m\sum _{k=1}^n\int _{I_{jm}^+}\int _{I_{kn}^+}\{h(u,v)- h(\eta _{jm},v)-h(u,\eta _{kn})+h(\eta _{jm},\eta _{kn})\}\\&\qquad \times D_m(x-u,x)D_n(y-v,y)dudv\\&\qquad +\sum _{j=1}^m\sum _{k=1}^n\int _{I_{jm}^+}\int _{I_{kn}^+}\{h(\eta _{jm},v) -h(\eta _{jm},\eta _{kn})\}D_m(x-u,x)D_n(y-v,y)dudv\\&\qquad +\sum _{j=1}^m\sum _{k=1}^n\int _{I_{jm}^+}\int _{I_{kn}^+}\{h(u,\eta _{kn}) -h(\eta _{jm},\eta _{kn})\}D_m(x-u,x)D_n(y-v,y)dudv\\&\qquad +\sum _{j=1}^m\sum _{k=1}^n\int _{I_{jm}^+}\int _{I_{kn}^+} h(\eta _{jm},\eta _{kn})D_m(x-u,x)D_n(y-v,y)dudv\\&\quad :=\sum _{k=1}^9 A_{1k}. \end{aligned}$$

We will mainly use the following inequalities

$$\begin{aligned} \mid D_m(x-u,x)\mid \le \frac{1+r}{1-r}(m+1);\quad -\pi \le u\le \pi \end{aligned}$$
(9)

and

$$\begin{aligned} \frac{1}{\mid \sin (u/2)\mid }\le \frac{\pi }{\mid u\mid };\quad u\in [-\pi ,0)\cup (0,\pi ]. \end{aligned}$$
(10)

Using (9), we have

$$\begin{aligned} \mid A_{11}\mid&\le osc_2(\psi ,I_{0m}^+\times I_{0n}^+)\int _{I_{0m}^+} \int _{I_{0n}^+} \left( \frac{1+r}{1-r}\right) ^2(m+1)(n+1)dudv\\&\le \pi ^2\left( \frac{1+r}{1-r}\right) ^2osc_2(\psi ,I_{0m}^+\times I_{0n}^+). \end{aligned}$$

Using (9) and (10), we get

$$\begin{aligned} \mid A_{12}\mid&\le \sum _{j=1}^m osc_2(\psi ,I_{jm}^+\times I_{0n}^+) \int _{I_{jm}^+}\int _{I_{0n}^+}\frac{\pi }{2\eta _{jm}}(n+1)\frac{1+r}{1-r}dudv\\&\le \pi ^2\frac{1+r}{1-r}\sum _{j=1}^m\frac{1}{2j} osc_2(I_{jm}^+\times I_{0n}^+)\\&\le \pi ^2\frac{1+r}{1-r}\sum _{j=1}^m\frac{1}{j+1} osc_2(I_{jm}^+\times I_{0n}^+). \end{aligned}$$

Similarly,

$$\begin{aligned} \mid A_{14}\mid \le \pi ^2\frac{1+r}{1-r}\sum _{k=1}^n \frac{1}{k+1}osc_2(I_{0m}^+\times I_{kn}^+). \end{aligned}$$

Let

$$\begin{aligned} R_{jm}^+=\int _{\eta _{jm}}^{\pi }D_m(x-u,x)du. \end{aligned}$$

Now, using summation by parts, we get

$$\begin{aligned} A_{13}&=\int _{I_{0n}^+}\left\{ \sum _{j=1}^m h(\eta _{jm},v) (R_{jm}^+-R_{(j+1)m}^+)\right\} D_n(x-v,v)dv\\&=\int _{I_{0n}^+}\left\{ \sum _{j=1}^m \left( h(\eta _{jm},v)-h(\eta _{(j-1)m},v)\right) (R_{jm}^+)\right\} D_n(x-v,v)dv. \end{aligned}$$

In view of [22, Lemma 2.3], we have for \(0<t<\pi \),

$$\begin{aligned} \Bigl |\int _t^{\pi }D_m(x-u,x)du \Big |\le \frac{\pi ^2(1+r)}{2m(1-r)t}. \end{aligned}$$

Thus,

$$\begin{aligned} \mid R_{jm}^+\mid \le \frac{\pi (1+r)}{j(1-r)}. \end{aligned}$$
(11)

Using (9) and (11), we get

$$\begin{aligned} \mid A_{13}\mid&\le \sum _{j=1}^m\frac{\pi (1+r)}{j(1-r)} osc_2 (\psi ,I_{(j-1)m}^+\times I_{0n}^+)\int _{I_{0n}^+}\frac{(1+r)(n+1)}{1-r}dv\\&\le \frac{\pi ^2(1+r)^2}{(1-r)^2}\sum _{j=1}^m\frac{1}{j}osc_2(\psi ,I_{(j-1)m}^+\times I_{0n}^+)\\&\le \frac{\pi ^2(1+r)^2}{(1-r)^2}\sum _{j=0}^{m-1}\frac{1}{j+1}osc_2(\psi ,I_{jm}^+\times I_{0n}^+). \end{aligned}$$

Similarly,

$$\begin{aligned} \mid A_{15}\mid \le \frac{\pi ^2(1+r)^2}{(1-r)^2}\sum _{k=0}^{n-1} \frac{1}{k+1}osc_2(\psi ,I_{0m}^+\times I_{kn}^+). \end{aligned}$$

Using (10), we get

$$\begin{aligned} \mid A_{16}\mid&\le \sum _{j=1}^m\sum _{k=1}^n osc_2(\psi ,I_{jm}^+ \times I_{kn}^+)\int _{I_{jm}^+}\int _{I_{kn}^+}\frac{\pi ^2}{4\eta _{jm}\eta _{kn}} du dv\\&\le \pi ^2\sum _{j=1}^m\sum _{k=1}^n\frac{1}{4jk} osc_2(\psi ,I_{jm}^+\times I_{kn}^+)\\&\le \pi ^2\sum _{j=1}^m\sum _{k=1}^n\frac{1}{(j+1)(k+1)} osc_2(\psi ,I_{jm}^+\times I_{kn}^+). \end{aligned}$$

Using summation by parts,

$$\begin{aligned} A_{17}&=\sum _{k=1}^n\int _{I_{kn}^+}\left\{ \sum _{j=1}^m\left( h(\eta _{jm},v) -h(\eta _{jm},\eta _{kn})\right) (R_{jm}^+-R_{(j+1)m}^+)\right\} \\&\qquad \qquad \times D_n(x-v,x)dv\\&=\sum _{k=1}^n\int _{I_{kn}^+}\Big \{\sum _{j=1}^m(h(\eta _{jm},v) -h(\eta _{jm},\eta _{kn})-h(\eta _{(j-1)m},v)\\&\qquad +h(\eta _{(j-1)m},\eta _{kn}))(R_{jm}^+)\Big \}D_n(x-v,x)dv. \end{aligned}$$

Using (10) and (11), we have

$$\begin{aligned} \mid A_{17}\mid&\le \sum _{j=1}^m\sum _{k=1}^n\frac{\pi (1+r)}{j(1-r)} osc_2(\psi ,I_{(j-1)m}^+\times I_{kn}^+)\int _{I_{kn}}\frac{\pi }{2\eta _{kn}}dv\\&\le \frac{\pi ^2(1+r)}{1-r}\sum _{j=1}^m\sum _{k=1}^n\frac{1}{2jk} osc_2(\psi ,I_{(j-1)m}^+\times I_{kn}^+)dv\\&\le \frac{\pi ^2(1+r)}{1-r}\sum _{j=0}^{m-1}\sum _{k=1}^n \frac{1}{(j+1)(k+1)} osc_2(\psi ,I_{jm}^+\times I_{kn}^+)dv. \end{aligned}$$

Similarly,

$$\begin{aligned} \mid A_{18}\mid \le \frac{\pi ^2(1+r)}{1-r}\sum _{j=1}^{m} \sum _{k=0}^{n-1}\frac{1}{(j+1)(k+1)} osc_2(\psi ,I_{jm}^+\times I_{kn}^+)dv. \end{aligned}$$

Using double summation by parts,

$$\begin{aligned} A_{19}&=\sum _{j=1}^m\sum _{k=1}^nh(\eta _{jm},\eta _{kn}) (R_{jm}^+-R_{(j+1)m}^+)(R_{km}^+-R_{(k+1)m}^+)\\&=\sum _{j=1}^m\sum _{k=1}^n \{h(\eta _{jm},\eta _{kn})-h(\eta _{(j-1)m},\eta _{kn})-h(\eta _{jm},\eta _{(k-1)n})\\&\qquad +h(\eta _{(j-1)m},\eta _{(k-1)n})\}R_{jm}^+ R_{kn}^+. \end{aligned}$$

Using (11), we get

$$\begin{aligned} \mid A_{19}\mid&\le \pi ^2\left( \frac{1+r}{1-r}\right) ^2\ \sum _{j=1}^m \sum _{k=1}^n \frac{1}{jk}osc_2(\psi ,I_{(j-1)m}^+\times I_{(k-1)n}^+)\\&\le \pi ^2\left( \frac{1+r}{1-r}\right) ^2\ \sum _{j=0}^{m-1}\sum _{k=0}^{n-1} \frac{1}{(j+1)(k+1)}osc_2(\psi ,I_{jm}^+\times I_{kn}^+). \end{aligned}$$

Now from all the inequalities, it is easy to deduce

$$\begin{aligned} \mid A_1\mid \le 4\left( \frac{1+r}{1-r}\right) ^2\ \sum _{j=0}^m\sum _{k=0}^n \frac{1}{(j+1)(k+1)}osc_2(\psi ,I_{jm}^+\times I_{kn}^+). \end{aligned}$$

Now, let

$$\begin{aligned} R_{jm}^-=\int ^{-\eta _{jm}}_{-\pi }D_m(x-u,x)du. \end{aligned}$$

In view of [22, Lemma 2.3], we have for \(0<t<\pi \),

$$\begin{aligned} \Big |\int ^{-t}_{-\pi }D_m(x-u,x)du\Big |\le \frac{\pi ^2(1+r)}{2m(1-r)t}. \end{aligned}$$

Thus,

$$\begin{aligned} \mid R_{jm}^-\mid \le \frac{\pi (1+r)}{j(1-r)}. \end{aligned}$$
(12)

Now using (9),(10),(12) and following similar steps as before, we get

$$\begin{aligned} \mid A_2\mid \le 4\left( \frac{1+r}{1-r}\right) ^2\sum _{j=0}^m \sum _{k=0}^n\frac{1}{(j+1)(k+1)}osc_2(\psi ,I_{jm}^-\times I_{kn}^+), \\ \mid A_3\mid \le 4\left( \frac{1+r}{1-r}\right) ^2\sum _{j=0}^m \sum _{k=0}^n\frac{1}{(j+1)(k+1)}osc_2(\psi ,I_{jm}^+\times I_{kn}^-) \end{aligned}$$

and

$$\begin{aligned} \mid A_4\mid \le 4\left( \frac{1+r}{1-r}\right) ^2\sum _{j=0}^m \sum _{k=0}^n\frac{1}{(j+1)(k+1)}osc_2(\psi ,I_{jm}^-\times I_{kn}^-). \end{aligned}$$

In view of [12, Theorem 1], we get

$$\begin{aligned} \mid A_5\mid \le 2\frac{1+r}{1-r}\sum _{j=0}^m\frac{1}{j+1} \{osc_1(\psi (.,0),I_{jm}^+)+osc_1(\psi (.,0),I_{jm}^-)\} \end{aligned}$$

and

$$\begin{aligned} \mid A_6\mid \le 2\frac{1+r}{1-r}\sum _{k=0}^n\frac{1}{k+1} \{osc_1(\psi (0,.),I_{kn}^+)+osc_1(\psi (0,.),I_{kn}^-)\}. \end{aligned}$$

Thus, the result is proved. \(\square \)

Remark 1

Theorem 1 is analogous result of [14, Theorem 2] for double rational Fourier series and it is extension of [12, Theorem 1] for two variable functions.

Theorem 2

If \(f\in \Lambda BV ([0,\pi ]\times [0,\pi ]);\ f\) is continuous in \([0,\pi ]\times [0,\pi ];\) for \(m,n\in {\mathbb {N}};\) \(\frac{\pi }{m+1}=a_m<a_{m-1}<...<a_0=\pi \) and \(\frac{\pi }{n+1}=b_n<b_{n-1}<...<b_0=\pi ;\) then

$$\begin{aligned}&\sum _{j=0}^m\sum _{k=0}^n\frac{1}{(j+1)(k+1)}osc_2(\psi ,I_{jm}^+\times I_{kn}^+)\nonumber \\&\le \frac{\pi ^2}{(m+1)(n+1)}\sum _{j=0}^{m-1}\sum _{k=0}^{n-1} \big \{ V_2 (\psi , [0,a_j]\times [0,b_k])(H(a_{j+1})-H(a_j))\nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \times (G(b_{k+1})-G(b_k))\big \}\nonumber \\&\qquad +\frac{\pi \lambda _{n+1}}{(m+1)(n+1)}\sum _{j=0}^{m-1} V_2(\psi ,[0,a_j]\times [0,\pi ])(H(a_{j+1})-H(a_j))\nonumber \\&\qquad +\frac{\pi \mu _{m+1}}{(m+1)(n+1)}\sum _{k=0}^{n-1} V_2(\psi ,[0,\pi ],[0,b_k])(G(b_{k+1})-G(b_k))\nonumber \\&\qquad +\frac{\lambda _{m+1}\mu _{n+1}}{(m+1)(n+1)}V_2(\psi ,[0,\pi ]\times [0,\pi ]). \end{aligned}$$
(13)

Proof

We will follow similar technique as in [5, 14, 27]. Let

$$\begin{aligned} M_{jk}=\sum _{i=0}^j\sum _{p=0}^k\frac{osc_2(\psi ,I_{im}^+\times I_{pn}^+)}{\lambda _{i+1}\mu _{p+1}}, \\ N_j=\sum _{i=0}^j\frac{osc_2(\psi ,I_{im}^+\times I_{nn}^+)}{\lambda _{i+1}\mu _{n+1}} \end{aligned}$$

and

$$\begin{aligned} R_k=\sum _{p=0}^k\frac{osc_2(\psi ,I_{mm}^+\times I_{pn}^+)}{\lambda _{m+1}\mu _{p+1}}. \end{aligned}$$

Now for \(\ j=0,1,\dots , m-1\) and \(k=0,1,\dots n-1,\) define M(u, v) on the rectangle \([\pi /(m+1),\pi )\times [\pi /(n+1),\pi )\) and N(u) and R(v) on the intervals \([\pi /(m+1),\pi )\) and \([\pi /(n+1),\pi )\) respectively as follows:

$$\begin{aligned} M(u,v)=M_{\left[ \frac{(m+1)u}{\pi }\right] -1, \left[ \frac{(n+1)v}{\pi }\right] -1}, \\ N(u)=N_{\left[ \frac{(m+1)u}{\pi }\right] -1} \end{aligned}$$

and

$$\begin{aligned} R(v)=R_{\left[ \frac{(n+1)v}{\pi }\right] -1}. \end{aligned}$$

Therefore,

$$\begin{aligned} M(u,v)=M_{jk};\ (u,v)\in [\eta _{(j+1)m},\eta _{(j+2)m}) \times [\eta _{(k+1)n},\eta _{(k+2)n}), \\ N(u)=N_{j};\ u\in [\eta _{(j+1)m},\eta _{(j+2)m}) \end{aligned}$$

and

$$\begin{aligned} R(v)=R_{k};\ v\in [\eta _{(k+1)n},\eta _{(k+2)n}). \end{aligned}$$

Now, applying double summation by parts, we have

$$\begin{aligned} \sum _{j=0}^m&\sum _{k=0}^n\frac{1}{(j+1)(k+1)}osc_2(\psi ,I_{jm}^+\times I_{kn}^+)\nonumber \\&=\sum _{j=0}^{m-1}\sum _{k=0}^{n-1} M_{jk}\left( \frac{\lambda _{j+1}}{j+1}- \frac{\lambda _{j+2}}{j+2}\right) \left( \frac{\mu _{k+1}}{k+1}-\frac{\mu _{k+2}}{k+2}\right) \nonumber \\&\qquad +\frac{\lambda _{n+1}}{n+1}\sum _{j=0}^{m-1} M_{jn} \left( \frac{\lambda _{j+1}}{j+1}-\frac{\lambda _{j+2}}{j+2}\right) \nonumber \\&\qquad +\frac{\mu _{m+1}}{m+1}\sum _{k=0}^{n-1} M_{mk} \left( \frac{\mu _{k+1}}{k+1}-\frac{\mu _{k+2}}{k+2}\right) \nonumber \\&\qquad +\frac{M_{mn}\lambda _{m+1}\mu _{n+1}}{(m+1)(n+1)}\nonumber \\&:=A_1+A_2+A_3+A_4. \end{aligned}$$
(14)

Note that, we have \(-H(u)\) and \(-G(v)\) as non decreasing and continuous function on \((0,\pi ]\). Thus, by properties of two dimensional Riemann-Stieltjes integrals we can estimate \(A_1,\ A_2\) and \(A_3\) in the following manner.

$$\begin{aligned} A_1&=\frac{\pi ^2}{(m+1)(n+1)}\sum _{j=0}^{m-1}\sum _{k=0}^{n-1}\int _{I_{(j+1)m}^+}\int _{I_{(k+1)n}^+} M(u,v)d(-H(u))d(-G(v))\\&=\frac{\pi ^2}{(m+1)(n+1)}\int _{\eta _{1m}}^{\pi }\int _{\eta _{1n}}^{\pi } M(u,v)d(-H(u))d(-G(v))\\&\le \frac{\pi ^2}{(m+1)(n+1)}\sum _{j=0}^{m-1}\sum _{k=0}^{n-1} M(a_j,b_k)(H(a_{j+1})-H(a_j))(G(b_{k+1})-G(b_k)). \end{aligned}$$

Now, consider \(A_2\).

$$\begin{aligned} A_2&=\frac{\lambda _{n+1}}{n+1}\sum _{j=0}^{m-1} \left( M_{j(n-1)}+N_j\right) \left( \frac{\lambda _{j+1}}{j+1}-\frac{\lambda _{j+2}}{j+2}\right) \\&=\frac{\pi \lambda _{n+1}}{(m+1)(n+1)}\sum _{j=0}^{m-1}\int _{I_{(j+1)m}^+} \left( M(u,\eta _{nn})+N(u)\right) d(-H(u))\\&=\frac{\pi \lambda _{n+1}}{(m+1)(n+1)}\int _{\eta _{1m}}^{\pi } \left( M(u,\eta _{nn})+N(u)\right) d(-H(u))\\&\le \frac{\pi \lambda _{n+1}}{(m+1)(n+1)}\sum _{j=0}^{m-1} \left( M(a_j,\eta _{nn})+N(a_j)\right) (H(a_{j+1})-H(a_j)). \end{aligned}$$

Similarly,

$$\begin{aligned} A_3\le \frac{\pi \mu _{m+1}}{(m+1)(n+1)}\sum _{k=0}^{n-1} \left( M(\eta _{mm},b_k)+R(b_k)\right) (G(b_{k+1})-G(b_k)). \end{aligned}$$

Since f is continuous in \([0,\pi ]\times [0,\pi ]\),we get,

$$\begin{aligned} V_2(\psi ,[0,a_j]\times [0,b_k])\ge M(a_j,b_k) \end{aligned}$$

and

$$\begin{aligned} V_2(\psi ,[0,\pi ]\times [0,\pi ])\ge M_{mn}. \end{aligned}$$

Thus,

$$\begin{aligned} A_1&\le \frac{\pi ^2}{(m+1)(n+1)}\sum _{j=0}^{m-1}\sum _{k=0}^{n-1} \big \{ V_2 (\psi , [0,a_j]\times [0,b_k])(H(a_{j+1})-H(a_j))\nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \times (G(b_{k+1})-G(b_k))\big \} \end{aligned}$$
(15)

and

$$\begin{aligned} A_4\le \frac{\lambda _{m+1}\mu _{n+1}}{(m+1)(n+1)}V_2(\psi ,[0,\pi ]\times [0,\pi ]). \end{aligned}$$
(16)

Consider,

$$\begin{aligned} M(a_j,\eta _{nn})+N(a_j)&=M_{\left[ \frac{(m+1) a_j}{\pi }\right] -1,n-1}+N_{\left[ \frac{(m+1)a_j}{\pi }\right] -1}\\&=M_{\left[ \frac{(m+1)a_j}{\pi }\right] -1,n}\\&\le V_2 (\psi , [0,a_j]\times [0,\pi ]). \end{aligned}$$

Thus, we get,

$$\begin{aligned} A_2\le \frac{\pi \lambda _{n+1}}{(m+1)(n+1)}\sum _{j=0}^{m-1} V_2(\psi ,[0,a_j]\times [0,\pi ])(H(a_{j+1})-H(a_j)). \end{aligned}$$
(17)

Similarly, we get,

$$\begin{aligned} A_3\le \frac{\pi \mu _{m+1}}{(m+1)(n+1)}\sum _{k=0}^{n-1} V_2(\psi ,[0,\pi ]\times [0,b_k])(G(b_{k+1})-G(b_k)). \end{aligned}$$
(18)

By substituting (15), (17), (18) and (16) in (14), we get the result. \(\square \)

Remark 2

Theorem 2 is extension of [27, Theorem on p. 52] for two variable continuous functions.

Corollary 1

If \(f\in \Lambda BV ([-\pi ,\pi ]\times [-\pi ,\pi ]);\ f\) is continuous in \([-\pi ,\pi ]\times [-\pi ,\pi ];\) for \(m,n\in {\mathbb {N}};\) \(\frac{\pi }{m+1}=a_m^{(1)}<a_{m-1}^{(1)}<...<a_0^{(1)}=\pi ,\ \frac{-\pi }{m+1}=a_m^{(2)}>a_{m-1}^{(2)}>...>a_0^{(2)}=-\pi ,\ \frac{\pi }{n+1}=b_n^{(1)}<b_{n-1}^{(1)}<...<b_0^{(1)}=\pi \) and \(\frac{-\pi }{n+1}=b_n^{(2)}>b_{n-1}^{(2)}>...>b_0^{(2)}=-\pi ,\) then

$$\begin{aligned} \mid S_{mn}(f;x,y)-f(x,y)\mid \le 2\left( \frac{1+r}{1-r}\right) \sum _{t=1}^2 (A_t+B_t)+ 4\left( \frac{1+r}{1-r}\right) ^2\sum _{p=1}^2\sum _{q=1}^2 C_{p,q}; \end{aligned}$$

where

$$\begin{aligned} A_t=&\frac{\lambda _{m+1} }{m+1}\ V_1\left( \psi (.,0),T\left( a_0^{(t)}\right) \right) \\&+\frac{\pi }{m+1}\sum _{i=0}^{m-1}V_1\left( \psi (.,0),T\left( a_i^{(t)}\right) \right) \mid H(a_{i+1}^{(t)})-H(a_i^{(t)})\mid , \end{aligned}$$
$$\begin{aligned} B_t=&\frac{\mu _{n+1} }{n+1}\ V_1\left( \psi (0,.),T\left( b_0^{(t)}\right) \right) \\&+\frac{\pi }{n+1}\sum _{i=0}^{n-1}V_1\left( \psi (0,.),T\left( b_i^{(t)}\right) \right) \mid G(b_{i+1}^{(t)})-G(a_i^{(t)})\mid , \end{aligned}$$
$$\begin{aligned} C_{p,q}&= \frac{\pi ^2}{(m+1)(n+1)}\sum _{j=0}^{m-1}\sum _{k=0}^{n-1} \big \{ V_2 \left( \psi ,T \left( a_j^{(p)}\right) \times T\left( b_k^{(q)}\right) \right) \\&\qquad \qquad \qquad \qquad \qquad \qquad \times \mid H(a_{j+1}^{(p)})-H(a_j^{(p)}) \mid \mid G(b_{k+1}^{(q)})-G(b_k^{(q)})\mid \big \}\nonumber \\&\quad +\frac{\pi \lambda _{n+1}}{(m+1)(n+1)}\sum _{j=0}^{m-1} V_2\left( \psi ,T \left( a_j^{(p)}\right) \times T\left( b_0^{(q)}\right) \right) \mid H(a_{j+1}^{(p)})-H(a_j^{(p)})\mid \\&\quad +\frac{\pi \mu _{m+1}}{(m+1)(n+1)}\sum _{k=0}^{n-1} V_2 \left( \psi ,T \left( a_0^{(p)}\right) \times T\left( b_k^{(q)}\right) \right) \mid G(b_{k+1}^{(q)})-G(b_k^{(q)})\mid \\&\quad +\frac{\lambda _{m+1}\mu _{n+1}}{(m+1)(n+1)}V_2 \left( \psi ,T \left( a_0^{(p)}\right) \times T\left( b_0^{(q)}\right) \right) , \end{aligned}$$

for \(a>0, \ T(a)=[0,a]\) and for \(a<0, \ T(a)=[a,0].\)

Proof

In view of Theorem 1, [27, Theorem on p. 52] and by proceeding in the similar manner as in the proof of Theorem 2, the result is proved. \(\square \)

Remark 3

In the above corollary, by setting \(\lambda _k=\mu _k=1,\) for all \(k\in {\mathbb {N}};\) \(a_i^{(1)}=\frac{\pi }{i+1},\ a_i^{(2)}=\frac{-\pi }{i+1},\) for \(i=0,1,...,m;\ b_j^{(1)}=\frac{\pi }{j+1},\ b_j^{(2)}=\frac{-\pi }{j+1}\) for \(j=0,1,...,n;\) and \(H(t)=G(t)=1/t;\) we get analogous result of [14, Theorem 3] for double rational Fourier series of continuous functions of bounded variation in two variables.