Trigonometric Approximation on the Hexagon

Ali Guven$^\ast $

November 20, 2020; accepted: December 21, 2020; published online: February 20, 2021.

The degree of trigonometric approximation of continuous functions, which are periodic with respect to the hexagon lattice, is estimated in uniform and Hölder norms. Approximating trigonometric polynomials are matrix means of hexagonal Fourier series.

MSC. 41A25, 41A63, 42B08

Keywords. Hexagonal Fourier series, Hölder class, matrix mean, regular hexagon.

Dedicated to the memory of Dr. Figen Kiraz.

$^\ast $Department of Mathematics, Balikesir University, 10145 Balikesir, Turkey, e-mail: guvennali@gmail.com.

1 Introduction

Approximation problems of functions of several variables defined on cubes of the Euclidean space are usually studied by assuming that the functions are periodic in each of their variables (see, for example [ 20 , § § 5.3, 6.3 ] , [ 23 , vol. II, ch. XVII ] , [ 22 , part 2 ] , [ 15 ] , [ 16 ] and [ 17 ] ). But, we need other definitions of periodicity to study approximation problems on non-tensor product domains, for example on hexagonal domains of $\mathbb {R}^{2}$. The periodicity defined by lattices is the most useful one.

In the Euclidean plane $\mathbb {R}^{2},$ besides the standard lattice $\mathbb {Z}^{2}$ and the rectangular domain $[ -\frac{1}{2},\frac{1}{2}) ^{2},$ the simplest lattice is the hexagon lattice and the simplest spectral set is the regular hexagon. The hexagon lattice has importance, since it offers the densest packing of the plane with unit circles. In this section we give basic information about hexagonal lattice and hexagonal Fourier series. More detailed information can be found in [ 11 ] and [ 21 ] .

The generator matrix and the spectral set of the hexagonal lattice $H\mathbb {Z}^{2}$ are given by

\begin{equation*} H=\left( \begin{array}{cc} \sqrt{3} & 0 \\ -1 & 2\end{array}\right) \end{equation*}

and

\begin{equation*} \Omega _{H}=\left\{ \left( x_{1},x_{2}\right) \in \mathbb {R}^{2}:-1\leq x_{2},\tfrac {\sqrt{3}}{2}x_{1}\pm \tfrac {1}{2}x_{2}{\lt}1\right\} . \end{equation*}

It is more convenient to use the homogeneous coordinates $\left( t_{1},t_{2},t_{3}\right) $ that satisfies $t_{1}+t_{2}+t_{3}=0$. As it is pointed out in [ 21 ] , using homogeneous coordinates reveals symmetry in various formulas. If we set

\begin{equation*} t_{1}:=-\tfrac {x_{2}}{2}+\tfrac {\sqrt{3}x_{1}}{2},\text{ }t_{2}:=x_{2},\text{ }t_{3}:=-\tfrac {x_{2}}{2}-\tfrac {\sqrt{3}x_{1}}{2}, \end{equation*}

the hexagon $\Omega _{H}$ becomes

\begin{equation*} \Omega =\left\{ \left( t_{1},t_{2},t_{3}\right) \in \mathbb {R}^{3}:-1\leq t_{1},t_{2},-t_{3}{\lt}1,\text{ }t_{1}+t_{2}+t_{3}=0\right\} , \end{equation*}

which is the intersection of the plane $t_{1}+t_{2}+t_{3}=0$ with the cube $\left[ -1,1\right] ^{3}.$

We use bold letters $\mathbf{t}$ for homogeneous coordinates and we set

\begin{equation*} \mathbb {R}_{H}^{3}=\left\{ \mathbf{t}=\left( t_{1},t_{2},t_{3}\right) \in \mathbb {R}^{3}:t_{1}+t_{2}+t_{3}=0\right\} ,\text{ }\mathbb {Z}_{H}^{3}:=\mathbb {Z}^{3}\cap \mathbb {R}_{H}^{3}. \end{equation*}

A function $f:\mathbb {R}^{2}\rightarrow \mathbb {C}$ is called $H$-periodic if

\begin{equation*} f\left( x+Hk\right) =f\left( x\right) \end{equation*}

for all $k\in \mathbb {Z}^{2}$ and $x\in \mathbb {R}^{2}.$ If we define $\mathbf{t}\equiv \mathbf{s}$ $\left( \operatorname {mod}3\right) $ as

\begin{equation*} t_{1}-s_{1}\equiv t_{2}-s_{2}\equiv t_{3}-s_{3}\text{ }\left( \operatorname {mod}3\right) \end{equation*}

for $\mathbf{t}=\left( t_{1},t_{2},t_{3}\right) ,$ $\mathbf{s}=\left( s_{1},s_{2},s_{3}\right) \mathbf{\in }$ $\mathbb {R}_{H}^{3},$ it follows that the function $f$ is $H$-periodic if and only if $f\left( \mathbf{t}\right) =f\left( \mathbf{t}+\mathbf{s}\right) $ whenever $\mathbf{s}\equiv \mathbf{0}$ $\left( \operatorname {mod}3\right) ,$ and

\begin{equation*} \underset {\Omega }{\int }f\left( \mathbf{t}+\mathbf{s}\right) d\mathbf{t=}\underset {\Omega }{\int }f\left( \mathbf{t}\right) d\mathbf{t}\text{ }\left( \mathbf{s}\in \mathbb {R}_{H}^{3}\right) \end{equation*}

for $H$-periodic integrable function $f$ [ 21 ] $.$

$L^{2}\left( \Omega \right) $ becomes a Hilbert space with respect to the inner product

\begin{equation*} \left\langle f,g\right\rangle _{H}:=\tfrac {1}{\left\vert \Omega \right\vert }\underset {\Omega }{\int }f\left( \mathbf{t}\right) \overline{g\left( \mathbf{t}\right) }d\mathbf{t}, \end{equation*}

where $\left\vert \Omega \right\vert $ denotes the area of $\Omega .$ The functions

\begin{equation*} \phi _{\mathbf{j}}\left( \mathbf{t}\right) :=e^{\frac{2\pi i}{3}\left\langle \mathbf{j},\mathbf{t}\right\rangle }\qquad ( \mathbf{t}\in \mathbb {R}_{H}^{3}) , \end{equation*}

where $\left\langle \mathbf{j},\mathbf{t}\right\rangle $ is the usual Euclidean inner product of $\mathbf{j}$ and $\mathbf{t},$ are $H$-periodic, and by a theorem of B. Fuglede the set

\begin{equation*} \left\{ \phi _{\mathbf{j}}:\mathbf{j}\in \mathbb {Z}_{H}^{3}\right\} \end{equation*}

becomes an orthonormal basis of $L^{2}\left( \Omega \right) $ [ 3 ] (see also [ 11 ] )$.$

For every natural number $n,$ we define a subset of $\mathbb {Z}_{H}^{3}$ by

\begin{equation*} \mathbb {H}_{n}:=\left\{ \mathbf{j}=\left( j_{1},j_{2},j_{3}\right) \in \mathbb {Z}_{H}^{3}:-n\leq j_{1},j_{2},j_{3}\leq n\right\} . \end{equation*}

The subspace

\begin{equation*} \mathcal{H}_{n}:=\text{span}\left\{ \phi _{\mathbf{j}}:\mathbf{j}\in \mathbb {H}_{n}\right\} \qquad \left( n\in \mathbb {N}\right) \end{equation*}

has dimension $\# \mathbb {H}_{n}=3n^{2}+3n+1,$ and its members are called hexagonal trigonometric polynomials of degree $n$.

The hexagonal Fourier series of an $H$-periodic function $f\in L^{1}\left( \Omega \right) $ is

\begin{equation} f\left( \mathbf{t}\right) \sim \underset {\mathbf{j}\in \mathbb {Z}_{H}^{3}}{\sum }\widehat{f}_{\mathbf{j}}\phi _{\mathbf{j}}\left( \mathbf{t}\right) , \label{fourier} \end{equation}

where

\begin{equation*} \widehat{f}_{\mathbf{j}}=\tfrac {1}{\left\vert \Omega \right\vert }\underset {\Omega }{\int }f\left( \mathbf{t}\right) \overline{\phi _{\mathbf{j}}\left( \mathbf{t}\right) }d\mathbf{t}\qquad ( \mathbf{j}\in \mathbb {Z}_{H}^{3}) . \end{equation*}

The $n$th hexagonal partial sum of the series $(\ref{fourier})$ is defined by

\begin{equation*} S_{n}\left( f\right) \left( \mathbf{t}\right) :=\underset {\mathbf{j}\in \mathbb {H}_{n}}{\sum }\widehat{f}_{\mathbf{j}}\phi _{\mathbf{j}}\left( \mathbf{t}\right) \qquad \left( n\in \mathbb {N}\right) . \end{equation*}

It is clear that

\begin{equation*} S_{n}\left( f\right) \left( \mathbf{t}\right) =\tfrac {1}{\left\vert \Omega \right\vert }\underset {\Omega }{\int }f\left( \mathbf{t}-\mathbf{u}\right) D_{n}\left( \mathbf{u}\right) d\mathbf{u}, \end{equation*}

where $D_{n}$ is the Dirichlet kernel, defined by

\begin{equation*} D_{n}\left( \mathbf{t}\right) :=\underset {\mathbf{j}\in \mathbb {H}_{n}}{\sum }\phi _{\mathbf{j}}\left( \mathbf{t}\right) . \end{equation*}

It is known that the Dirichlet kernel can be expressed as

\begin{equation} D_{n}\left( \mathbf{t}\right) =\Theta _{n}\left( \mathbf{t}\right) -\Theta _{n-1}\left( \mathbf{t}\right) \qquad \left( n\geq 1\right) , \label{dirichlet} \end{equation}

where

\begin{equation} \Theta _{n}\left( \mathbf{t}\right) :=\tfrac {\sin \frac{\left( n+1\right) \left( t_{1}-t_{2}\right) \pi }{3}\sin \frac{\left( n+1\right) \left( t_{2}-t_{3}\right) \pi }{3}\sin \frac{\left( n+1\right) \left( t_{3}-t_{1}\right) \pi }{3}}{\sin \frac{\left( t_{1}-t_{2}\right) \pi }{3}\sin \frac{\left( t_{2}-t_{3}\right) \pi }{3}\sin \frac{\left( t_{3}-t_{1}\right) \pi }{3}} \label{thetan} \end{equation}

for $\mathbf{t}=\left( t_{1},t_{2},t_{3}\right) \in \mathbb {R}_{H}^{3}$ [ 11 ] .

The degree of approximation of $H$-periodic continuous functions by Cesàro, Riesz and Nörlund means of their hexagonal Fourier series was investigated by us in [ 4 ] , [ 5 ] , [ 6 ] , [ 7 ] and [ 8 ] . In this paper, we studied the degree of approximation by matrix means of hexagonal Fourier and we obtained generalizations of previous results.

2 Main Results

Let $C_{H}( \overline{\Omega }) $ be the Banach space of complex valued $H$-periodic continuous functions defined on $\mathbb {R}_{H}^{3},$ whose norm is the uniform norm:

\begin{equation*} \left\Vert f\right\Vert _{C_{H}\left( \overline{\Omega }\right) }:=\sup \left\{ \left\vert f\left( \mathbf{t}\right) \right\vert :\mathbf{t}\in \overline{\Omega }\right\} . \end{equation*}

The modulus of continuity of the function $f\in C_{H}( \overline{\Omega }) $ is defined by

\begin{equation*} \omega _{H}\left( f,\delta \right) :=\underset {0{\lt}\left\Vert \mathbf{t}\right\Vert \leq \delta }{\sup }\left\Vert f-f\left( \cdot +\mathbf{t}\right) \right\Vert _{C_{H}( \overline{\Omega }) }, \end{equation*}

where

\begin{equation*} \left\Vert \mathbf{t}\right\Vert :=\max \left\{ \left\vert t_{1}\right\vert ,\left\vert t_{2}\right\vert ,\left\vert t_{3}\right\vert \right\} \end{equation*}

for $\mathbf{t}=\left( t_{1},t_{2},t_{3}\right) \in \mathbb {R}_{H}^{3}.$ $\omega _{H}\left( f,\cdot \right) $ is a nonnegative and nondecreasing function, and satisfies

\begin{equation} \omega _{H}\left( f,\lambda \delta \right) \leq \left( 1+\lambda \right) \omega _{H}\left( f,\delta \right) \label{modc} \end{equation}

for $\lambda {\gt}0$ [ 21 ] $.$

A function $f\in C_{H}( \overline{\Omega }) $ is said to belong to the Hölder space $H^{\alpha }(\overline{\Omega }) $ $\left( 0{\lt}\alpha \leq 1\right) $ if

\begin{equation*} \Lambda ^{\alpha }\left( f\right) :=\underset {\mathbf{t}\neq \mathbf{s}}{\sup }\tfrac {\left\vert f\left( \mathbf{t}\right) -f\left( \mathbf{s}\right) \right\vert }{\left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{\alpha }}{\lt}\infty . \end{equation*}

$H^{\alpha }( \overline{\Omega }) $ becomes a Banach space with respect to the Hölder norm

\begin{equation*} \left\Vert f\right\Vert _{H^{\alpha }\left( \overline{\Omega }\right) }:=\left\Vert f\right\Vert _{C_{H}\left( \overline{\Omega }\right) }+\Lambda ^{\alpha }\left( f\right) . \end{equation*}

Let $A=\left( a_{n,k}\right) $ $\left( n,k=0,1,\ldots \right) $ be a lower triangular infinite matrix of real numbers. The $A$-transform of the sequence $\left( S_{n}\left( f\right) \right) $ of partial sums the series (1) is defined by

\begin{equation*} T_{n}^{\left( A\right) }\left( f\right) \left( \mathbf{t}\right) :=\overset {n}{\underset {k=0}{\sum }}a_{n,k}S_{k}\left( f\right) \left( \mathbf{t}\right) \qquad \left( n\in \mathbb {N}\right) . \end{equation*}

We shall assume that the lower triangular matrix $A=\left( a_{n,k}\right) $ satisfies the conditions

\begin{equation} a_{n,k}\geq 0\qquad \left( n=0,1,\ldots ,0\leq k\leq n\right) , \label{m1} \end{equation}

\begin{equation} a_{n,k+1}\geq a_{n,k}\qquad \left( n=0,1,\ldots ,0\leq k\leq n-1\right) , \label{m2} \end{equation}

and

\begin{equation} \underset {k=0}{\overset {n}{\sum }}a_{n,k}=1\qquad \left( n=0,1,\ldots \right) . \label{m3} \end{equation}

Also we use the notations

\begin{equation*} A_{n,k}^{\ast }:=\underset {\nu =k}{\overset {n}{\sum }}a_{n,\nu }\text{ }\left( 0\leq k\leq n\right) ,\text{ }A_{n}^{\ast }\left( u\right) :=A_{n,n-\left[ u\right] }^{\ast }\text{ , }a_{n}^{\ast }\left( u\right) :=a_{n,n-\left[ u\right] }\text{ }\left( u{\gt}0\right) , \end{equation*}

where $\left[ u\right] $ is the integer part of $u.$

Hereafter, the relation $x\lesssim y$ will mean that there exists an absolute constant $c{\gt}0$ such that $x\leq cy$ holds for quantities $x$ and $y$.

Theorem 1

Let $f\in C_{H}( \overline{\Omega }) $ and let $A=\left( a_{n,k}\right) $ $\left( n,k=0,1,\ldots \right) $ be a lower triangular infinite matrix of real numbers which satisfies $\left( \ref{m1}\right) $, $\left( \ref{m2}\right) $ and $\left( \ref{m3}\right) $. Then the estimate

\begin{equation} \left\Vert f-T_{n}^{\left( A\right) }\left( f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\lesssim \log \left( n+1\right) \left\{ \omega _{H}\left( f,a_{n,n}\right) +\underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}\left( f,\frac1k\right) }{k}A_{n,n-k}^{\ast }\right\} \label{uniform} \end{equation}

holds$.$

Corollary 2

If $f\in H^{\alpha }( \overline{\Omega }) $ $\left( 0{\lt}\alpha \leq 1\right) $ and $A=\left( a_{n,k}\right) $ $\left( n,k=0,1,\ldots \right) $ as in theorem 1, then

\begin{equation} \left\Vert f-T_{n}^{\left( A\right) }\left( f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\lesssim \log \left( n+1\right) \left\{ a_{n,n}^{\alpha }+\underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k^{1+\alpha }}\right\} \text{.} \label{uniformx} \end{equation}

Theorem 3

Let $0\leq \beta {\lt}\alpha \leq 1$, $f\in H^{\alpha }( \overline{\Omega }) $ and let $A=\left( a_{n,k}\right) $ $\left( n,k=0,1,\ldots \right) $ be a lower triangular infinite matrix of real numbers which satisfies $\left( \ref{m1}\right) $, $\left( \ref{m2}\right) $ and $\left( \ref{m3}\right) $. Then

\begin{equation} \left\Vert f-T_{n}^{\left( A\right) }\left( f\right) \right\Vert _{H^{\beta }\left( \overline{\Omega }\right) }\lesssim \log \left( n+1\right) \left( \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k}\right) ^{\frac\beta \alpha }\left\{ a_{n,n}^{\alpha -\beta }\! +\! \left( \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k^{1+\alpha }}\right) ^{1-\frac\beta \alpha }\right\} . \label{holderbeta} \end{equation}

Analogues of these results were obtained in [ 10 ] , [ 13 ] , [ 14 ] and [ 2 ] for matrix means of trigonometric Fourier series of $2\pi $-periodic continuous functions.

3 Proofs of Main Results

Proof â–¼

[Proof of theorem 1] It is clear that

\begin{eqnarray*} \left\vert f\left( \mathbf{t}\right) -T_{n}^{\left( A\right) }\left( f\right) \left( \mathbf{t}\right) \right\vert & \leq & \tfrac {1}{\left\vert \Omega \right\vert }\underset {\Omega }{\int }\left\vert f\left( \mathbf{t}\right) -f\left( \mathbf{t}-\mathbf{u}\right) \right\vert \left\vert \overset {n}{\underset {k=0}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u} \\ & \lesssim & \tfrac {1}{\left\vert \Omega \right\vert }\underset {\Omega }{\int }\omega _{H}\left( f,\left\Vert \mathbf{u}\right\Vert \right) \left\vert \overset {n}{\underset {k=0}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u}. \end{eqnarray*}

If we set $\Theta _{-1}\left( \mathbf{u}\right) :=0,$ by $\left( \ref{dirichlet}\right) $ we get

\begin{align*} & \underset {\Omega }{\int }\omega _{H}\left( f,\left\Vert \mathbf{u}\right\Vert \right) \left\vert \overset {n}{\underset {k=0}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u}= \\ & =\underset {\Omega }{\int }\omega _{H}\left( f,\left\Vert \mathbf{u}\right\Vert \right) \left\vert \overset {n}{\underset {k=0}{\sum }}a_{n,k}\Big( \Theta _{k}\left( \mathbf{u}\right) \! -\! \Theta _{k-1}\left( \mathbf{u}\right) \Big) \right\vert \! d\mathbf{u}. \end{align*}

The function

\begin{equation*} \mathbf{t}\rightarrow \omega _{H}\left( f,\left\Vert \mathbf{t}\right\Vert \right) \left\vert \overset {n}{\underset {k=0}{\sum }}a_{n,k}\left( \Theta _{k}\left( \mathbf{t}\right) -\Theta _{k-1}\left( \mathbf{t}\right) \right) \right\vert \end{equation*}

is symmetric with respect to variables $t_{1},t_{2}$ and $t_{3}$, where $\mathbf{t}=\left( t_{1},t_{2},t_{3}\right) \in \Omega .$ Hence it is sufficient to estimate the integral over the triangle

\begin{align*} \Delta :=& \left\{ \mathbf{t}=\left( t_{1},t_{2},t_{3}\right) \in \mathbb {R}_{H}^{3}:0\leq t_{1},t_{2},-t_{3}\leq 1\right\} \\ =& \left\{ \left( t_{1},t_{2}\right) :t_{1}\geq 0,\text{ }t_{2}\geq 0,\text{ }t_{1}+t_{2}\leq 1\right\} , \end{align*}

which is one of the six equilateral triangles in $\overline{\Omega }.$ By considering the formula $(\ref{thetan})$, we obtain

\begin{align*} & \underset {\Delta }{\int }\omega _{H}\left( f,\left\Vert \mathbf{t}\right\Vert \right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}\left( \Theta _{k}\left( \mathbf{t}\right) -\Theta _{k-1}\left( \mathbf{t}\right) \right) \right\vert d\mathbf{t}= \\ & =\underset {\Delta }{\int }\omega _{H}\left( f,t_{1}+t_{2}\right) \bigg\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}\bigg( \tfrac {\sin \frac{\left( k+1\right) \left( t_{1}-t_{2}\right) \pi }{3}\sin \frac{\left( k+1\right) \left( t_{2}-t_{3}\right) \pi }{3}\sin \frac{\left( k+1\right) \left( t_{3}-t_{1}\right) \pi }{3}}{\sin \frac{\left( t_{1}-t_{2}\right) \pi }{3}\sin \frac{\left( t_{2}-t_{3}\right) \pi }{3}\sin \frac{\left( t_{3}-t_{1}\right) \pi }{3}} \\ & \hspace{5cm}-\tfrac {\sin \frac{k\left( t_{1}-t_{2}\right) \pi }{3}\sin \frac{k\left( t_{2}-t_{3}\right) \pi }{3}\sin \frac{k\left( t_{3}-t_{1}\right) \pi }{3}}{\sin \frac{\left( t_{1}-t_{2}\right) \pi }{3}\sin \frac{\left( t_{2}-t_{3}\right) \pi }{3}\sin \frac{\left( t_{3}-t_{1}\right) \pi }{3}}\bigg) \bigg\vert d\mathbf{t}. \end{align*}

If we use the change of variables

\begin{equation} s_{1}:=\tfrac {t_{1}-t_{3}}{3}=\tfrac {2t_{1}+t_{2}}{3},\text{ }s_{2}:=\tfrac {t_{2}-t_{3}}{3}=\tfrac {t_{1}+2t_{2}}{3}, \label{change1} \end{equation}

the integral becomes

\begin{align*} 3\underset {\widetilde{\Delta }}{\int }\omega _{H}\left( f,s_{1}+s_{2}\right) \bigg\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}\Big( & \tfrac {\sin \left( \left( k+1\right) \left( s_{1}-s_{2}\right) \pi \right) \sin \left( \left( k+1\right) s_{2}\pi \right) \sin \left( \left( k+1\right) \left( -s_{1}\pi \right) \right) }{\sin \left( \left( s_{1}-s_{2}\right) \pi \right) \sin \left( s_{2}\pi \right) \sin \left( -s_{1}\pi \right) } \\ & -\tfrac {\sin \left( k\left( s_{1}-s_{2}\right) \pi \right) \sin \left( ks_{2}\pi \right) \sin \left( k\left( -s_{1}\pi \right) \right) }{\sin \left( \left( s_{1}-s_{2}\right) \pi \right) \sin \left( s_{2}\pi \right) \sin \left( -s_{1}\pi \right) }\Big) \bigg\vert ds_{1}ds_{2}, \end{align*}

where $\widetilde{\Delta }$ is the image of $\Delta $ in the plane, that is

\begin{equation*} \widetilde{\Delta }:=\left\{ \left( s_{1},s_{2}\right) :0\leq s_{1}\leq 2s_{2},\text{ }0\leq s_{2}\leq 2s_{1},\text{ }s_{1}+s_{2}\leq 1\right\} . \end{equation*}

Since the integrated function is symmetric with respect to $s_{1}$ and $s_{2},$ estimating the integral over the triangle

\begin{equation*} \Delta ^{\ast }:=\left\{ \left( s_{1},s_{2}\right) \in \widetilde{\Delta }:s_{1}\leq s_{2}\right\} =\left\{ \left( s_{1},s_{2}\right) :s_{1}\leq s_{2}\leq 2s_{1},\text{ }s_{1}+s_{2}\leq 1\right\} , \end{equation*}

which is the half of $\widetilde{\Delta },$ will be sufficient$.$ The change of variables

\begin{equation} s_{1}:=\tfrac {u_{1}-u_{2}}{2},\text{ }s_{2}:=\tfrac {u_{1}+u_{2}}{2} \label{change2} \end{equation}

transforms the triangle $\Delta ^{\ast }$ to the triangle

\begin{equation*} \Gamma :=\left\{ \left( u_{1},u_{2}\right) :0\leq u_{2}\leq \tfrac {u_{1}}{3},\text{ }0\leq u_{1}\leq 1\right\} . \end{equation*}

Thus we have to estimate the integral

\begin{equation*} I_{n}:=\underset {\Gamma }{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}, \end{equation*}

where

\begin{eqnarray*} D_{k}^{\ast }\left( u_{1},u_{2}\right) & =& \tfrac {\sin \left( \left( k+1\right) \left( u_{2}\right) \pi \right) \sin \left( \left( k+1\right) \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \left( k+1\right) \left( \frac{u_{1}-u_{2}}{2}\pi \right) \right) }{\sin \left( \left( u_{2}\right) \pi \right) \sin \left( \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \frac{u_{1}-u_{2}}{2}\pi \right) } \\ & & -\tfrac {\sin \left( k\left( u_{2}\right) \pi \right) \sin \left( k\frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( k\left( \frac{u_{1}-u_{2}}{2}\pi \right) \right) }{\sin \left( \left( u_{2}\right) \pi \right) \sin \left( \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \frac{u_{1}-u_{2}}{2}\pi \right) }. \end{eqnarray*}

By elementary trigonometric identities, we obtain

\begin{equation} D_{k}^{\ast }\left( u_{1},u_{2}\right) =D_{k,1}^{\ast }\left( u_{1},u_{2}\right) +D_{k,2}^{\ast }\left( u_{1},u_{2}\right) +D_{k,3}^{\ast }\left( u_{1},u_{2}\right) , \label{expression1} \end{equation}

where

\begin{align*} D_{k,1}^{\ast }\left( u_{1},u_{2}\right) & :=2\cos \left( ( k+\tfrac {1}{2}) u_{2}\pi \right) \tfrac {\sin \left( \frac{1}{2}u_{2}\pi \right) \sin \left( \left( k+1\right) \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \left( k+1\right) \frac{u_{1}-u_{2}}{2}\pi \right) }{\sin \left( u_{2}\pi \right) \sin \left( \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \frac{u_{1}-u_{2}}{2}\pi \right) }, \\ D_{k,2}^{\ast }\left( u_{1},u_{2}\right) & :=2\cos \left( ( k+\tfrac {1}{2}) \tfrac {u_{1}+u_{2}}{2}\pi \right) \tfrac {\sin \left( ku_{2}\pi \right) \sin \left( \frac{1}{2}\frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \left( k+1\right) \frac{u_{1}-u_{2}}{2}\pi \right) }{\sin \left( u_{2}\pi \right) \sin \left( \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \frac{u_{1}-u_{2}}{2}\pi \right) } \end{align*}

and

\begin{eqnarray*} D_{k,3}^{\ast }\left( u_{1},u_{2}\right) :=2\cos \left( ( k+\tfrac {1}{2}) \tfrac {u_{1}-u_{2}}{2}\pi \right) \tfrac {\sin \left( ku_{2}\pi \right) \sin \left( k\frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \frac{1}{2}\frac{u_{1}-u_{2}}{2}\pi \right) }{\sin \left( u_{2}\pi \right) \sin \left( \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \frac{u_{1}-u_{2}}{2}\pi \right) }. \end{eqnarray*}

We partition the triangle $\Gamma $ as $\Gamma =\Gamma _{1}\cup \Gamma _{2}\cup \Gamma _{3},$ where

\begin{align*} \Gamma _{1} & :=\left\{ \left( u_{1},u_{2}\right) \in \Gamma :u_{1}\leq a_{n,n}\right\} , \\ \Gamma _{2} & :=\left\{ \left( u_{1},u_{2}\right) \in \Gamma :u_{1}\geq a_{n,n},\text{ }u_{2}\leq \tfrac {a_{n,n}}{3}\right\} , \\ \Gamma _{3} & :=\left\{ \left( u_{1},u_{2}\right) \in \Gamma :u_{1}\geq a_{n,n},\text{ }u_{2}\geq \tfrac {a_{n,n}}{3}\right\} . \end{align*}

Hence $I_{n}=I_{n,1}+I_{n,2}+I_{n,3},$ where

\begin{equation*} I_{n,j}:=\underset {\Gamma _{j}}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\qquad \left( j=1,2,3\right) . \end{equation*}

We need the inequalities

\begin{equation} \left\vert \frac{\sin nt}{\sin t}\right\vert \leq n,\qquad \left( n\in \mathbb {N}\right) , \label{sin1} \end{equation}

and

\begin{equation} \sin t\geq \tfrac {2}{\pi }t,\qquad \left( 0\leq t\leq \tfrac {\pi }{2}\right) \label{sin2} \end{equation}

to estimate integrals $I_{n,1},I_{n,2}$ and $I_{n,3}.$

We divide $\Gamma _{1}$ into three parts to estimate $I_{n,1}$ as follows:

\begin{align*} \Gamma _{1}^{\prime } & :=\left\{ \left( u_{1},u_{2}\right) \in \Gamma _{1}:u_{1}\leq \tfrac {1}{n+1}\right\} , \\ \Gamma _{1}^{\prime \prime } & :=\left\{ \left( u_{1},u_{2}\right) \in \Gamma _{1}:u_{1}\geq \tfrac {1}{n+1},u_{2}\leq \tfrac {1}{3\left( n+1\right) }\right\} , \\ \Gamma _{1}^{\prime \prime \prime } & :=\left\{ \left( u_{1},u_{2}\right) \in \Gamma _{1}:u_{1}\geq \tfrac {1}{n+1},u_{2}\geq \tfrac {1}{3\left( n+1\right) }\right\} . \end{align*}

Hence we have

\begin{equation*} I_{n,1}=\left( \underset {\Gamma _{1}^{\prime }}{\int }+\underset {\Gamma _{1}^{\prime \prime }}{\int }+\underset {\Gamma _{1}^{\prime \prime \prime }}{\int }\right) \omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}. \end{equation*}

By (14),

\begin{align*} & \underset {\Gamma _{1}^{\prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \lesssim \\ & \lesssim \underset {\Gamma _{1}^{\prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left( \underset {k=0}{\overset {n}{\sum }}\left( k+1\right) ^{2}a_{n,k}\right) du_{1}du_{2} \\ & \leq \left( n+1\right) ^{2}\overset {\frac{1}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {3u_{2}}{\overset {\frac{1}{n+1}}{\int }}\omega _{H}\left( f,u_{1}\right) du_{1}du_{2} \\ & \leq \omega _{H}\left( f,\tfrac {1}{n+1}\right) \leq \omega _{H}\left( f,a_{n,n}\right) . \end{align*}

By $\left( \ref{sin2}\right) ,$

\begin{align*} & \underset {\Gamma _{1}^{\prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k,1}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \lesssim \\ & \lesssim \overset {\frac{1}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {\frac{1}{n+1}}{\overset {a_{n,n}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}du_{1}du_{2} \leq \omega _{H}\left( f,a_{n,n}\right) \overset {\frac{1}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {\frac{1}{n+1}}{\overset {a_{n,n}}{\int }}\tfrac {1}{u_{1}^{2}}du_{1}du_{2} \leq \omega _{H}\left( f,a_{n,n}\right) . \end{align*}

$\left( \ref{sin1}\right) $ and $\left( \ref{sin2}\right) $ gives for $j=1,2, $

\begin{eqnarray*} & & \underset {\Gamma _{1}^{\prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k,j}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\lesssim \\ & \lesssim & \left( n+1\right) \overset {\frac{1}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {\frac{1}{n+1}}{\overset {a_{n,n}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}du_{1}du_{2}\leq \left( n+1\right) \omega _{H}\left( f,a_{n,n}\right) \overset {\frac{1}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {\frac{1}{n+1}}{\overset {a_{n,n}}{\int }}\tfrac {1}{u_{1}}du_{1}du_{2} \\ & \leq & \log \left( n+1\right) \omega _{H}\left( f,a_{n,n}\right) . \end{eqnarray*}

Since

\begin{equation*} \sin 2x+\sin 2y+\sin 2z=-4\sin x\sin y\sin z \end{equation*}

for $x+y+z=0,$ we also get the expression

\begin{equation} D_{k}^{\ast }\left( u_{1},u_{2}\right) =H_{k,1}\left( u_{1},u_{2}\right) +H_{k,2}\left( u_{1},u_{2}\right) +H_{k,3}\left( u_{1},u_{2}\right) , \label{expression2} \end{equation}

where

\begin{align*} H_{k,1}\left( u_{1},u_{2}\right) & :=\tfrac {1}{2}\tfrac {\cos \left( \left( 2k+1\right) u_{2}\pi \right) }{\sin \left( \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \frac{u_{1}-u_{2}}{2}\pi \right) }, \\ H_{k,2}\left( u_{1},u_{2}\right) & :=-\tfrac {1}{2}\tfrac {\cos \left( \left( 2k+1\right) \frac{u_{1}+u_{2}}{2}\pi \right) }{\sin \left( u_{2}\pi \right) \sin \left( \frac{u_{1}-u_{2}}{2}\pi \right) }, \end{align*}

\begin{align*} H_{k,3}\left( u_{1},u_{2}\right) & :=\tfrac {1}{2}\tfrac {\cos \left( \left( 2k+1\right) \frac{u_{1}-u_{2}}{2}\pi \right) }{\sin \left( u_{2}\pi \right) \sin \left( \frac{u_{1}+u_{2}}{2}\pi \right) }. \end{align*}

By the method used in [ 12 , p. 179 ] we get

\begin{align} \label{Ant1} \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}\cos \left( 2k+1\right) t\right\vert \lesssim A_{n}^{\ast }\left( \tfrac {1}{t}\right) +\tfrac {1}{\sin t }a_{n}^{\ast }\left( \tfrac {1}{t}\right) \quad \text{ }\left( 0{\lt}t{\lt}\pi \right) \end{align}

and

\begin{align} \label{Ant2} \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}\cos \left( 2k+1\right) t\right\vert \lesssim A_{n}^{\ast }\left( \tfrac {1}{t}\right) \qquad \left( 0{\lt}t\leq \tfrac {\pi }{2}\right). \end{align}

By aim of (18) and $\left( \ref{sin2}\right) $ we obtain

\begin{equation} \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,1}\left( u_{1},u_{2}\right) \right\vert \lesssim \tfrac {1}{u_{1}^{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) \label{hk1} \end{equation}

and

\begin{equation} \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,3}\left( u_{1},u_{2}\right) \right\vert \lesssim \tfrac {1}{u_{1}u_{2}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) , \label{hk3} \end{equation}

where both of for $u_{1}$ and $u_{2}$ are away from the origin. Also, for such $u_{1}$ and $u_{2},$ it follows from $\left( \ref{Ant1}\right) ,$ $\left( \ref{sin2}\right) $ and from the fact

\begin{equation*} \sin \left( \tfrac {u_{1}\pi }{2}\right) \lesssim \sin \left( \tfrac {\left( u_{1}+u_{2}\right) \pi }{2}\right) \end{equation*}

that

\begin{equation} \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,2}\left( u_{1},u_{2}\right) \right\vert \lesssim \tfrac {1}{u_{1}u_{2}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) . \label{hk2} \end{equation}

Using $\left( \ref{hk1}\right) $ and the inequality

\begin{equation} \frac{\omega _{H}\left( f,\delta _{2}\right) }{\delta _{2}}\leq 2\frac{\omega _{H}\left( f,\delta _{1}\right) }{\delta _{1}}\qquad \left( \delta _{1}<\delta _{2}\right) , \label{modc2} \end{equation}

which is obtained from $\left( \ref{modc}\right) ,$ and considering that the function $A_{n}^{\ast }$ is nondecreasing yield

\begin{align*} & \underset {\Gamma _{1}^{\prime \prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,1}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\lesssim \end{align*}

\begin{align*} & \lesssim \overset {\frac{a_{n,n}}{3}}{\underset {\frac{1}{3\left( n+1\right) }}{\int }}\underset {3u_{2}}{\overset {a_{n,n}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{1}du_{2} \\ & \leq 2\overset {\frac{a_{n,n}}{3}}{\underset {\frac{1}{3\left( n+1\right) }}{\int }}\underset {3u_{2}}{\overset {a_{n,n}}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{3u_{1}u_{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{1}du_{2} \\ & = \tfrac {2}{3}\overset {\frac{a_{n,n}}{3}}{\underset {\frac{1}{3\left( n+1\right) }}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{2}}\log \left( \tfrac {a_{n,n}}{3u_{2}}\right) A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{2} \\ & \leq \log \left( \left( n+1\right) a_{n,n}\right) \overset {\frac{a_{n,n}}{3}}{\underset {\frac{1}{3\left( n+1\right) }}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{2} \\ & \leq \log \left( n+1\right) \overset {\frac{a_{n,n}}{3}}{\underset {\frac{1}{3\left( n+1\right) }}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{2} =\log \left( n+1\right) \overset {\frac{3}{\pi }\left( n+1\right) }{\underset {\frac{3}{\pi }}{\int }}\tfrac {\omega _{H}( f,\frac{3}{\pi t}) }{t}A_{n}^{\ast }\left( t\right) dt \\ & =\log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\left( \overset {\frac{3}{\pi }\left( k+1\right) }{\underset {\frac{3}{\pi }k}{\int }}\tfrac {\omega _{H}\left( f,\frac{3}{\pi t}\right) }{t}A_{n}^{\ast }\left( t\right) dt\right) \\ & \leq \log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}( f,\frac{1}{k}) }{k}A_{n}^{\ast }\left( k+1\right) . \end{align*}

For $j=2,3,$ by $\left( \ref{hk3}\right) $ and (21)

\begin{align*} & \underset {\Gamma _{1}^{\prime \prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,j}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \lesssim \\ & \lesssim \overset {a_{n,n}}{\underset {\frac{1}{n+1}}{\int }}\underset {\frac{1}{3\left( n+1\right) }}{\overset {\frac{u_{1}}{3}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}u_{2}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{2}du_{1} \\ & =\overset {a_{n,n}}{\underset {\frac{1}{n+1}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}\log \left( \left( n+1\right) u_{1}\right) A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{1} \\ & \leq \log \left( \left( n+1\right) a_{n,n}\right) \overset {a_{n,n}}{\underset {\frac{1}{n+1}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{1}= \end{align*}

\begin{align*} & =\log \left( \left( n+1\right) a_{n,n}\right) \overset {\frac{3}{\pi }\left( n+1\right) }{\underset {\frac{3}{\pi }}{\int }}\tfrac {\omega _{H}\left( f,\frac{3}{\pi t}\right) }{t}A_{n}^{\ast }\left( t\right) dt \\ & \leq \log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}\left( f,\frac{1}{k}\right) }{k}A_{n}^{\ast }\left( k+1\right) . \end{align*}

Thus we get the estimate

\begin{equation} I_{n,1}\lesssim \log \left( n+1\right) \left\{ \omega _{H}\left( f,a_{n,n}\right) +\underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}\left( f,\frac{1}{k}\right) }{k}A_{n}^{\ast }\left( k+1\right) \right\} . \label{In1} \end{equation}

We divide $\Gamma _{2}$ into two domains as

\begin{align*} \Gamma _{2}^{\prime } & :=\left\{ \left( u_{1},u_{2}\right) \in \Gamma _{2}:u_{2}\leq \tfrac {a_{n,n}}{3\left( n+1\right) }\right\} , \\ \Gamma _{2}^{\prime \prime } & :=\left\{ \left( u_{1},u_{2}\right) \in \Gamma _{2}:u_{2}\geq \tfrac {a_{n,n}}{3\left( n+1\right) }\right\} \end{align*}

to estimate $I_{n,2}.$ By $\left( \ref{sin2}\right) $ and $\left( \ref{modc2}\right) ,$

\begin{eqnarray*} & & \underset {\Gamma _{2}^{\prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k,1}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \lesssim \\ & \lesssim & \overset {\frac{a_{n,n}}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {a_{n,n}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}du_{1}du_{2}\leq 2\tfrac {\omega _{H}\left( f,a_{n,n}\right) }{a_{n,n}}\overset {\frac{a_{n,n}}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {a_{n,n}}{\overset {1}{\int }}\tfrac {1}{u_{1}}du_{1}du_{2} \\ & =& 2\tfrac {\omega _{H}\left( f,a_{n,n}\right) }{a_{n,n}}\log \left( \tfrac {1}{a_{n,n}}\right) \tfrac {a_{n,n}}{3\left( n+1\right) }\leq \log \left( n+1\right) \omega _{H}\left( f,a_{n,n}\right) . \end{eqnarray*}

$\left( \ref{sin1}\right) ,$ $\left( \ref{sin2}\right) $ and $\left( \ref{modc2}\right) $ yield

\begin{eqnarray*} & & \underset {\Gamma _{2}^{\prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k,j}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \lesssim \\ & \lesssim & \left( n+1\right) \overset {\frac{a_{n,n}}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {a_{n,n}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}du_{1}du_{2}\leq 2\left( n+1\right) \tfrac {\omega _{H}\left( f,a_{n,n}\right) }{a_{n,n}}\overset {\frac{a_{n,n}}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {a_{n,n}}{\overset {1}{\int }}du_{1}du_{2} \\ & \leq & \omega _{H}\left( f,a_{n,n}\right) \end{eqnarray*}

for $j=2,3.$ Since $\left( \ref{sin2}\right) $ implies $\left\vert H_{k,1}\left( u_{1},u_{2}\right) \right\vert \lesssim \frac{1}{u_{1}^{2}}$ for $\left( u_{1},u_{2}\right) \in \Gamma _{2}^{\prime \prime },$ we get

\begin{align*} & \underset {\Gamma _{2}^{\prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,1}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\lesssim \end{align*}

\begin{align*} & \lesssim \overset {\frac{a_{n,n}}{3}}{\underset {\frac{a_{n,n}}{3\left( n+1\right) }}{\int }}\underset {a_{n,n}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}du_{1}du_{2}\leq 2\tfrac {\omega _{H}\left( f,a_{n,n}\right) }{a_{n,n}}\overset {\frac{a_{n,n}}{3}}{\underset {\frac{a_{n,n}}{3\left( n+1\right) }}{\int }}\underset {a_{n,n}}{\overset {1}{\int }}\tfrac {1}{u_{1}}du_{1}du_{2} \\ & =2\tfrac {\omega _{H}\left( f,a_{n,n}\right) }{a_{n,n}}\log \left( \tfrac {1}{a_{n,n}}\right) \tfrac {a_{n,n}}{3}\left( 1-\tfrac {1}{n+1}\right) \leq \log \left( n+1\right) \omega _{H}\left( f,a_{n,n}\right) . \end{align*}

By considering $\left( \ref{hk3}\right) $ and $\left( \ref{hk2}\right) $ we obtain

\begin{eqnarray*} & & \underset {\Gamma _{2}^{\prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,j}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \lesssim \\ & \lesssim & \overset {\frac{a_{n,n}}{3}}{\underset {\frac{a_{n,n}}{3\left( n+1\right) }}{\int }}\underset {a_{n,n}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}u_{2}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{1}du_{2}=\log \left( n+1\right) \underset {a_{n,n}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{1} \\ & \leq & \log \left( n+1\right) \underset {\frac{1}{n+1}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{1}\leq \log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}\left( f,\frac{1}{k}\right) }{k}A_{n}^{\ast }\left( k+1\right) \end{eqnarray*}

for $j=2,3.$ Thus we obtain

\begin{equation} I_{n,2}\lesssim \log \left( n+1\right) \left\{ \omega _{H}\left( f,a_{n,n}\right) +\underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}\left( f,\frac{1}{k}\right) }{k}A_{n}^{\ast }\left( k+1\right) \right\} . \label{In2} \end{equation}

If we use $\left( \ref{hk1}\right) $ and $\left( \ref{modc2}\right) ,$

\begin{align*} & \underset {\Gamma _{3}}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,1}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \lesssim \\ & \lesssim \overset {\frac{1}{3}}{\underset {\frac{a_{n,n}}{3}}{\int }}\underset {3u_{2}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{1}du_{2} \\ & \leq \tfrac {2}{3}\overset {\frac{1}{3}}{\underset {\frac{a_{n,n}}{3}}{\int }}\underset {3u_{2}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{1}u_{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{1}du_{2} \\ & =\tfrac {2}{3}\overset {\frac{1}{3}}{\underset {\frac{a_{n,n}}{3}}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{2}}\log \left( \tfrac {1}{3u_{2}}\right) A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{2} \\ & \leq \tfrac {2}{3}\log \left( \tfrac {1}{a_{n,n}}\right) \overset {\frac{1}{3}}{\underset {\frac{a_{n,n}}{3}}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{2} \leq \end{align*}

\begin{align*} & \leq \log \left( n+1\right) \overset {\frac{1}{3}}{\underset {\frac{1}{3\left( n+1\right) }}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{2} \\ & =\log \left( n+1\right) \overset {\frac{3}{\pi }\left( n+1\right) }{\underset {\frac{3}{\pi }}{\int }}\tfrac {\omega _{H}\left( f,\frac{3}{\pi t}\right) }{t}A_{n}^{\ast }\left( t\right) dt \\ & \leq \log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}\left( f,\frac{1}{k}\right) }{k}A_{n}^{\ast }\left( k+1\right) . \end{align*}

By $\left( \ref{hk3}\right) $ and $\left( \ref{hk2}\right) $ we get

\begin{eqnarray*} & & \underset {\Gamma _{3}}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,j}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \lesssim \\ & \lesssim & \overset {1}{\underset {a_{n,n}}{\int }}\underset {\frac{a_{n,n}}{3}}{\overset {\frac{u_{1}}{3}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}u_{2}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{2}du_{1}=\overset {1}{\underset {a_{n,n}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}\log \left( \tfrac {u_{1}}{a_{n,n}}\right) A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{1} \\ & \leq & \log \left( \tfrac {1}{a_{n,n}}\right) \overset {1}{\underset {\frac{1}{n+1}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{1}\leq \log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}\left( f,\frac{1}{k}\right) }{k}A_{n}^{\ast }\left( k+1\right) \end{eqnarray*}

for $j=2,3.$ Thus, we have

\begin{equation} I_{n,3}\lesssim \log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}\left( f,\frac{1}{k}\right) }{k}A_{n}^{\ast }\left( k+1\right) . \label{In3} \end{equation}

Inequalities $\left( \ref{In1}\right) $, $\left( \ref{In2}\right) $, $\left( \ref{In3}\right) $ and the fact $A_{n}^{\ast }\left( k+1\right) \lesssim A_{n,n-k}^{\ast }$ imply $\left( \ref{uniform}\right) $.

Proof â–¼

[Proof of theorem 3] The method used in proof of theorem 1 gives

\begin{equation} \underset {\Omega }{\int }\left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u\lesssim }\log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k} \label{d1} \end{equation}

and

\begin{equation} \underset {\Omega }{\int }\left\Vert \mathbf{u}\right\Vert ^{\alpha }\left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u\lesssim }\log \left( n+1\right) \left\{ a_{n,n}^{\alpha }+\underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,k}}{k^{1+\alpha }}\right\} \qquad \left( 0<\alpha \leq 1\right) . \label{d2} \end{equation}

If we set $e_{n}\left( \mathbf{t}\right) :=f\left( \mathbf{t}\right) -T_{n}^{\left( A\right) }\left( f\right) \left( \mathbf{t}\right) ,$ we have

\begin{equation} \left\Vert f-T_{n}^{\left( A\right) }\left( f\right) \right\Vert _{H^{\beta }\left( \overline{\Omega }\right) }=\left\Vert f-T_{n}^{\left( A\right) }\left( f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }+\Lambda ^{\beta }\left( e_{n}\right) . \label{en} \end{equation}

Since

\begin{equation*} \left\vert e_{n}\left( \mathbf{t}\right) -e_{n}\left( \mathbf{s}\right) \right\vert \leq \tfrac {1}{\left\vert \Omega \right\vert }\! \underset {\Omega }{\int }\! \Big\vert f\left( \mathbf{t}\right) -f\left( \mathbf{t}\! -\! \mathbf{u}\right) -f\left( \mathbf{s}\right) +f\left( \mathbf{s}\! -\! \mathbf{u}\right) \Big\vert \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u}, \end{equation*}

we have to estimate the integral

\begin{equation*} J_{n}:=\underset {\Omega }{\int }\Big\vert f\left( \mathbf{t}\right) -f\left( \mathbf{t}-\mathbf{u}\right) -f\left( \mathbf{s}\right) +f\left( \mathbf{s}-\mathbf{u}\right) \Big\vert \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u}. \end{equation*}

Since $f\in H^{\alpha }( \overline{\Omega }) $ we have

\begin{equation*} \Big\vert f\left( \mathbf{t}\right) -f\left( \mathbf{t}-\mathbf{u}\right) -f\left( \mathbf{s}\right) +f\left( \mathbf{s}-\mathbf{u}\right) \Big\vert \lesssim \left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{\alpha }, \end{equation*}

hence by $\left( \ref{d1}\right) $ we get

\begin{eqnarray*} \left( J_{n}\right) ^{\frac{\beta }{\alpha }} & =& \left( \underset {\Omega }{\int }\Big\vert f\left( \mathbf{t}\right) -f\left( \mathbf{t}-\mathbf{u}\right) -f\left( \mathbf{s}\right) +f\left( \mathbf{s}-\mathbf{u}\right) \Big\vert \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u}\right) ^{\frac{\beta }{\alpha }} \\ & \lesssim & \left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{\beta }\left( \underset {\Omega }{\int }\left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u}\right) ^{\frac{\beta }{\alpha }} \\ & \lesssim & \left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{\beta }\left( \log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k}\right) ^{\frac{\beta }{\alpha }}. \end{eqnarray*}

We also have

\begin{equation*} \left\vert f\left( \mathbf{t}\right) -f\left( \mathbf{t}-\mathbf{u}\right) -f\left( \mathbf{s}\right) +f\left( \mathbf{s}-\mathbf{u}\right) \right\vert \lesssim \left\Vert \mathbf{u}\right\Vert ^{\alpha } \end{equation*}

for $f\in H^{\alpha }( \overline{\Omega }) .$ Thus by $\left( \ref{d2}\right) $ we obtain

\begin{align*} \left( J_{n}\right) ^{1-\frac{\beta }{\alpha }} & \lesssim \left( \underset {\Omega }{\int }\left\Vert \mathbf{u}\right\Vert ^{\alpha }\left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u}\right) ^{1-\frac{\beta }{\alpha }} \\ & \lesssim \left( \log \left( n+1\right) \left\{ a_{n,n}^{\alpha }+\underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k^{1+\alpha }}\right\} \right) ^{1-\frac{\beta }{\alpha }}. \end{align*}

Since

\begin{align*} & \left\vert e_{n}\left( \mathbf{t}\right) -e_{n}\left( \mathbf{s}\right) \right\vert \leq \\ & \leq J_{n}= \left( J_{n}\right) ^{\frac{\beta }{\alpha }}\left( J_{n}\right) ^{1-\frac{\beta }{\alpha }} \\ & \lesssim \left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{\beta }\log \left( n+1\right) \left( \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k}\right) ^{\frac{\beta }{\alpha }}\left\{ a_{n,n}^{\alpha -\beta }+\left( \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k^{1+\alpha }}\right) ^{1-\frac{\beta }{\alpha }}\right\} , \end{align*}

we get

\begin{equation*} \tfrac {\left\vert e_{n}\left( \mathbf{t}\right) -e_{n}\left( \mathbf{s}\right) \right\vert }{\left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{\beta }}\lesssim \log \left( n+1\right) \left( \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k}\right) ^{\frac{\beta }{\alpha }}\left\{ a_{n,n}^{\alpha -\beta }+\left( \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k^{1+\alpha }}\right) ^{1-\frac{\beta }{\alpha }}\right\} \text{ }\left( \mathbf{t}\neq \mathbf{s}\right) , \end{equation*}

which implies

\begin{equation*} \Lambda ^{\beta }\left( e_{n}\right) \lesssim \log \left( n+1\right) \left( \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k}\right) ^{\frac{\beta }{\alpha }}\left\{ a_{n,n}^{\alpha -\beta }+\left( \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k^{1+\alpha }}\right) ^{1-\frac{\beta }{\alpha }}\right\} . \end{equation*}

The proof is completed by combining this last estimate, $\left( \ref{uniformx}\right) $ and $\left( \ref{en}\right) $.

Proof â–¼

4 Remarks

The degree of approximation by $\left( C,1\right) ,$ Riesz and Nörlund means of trigonometric Fourier series was investigated in [ 18 ] , [ 19 ] , [ 9 ] and [ 1 ] . In this section we conclude from ?? results about the degree of approximation of these means of hexagonal Fourier series.

Remark 4

Let $p=\left( p_{k}\right) $ be a nondecreasing sequence of numbers positive real numbers. If we take

\begin{equation*} a_{n,k}:= \begin{cases} \frac{p_{k}}{P_{n}}, & 0\leq k\leq n \\ 0, & k{\gt}n\end{cases}\end{equation*}

where $P_{n}:=\underset {k=0}{\overset {n}{\sum }}p_{k},$ then $A=\left( a_{n,k}\right) $ satisfies $\left( \ref{m1}\right) $, $\left( \ref{m2}\right) $ and $\left( \ref{m3}\right) $, and $T_{n}^{\left( A\right) }$ becomes the Riesz mean

\begin{equation*} R_{n}\left( p;f\right) =\tfrac {1}{P_{n}}\underset {k=0}{\overset {n}{\sum }}p_{k}S_{k}\left( f\right) . \end{equation*}

theorem 1 gives

\begin{equation} \left\Vert f-R_{n}\left( p;f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\lesssim \log \left( \tfrac {P_{n}}{p_{0}}\right) \left\{ \omega _{H}\left( f,\tfrac {p_{n}}{P_{n}}\right) +\tfrac {1}{P_{n}}\underset {k=1}{\overset {n}{\sum }}\tfrac {Q_{n,k}\omega _{H}\left( f,1/k\right) }{k}\right\} \label{riesz1} \end{equation}

for $f\in C_{H}( \overline{\Omega }) ,$ and theorem 3 yields

\begin{equation} \left\Vert f-R_{n}\left( p;f\right) \right\Vert _{H^{\beta }\left( \overline{\Omega }\right) }\lesssim \log \left( \tfrac {P_{n}}{p_{0}}\right) \! \! \left( \! \tfrac {1}{P_{n}}\underset {k=1}{\overset {n}{\sum }}\tfrac {Q_{n,k}}{k}\right) ^{\frac{\beta }{\alpha }}\! \! \left\{ \left(\! \tfrac {p_{n}}{P_{n}}\right) ^{\alpha -\beta }\! \! \! +\! \! \left( \tfrac {1}{P_{n}}\underset {k=1}{\overset {n}{\sum }}\tfrac {Q_{n,k}}{k^{1+\alpha }}\right) ^{1-\frac{\beta }{\alpha }}\right\} \label{riesz2} \end{equation}

for $f\in H^{\alpha }( \overline{\Omega }) ,$ $\left( 0\leq \beta {\lt}\alpha \leq 1\right) ,$ where $Q_{n,k}:=\underset {\nu =n-k}{\overset {n}{\sum }}p_{\nu }.$ â–¡

Remark 5

Let $\left( p_{k}\right) $ be a nonincreasing sequence of positive real numbers. In this case the matrix $A=\left( a_{n,k}\right) $ with entries

\begin{equation*} a_{n,k}= \begin{cases} \frac{p_{n-k}}{P_{n}}, & 0\leq k\leq n , \\ 0, & k{\gt}n, \end{cases}\end{equation*}

satisfies $\left( \ref{m1}\right)$, $\left( \ref{m2}\right) $ and $\left( \ref{m3}\right) ,$ and $T_{n}^{\left( A\right) }$ becomes the Nörlund mean

\begin{equation*} N_{n}\left( p;f\right) =\tfrac {1}{P_{n}}\underset {k=0}{\overset {n}{\sum }}p_{n-k}S_{k}\left( f\right) . \end{equation*}

By theorem 1 we conclude

\begin{equation} \left\Vert f-N_{n}\left( p;f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\lesssim \log \left( \tfrac {P_{n}}{p_{n}}\right) \left\{ \omega _{H}\left( f,\tfrac {p_{0}}{P_{n}}\right) +\tfrac {1}{P_{n}}\underset {k=1}{\overset {n}{\sum }}\tfrac {P_{k}\omega _{H}\left( f,1/k\right) }{k}\right\} \label{norlund1} \end{equation}

for $f\in C_{H}\left( \overline{\Omega }\right) ,$ and by theorem 3

\begin{equation} \left\Vert f-N_{n}\left( p;f\right) \right\Vert _{H^{\beta }\left( \overline{\Omega }\right) }\lesssim \log \left( \tfrac {P_{n}}{p_{n}}\right) \! \! \left( \tfrac {1}{P_{n}}\underset {k=1}{\overset {n}{\sum }}\tfrac {P_{k}}{k}\right) ^{\frac{\beta }{\alpha }}\! \left\{ \left( \tfrac {p_{0}}{P_{n}}\right) ^{\alpha -\beta }\! \! \! +\! \! \left( \tfrac {1}{P_{n}}\underset {k=1}{\overset {n}{\sum }}\tfrac {P_{k}}{k^{1+\alpha }}\right) ^{1-\frac{\beta }{\alpha }}\right\} \label{norlund2} \end{equation}

for $f\in H^{\alpha }( \overline{\Omega }) ,$ $\left( 0\leq \beta {\lt}\alpha \leq 1\right) .$ â–¡

Remark 6

f we take $p_{k}=1$ $\left( k=0,1,\ldots \right) ,$ $R_{n}\left( p;f\right) $ and $N_{n}\left( p;f\right) $ become $\left( C,1\right) $ means $S_{n}^{\left( 1\right) }\left( f\right) ,$ and both of $(\ref{riesz1})$ and $\left( \ref{norlund1}\right) $ reduce to

\begin{equation*} \left\Vert f-S_{n}^{\left( 1\right) }\left( f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\lesssim \tfrac {\log \left( n+1\right) }{n+1}\underset {k=1}{\overset {n}{\sum }}\omega _{H}\left( f,\tfrac {1}{k}\right) \end{equation*}

for $f\in C_{H}( \overline{\Omega }) .$ Furthermore, $\left( \ref{riesz2}\right) $ and $\left( \ref{norlund2}\right) $ give the estimate

\begin{equation*} \left\Vert f-S_{n}^{\left( 1\right) }\left( f\right) \right\Vert _{H^{\beta }\left( \overline{\Omega }\right) }\lesssim \left\{ \begin{array}{cc} \frac{\log \left( n+1\right) }{n^{\alpha -\beta }}, & \alpha {\lt}1 \\[2mm] \frac{\left( \log \left( n+1\right) \right) ^{2-\beta }}{n^{1-\beta }}, & \alpha =1\end{array}\right. \end{equation*}

for $\left( C,1\right) $ means of $f\in H^{\alpha }( \overline{\Omega }) $ $\left( 0\leq \beta {\lt}\alpha \leq 1\right) .$ â–¡

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