${}^1$This work has been supported by Balikesir University Scientific Research Project 2016/58.

${}^2$ Results of this work were presented in the International Conference on Analysis and Its Applications (ICAA-2016), which held in 12-15 of July at the Ahi Evran University, Kirsehir, Turkey.

${}^3$Department of Mathematics, Faculty of Arts and Sciences, University of Balikesir, Cagis Yerleskesi, 10145, Balikesir, Turkey, e-mail: rakgun@balikesir.edu.tr.

1 Introduction

The main aim of this research is to investigate the approximation properties of some means of two dimensional Fourier series in Lebesgue spaces $L_{\omega }^p\left({\mathbb{T}}^2\right)$ with weights $\omega$ in the Muckenhoupt’s class $A_p({\mathbb{T}}^2,\mathbb{J})$, where $\mathbb{J}$ is the set of rectangles in ${\mathbb{T}}^2:=\mathbb{T}\times \mathbb{T}$, $\mathbb{T}:=[0,2\pi ]$ with sides parallel to coordinate axes. Trigonometric approximation by "angle" and mixed K-functional will be the main tools. We obtain the main properties of the weighted mixed modulus of smoothness ${\mathrm{\Omega }}_r{(f,{\delta }_1,{\delta }_2)}_{p,\omega }$ in $L_{\omega }^p\left({\mathbb{T}}^2\right)$, $\omega \in A_p({\mathbb{T}}^2,\mathbb{J})$, $1<p<\mathrm{\infty }$. Note that, in general, in weighted spaces, such as $L_{\omega }^p\left({\mathbb{T}}^2\right),$ the classical translation operators are not bounded. Instead of classical translation operators we use Steklov type operators to define the weighted mixed modulus of smoothness ${\mathrm{\Omega }}_r{(f,{\delta }_1,{\delta }_2)}_{p,\omega }$ in $L_{\omega }^p\left({\mathbb{T}}^2\right)$ (see [ 2 ] ). Starting from 70s, in the classical nonweighted Lebesgue spaces $L^p\left({\mathbb{T}}^2\right)$ (defined on ${\mathbb{T}}^2$ or ${\mathbb{T}}^d$, $d\ge 1$), some problems related to the classical nonweighted mixed modulus of smoothness ${\omega }_r{(f,{\delta }_1,{\delta }_2)}_p$ have been actively studied by mathematicians: M. K. Potapov [ 14 , 17 ] , [ 18 ] , [ 15 , 16 ] ; Potapov, Simonov, Lakovich [ 20 ] ; Potapov, Simonov, Tikhonov [ 22 ] , [ 19 , 21 ] ; A. F. Timan [ 27 ] ; M. F. Timan [ 28 , Chapter 2 ] . Among these problems we mention direct and inverse theorems of angular approximation [ 14 , 17 ] , [ 18 ] ; Hardy-Littlewood, Marcinkiewicz-Littlewood-Paley and embedding results [ 15 , 16 ] ; transformed Fourier series; embedding results of the Besov-Nikolski and Weyl-Nikolskii classes [ 19 , 21 ] , Ulyanov type inequalities [ 23 ] ; mixed $K$-functionals [ 6 ] , [ 25 ] ; fractional order classical mixed modulus of smoothness [ 24 ] .

In what follows, $A\lesssim B$ will mean that, there exists a positive constant $C_{u,v,\dots }$, depending only on the parameters $u,v,\dots$ and can be different in different places, such that the inequality $A\le CB$ holds. If $A\lesssim B$ and $B\lesssim A$ we will write $A\approx B$.

It is well known that the main property of modulus of smoothness
${\mathrm{\Omega }}_r{(\cdot ,{\delta }_1,{\delta }_2)}_{p,w}$ is that it decreases to zero as $max\{{\delta }_1,{\delta }_2\}\to 0$. This rate can be characterized by some class ${\mathrm{\Phi }}_{a_1,a_2}$ defined below: the class ${\mathrm{\Phi }}_{a_1,a_2}$ $(a_1,a_2\in \mathbb{R}\times \mathbb{R})$ consists of functions $\psi (\cdot ,\cdot )$ satisfying conditions

• $\psi (t_1,t_2)\ge 0$ bounded on $(0,\mathrm{\infty })\times (0,\mathrm{\infty }),$

• $\psi (t_1,t_2)\to 0$ as $max\{t_1,t_2\}\to 0,$

• $\psi (t_1,t_2)$ is non-decreasing in $t_1$ and $t_2$, (d) $t_i^{-a_i}\psi \left(t_i\right)$ is non-increasing in $t_i$ $(i=1,2)$.

We suppose that $\mathbb{J}$ is the set of rectangles in ${\mathbb{T}}^2$ with the sides parallel to coordinate axes. A function $\omega :{\mathbb{T}}^2\to {\mathbb{R}}^{\ge }:=[0,\mathrm{\infty })$ is called a weight on ${\mathbb{T}}^2$ if $\omega (x_1,x_2)$ is measurable and positive almost everywhere on ${\mathbb{T}}^2$. We denote by $A_p({\mathbb{T}}^2,\mathbb{J})$, ($1<p<\mathrm{\infty }$) the collection of locally integrable weights $\omega :{\mathbb{T}}^2\to {\mathbb{R}}^{\ge }$ such that $\omega (x_1,x_2)$ is $2\pi$-periodic with respect to each variable $x,y$ and

The least constant $C$ in (1) will be called the Muckenhoupt’s constant of $\omega$ and denoted by ${\left[\omega \right]}_{A_p}$.

The main result of this work is the characterization of the modulus of smoothness, given in the following theorem.

For functions in $L_{\omega }^p\left(\mathbb{T}\right),$ $p\in (1,\mathrm{\infty }),$ $\omega \in A_p\left(\mathbb{T}\right)$ Theorem 1 was obtained by the author in [ 1 ] . In this work we simplify the (long) proof given in [ 1 ] .

This type characterization theorem was proved in [ 26 ] (one dimensional case) for the spaces $L^p\left(\mathbb{T}\right),$ $p\in [1,\mathrm{\infty })$, with classical moduli of smoothness of fractional order. The class ${\mathrm{\Phi }}_{\varrho }$ describes completely the class of all majorants for the moduli of smoothness ${\omega }_r{(\cdot ,\delta )}_p$ in the space $L^p\left(\mathbb{T}\right),$ $p\in [1,\mathrm{\infty })$. For ${\omega }_r{(\cdot ,\delta )}_p$, $r\in \mathbb{N}$ the characterization problem was investigated by O. V. Besov, S. B. Stechkin [ 4 ] , V. I. Kolyada [ 12 ] ; for ${\omega }_r{(\cdot ,\delta )}_p$, $r>0$ the characterization theorem was obtained by S. Tikhonov [ 26 ] .

2 Preliminaries

Let $L^1\left({\mathbb{T}}^2\right)$ be the collection of Lebesgue integrable functions $f(x_1,x_2):{\mathbb{T}}^2\to \mathbb{R}$ such that $f(x_1,x_2)$ is $2\pi$-periodic with respect to each variable $x_1,x_2.$ Let $1<p<\mathrm{\infty },$ $\omega (x_1,x_2)\in A_p({\mathbb{T}}^2,\mathbb{J})$, and let $L_{\omega }^p\left({\mathbb{T}}^2\right)$ be the collection of Lebesgue integrable functions $f(x_1,x_2):{\mathbb{T}}^2\to \mathbb{R}$ such that $f(x_1,x_2)$ is $2\pi$-periodic with respect to each variable $x_1,x_2$ and

When $\omega (x_1,x_2)\equiv 1$ we denote ${\Vert f\Vert }_{p,1}=:{\Vert f\Vert }_p$ and $L_1^p\left({\mathbb{T}}^2\right)=:L^p\left({\mathbb{T}}^2\right)$ for $1\le p<\mathrm{\infty }$; $L_1^{\mathrm{\infty }}\left({\mathbb{T}}^2\right)=:L^{\mathrm{\infty }}\left({\mathbb{T}}^2\right)$.

We define Steklov type averages by

For $1<p<\mathrm{\infty },$ $\omega \in A_p({\mathbb{T}}^2,\mathbb{J})$, $f\in L_{\omega }^p\left({\mathbb{T}}^2\right)$, $h_1,h_2,$ $r\in \mathbb{N}$, we define the mixed differences by

where $\mathbb{I}$ is identity operator on ${\mathbb{T}}^2$. Using the inequalities (4) we get

for $1<p<\mathrm{\infty },$ $\omega \in A_p({\mathbb{T}}^2,\mathbb{J})$, $f\in L_{\omega }^p\left({\mathbb{T}}^2\right),$ $r\in \mathbb{N}$, with constants depending only on ${\left[\omega \right]}_{A_p}$ and $p,r$.

The mixed modulus of smoothness of $f\in L_{\omega }^p\left({\mathbb{T}}^2\right),$ $1<p<\mathrm{\infty },$ $\omega (x,y)\in A_p({\mathbb{T}}^2,\mathbb{J})$, $r\in \left\{0\right\}\cup \mathbb{N}$, can be defined as

If $1<p<\mathrm{\infty },$ $\omega \in A_p({\mathbb{T}}^2,\mathbb{J})$, $f\in L_{\omega }^p\left({\mathbb{T}}^2\right)$, $r\in \mathbb{N}$, then from (6) and (5) ${\mathrm{\Omega }}_r{(f,{\delta }_1,{\delta }_2)}_{p,\omega }\lesssim {\Vert f\Vert }_{p,\omega }$ with constant depending only on ${\left[\omega \right]}_{A_p}$ and $p,r$.

Note that from the definition of ${\mathrm{\Omega }}_r{(f,\cdot ,\cdot )}_{p,\omega }$, it has the following properties when $1<p<\mathrm{\infty },$ $\omega \in A_p({\mathbb{T}}^2,\mathbb{J})$, $f\in L_{\omega }^p\left({\mathbb{T}}^2\right)$, $r\in \mathbb{N}$:

• ${\mathrm{\Omega }}_r{(f,0,0)}_{p,\omega }=0.$

• ${\mathrm{\Omega }}_r{(f,{\delta }_1,{\delta }_2)}_{p,\omega }$ is subadditive with respect to $f$.

• ${\mathrm{\Omega }}_r{(f,{\delta }_1,{\delta }_2)}_{p,\omega }\le {\mathrm{\Omega }}_r{(f,t_1,t_2)}_{p,\omega }$ for $0\le {\delta }_i\le t_i;i=1,2.$

When $\omega (x_1,x_2)\equiv 1$ we denote ${\mathrm{\Omega }}_r{(f,{\delta }_1,{\delta }_2)}_{p,1}=:{\mathrm{\Omega }}_r{(f,{\delta }_1,{\delta }_2)}_p$ for $1\le p<\mathrm{\infty }$; ${\mathrm{\Omega }}_r{(f,{\delta }_1,{\delta }_2)}_{\mathrm{\infty },1}=:{\mathrm{\Omega }}_r{(f,{\delta }_1,{\delta }_2)}_{\mathrm{\infty }}$.

Let $1<p<\mathrm{\infty },$ $\omega \in A_p({\mathbb{T}}^2,\mathbb{J})$, and $f\in L_{\omega }^p\left({\mathbb{T}}^2\right),$ then there is $\lambda \in (1,\mathrm{\infty })$ such that $f\in L^{\lambda }\left({\mathbb{T}}^2\right)$, namely, we have $L_{\omega }^p\left({\mathbb{T}}^2\right)\subset L^{\lambda }\left({\mathbb{T}}^2\right)$ and this gives possibility to define the corresponding Fourier series of $f$.

We define ${\mathcal{T}}_{m,n}$ as the set of all trigonometric polynomials of degree at most $m$ with respect to variable $x_1$ and of degree at most $n$ with respect to variable $x_2.$ Then

where ${\mathcal{T}}_{m_i}$ is the set of all two dimensional trigonometric polynomials of degree at most $m_i$ with respect to variable $x_i$ ($i=1,2$)

Let $1<p<\mathrm{\infty },$ $\omega \in A_p({\mathbb{T}}^2,\mathbb{J})$ and $\sum _{n_1=0}^{\mathrm{\infty }}\sum _{n_2=0}^{\mathrm{\infty }}{A}_{n_1,n_2}(x_1,x_2)$ be the corresponding Fourier series for $f\in L_{\omega }^p\left({\mathbb{T}}^2\right)$. We define the partial sums of Fourier series of $f\in L_{\omega }^p\left({\mathbb{T}}^2\right),$ $1<p<\mathrm{\infty },$ $\omega \in A_p({\mathbb{T}}^2,\mathbb{J})$ as

Define the partial de la Valleè Poussin means of $f$ as

By Theorem 6 of [ 13 ]

as $m_1,m_2\to \mathrm{\infty }$ where $C_{m_1,m_2}^{\alpha }f$ is $\alpha$th Cesàro mean of $f$. From this we can deduce that $C\left({\mathbb{T}}^2\right)$ is a dense subset of $L_{\omega }^p\left({\mathbb{T}}^2\right)$ for $1<p<\mathrm{\infty },$ $\omega \in A_p({\mathbb{T}}^2,\mathbb{J})$. Then $Y_{m_1,m_2}{\left(f\right)}_{p,\omega }\lesssim {\Vert f-C_{m_1,m_2}^{\alpha }f\Vert }_{p,\omega }\to 0$ and $Y_{m_1,m_2}{\left(f\right)}_{p,\omega }\to 0$ as $m_1,m_2\to \mathrm{\infty }$.

Let $W_{p,\omega }^{r,s}$, $r,s\in \mathbb{N}$, (respectively $W_{p,\omega }^{r,\circ };W_{p,\omega }^{\circ ,s}$) denote the collection of functions $f\in L^1\left({\mathbb{T}}^d\right)$ such that $f^{(r,s)}\in L_{\omega }^p\left({\mathbb{T}}^d\right)$ (respectively $f^{(r,\circ )}\in L_{\omega }^p\left({\mathbb{T}}^d\right);$ $f^{(\circ ,s)}\in L_{\omega }^p\left({\mathbb{T}}^d\right)$).

The following inequalities can be obtained by the method given in [ 2 ] . For $1<p<\mathrm{\infty },$ $\omega \in A_p({\mathbb{T}}^2,\mathbb{J})$, $r\in \mathbb{N}$, there exist constants depending only on ${\left[\omega \right]}_{A_p}$ and $p,r$ so that

(i) (Jackson inequalities of Favard type)

(ii) if ${\delta }_1,{\delta }_2>0$ then

and hence

(iii) If $1<p<\mathrm{\infty },$ $\omega \in A_p({\mathbb{T}}^2,\mathbb{J})$, $f\in L_{\omega }^p\left({\mathbb{T}}^2\right)$ and $r\in \mathbb{N}$, then there exist constants depending only on Muckenhoupt’s constant ${\left[\omega \right]}_{A_p}$ of $\omega$ and $p,r$ so that the equivalence

and the properties

hold for ${\delta }_1,{\delta }_2>0$, where $\lfloor x\rfloor :=max\{z\in \mathbb{Z}:z\le x\}.$

(iv) (11) can be refined by the inequality (19) below [ 2 ] . If $1<p<\mathrm{\infty },$ $\omega \in A_p({\mathbb{T}}^2,\mathbb{J})$, $f\in L_{\omega }^p\left({\mathbb{T}}^2\right)$ and $r\in \mathbb{N}$, then there exists $C_{{\left[\omega \right]}_{A_p},p,r}$ depending only on Muckenhoupt’s constant ${\left[\omega \right]}_{A_p}$ of $\omega$ and $p,r$ so that

where $m_1,m_2\in \mathbb{N}$.

3 Proof of Theorem 1

Let ${\omega }_r{(\cdot ,{\delta }_1,{\delta }_2)}_p$, $1\le p\le \mathrm{\infty }$, be the usual nonweighted mixed modulus of smoothness:

where $T_{h_1,\circ }g(x_1,x_2):=g(x_1+h_1,x_2)$; $T_{\circ ,h_2}g(x_1,x_2):=g(x_1,x_2+h_2)$. From [ 25 ] ($1\le p<\mathrm{\infty }$) and [ 6 ] ($p=\mathrm{\infty }$) and (18) there exist positive constants, depending only $r,p$, such that

holds for $1\le p\le \mathrm{\infty }$ and $g\in L^p\left({\mathbb{T}}^2\right).$

Theorem 2.5 of [ 26 ] give that: Let $r\in \mathbb{N},$ $p\in [1,\mathrm{\infty }].$

(a) If $f\in L^p\left({\mathbb{T}}^2\right)$, then there exists $\psi \in {\mathrm{\Phi }}_{r,r}$ such that

holds for all $t_1,t_2\in (0,\mathrm{\infty })\times (0,\mathrm{\infty })$ with equivalence constants depending only on $r$.

(b) If $\psi \in {\mathrm{\Phi }}_{r,r}$ then there exist $f_0\in L^p\left({\mathbb{T}}^2\right)$ and the positive real numbers $t_0,t_3$ such that

holds for all ${\delta }_1,{\delta }_2\in (0,t_0)\times (0,t_3)$ with equivalence constants depending only on $r.$

[Proof of Theorem 1] (i) Note that if $F\in C\left({\mathbb{T}}^2\right)$ then from (7)

Using Theorem 2.5 (A) of [ 26 ] , (7), (20), (18), (23) there exists $\psi \in {\mathrm{\Phi }}_{2r}$ such that

If $p\in (1,\mathrm{\infty })$, $A_p({\mathbb{T}}^2,\mathbb{J})$, $f\in L_{\omega }^p\left({\mathbb{T}}^2\right)$, then, by (11), for any $\epsilon >0$ there exists $F\in C\left({\mathbb{T}}^2\right)$ such that $\Vert f-F{\Vert }_{p,w}<\epsilon$. Thus

Letting $\epsilon \to 0^{+}$ we get

On the other hand, from (18), (20), (7) and Theorem 2.5 (A) of [ 26 ]

and the equivalence (2) is established.

(ii) For the equivalence (3) let $\psi \in {\mathrm{\Phi }}_{2r}.$ By Theorem 2.5 (B) and Remark 2.7 (1) of [ 26 ] there exist $f\in L^{\mathrm{\infty }}\left({\mathbb{T}}^2\right)$ and the positive real numbers $t_0,t_3$ such that

holds for all ${\delta }_1,{\delta }_2\in (0,t_0)\times (0,t_3)$ with equivalence constants depending only on $r$. Then by (18), (20) we get

for all ${\delta }_1,{\delta }_2\in (0,t_0)\times (0,t_3)$.