Approximation of continuous functions on hexagonal domains

Ali Guven$^\ast $

September 6, 2017. Accepted: March 26, 2018. Published online: August 6, 2018.

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

Some approximation properties of hexagonal Fourier series are investigated. The order of approximation by Nörlund means of hexagonal Fourier series is estimated in terms of modulus of continuity.

MSC. 41A25, 41A63, 42B08.

Keywords. Hexagonal Fourier series, modulus of continuity, Nörlund means, order of approximation.

1 Introduction

Let $C_{2\pi }$ be the Banach space of $2\pi $-periodic continuous functions on the real line, equipped with the norm

\begin{equation*} \left\Vert f\right\Vert _{C_{2\pi }}:=\underset {0\leq x\leq 2\pi }{\sup }\left\vert f\left( x\right) \right\vert . \end{equation*}

The modulus of continuity of a function $f\in C_{2\pi }$ is defined by

\begin{equation*} \omega \left( f,\delta \right) :=\underset {0{\lt}\left\vert h\right\vert \leq \delta }{\sup }\left\Vert f-T_{h}\left( f\right) \right\Vert _{C_{2\pi }},\text{ }\left( \delta {\gt}0\right) , \end{equation*}

where $T_{h}\left( f\right) \left( x\right) :=f\left( x+h\right) .$ For $0{\lt}\alpha \leq 1,$ we denote by $H_{2\pi }^{\alpha }$ the Hölder class of functions $f\in C_{2\pi }$ such that $\omega \left( f,\delta \right) \ll \delta ^{\alpha },$ where $A\ll B$ means that there exists a constant $K{\gt}0$ such that $A\leq KB$ holds.

Approximation of functions belonging the space $C_{2\pi }$ by trigonometric polynomials is one of the most important topics in approximation theory and it has a very rich history. Especially, the order of approximation of functions in $H_{2\pi }^{\alpha }$ classes was studied by several mathematicians. Linear summation methods of Fourier series are mostly used tools in these studies.

Let $f\in C_{2\pi }$ has the Fourier series

\begin{equation} f\left( x\right) \sim \underset {k=-\infty }{\overset {\infty }{\sum }}\widehat{f}_{k}e^{ikx}, \label{FS} \end{equation}

with partial sums

\begin{equation*} S_{n}\left( f\right) \left( x\right) :=\underset {k=-n}{\overset {n}{\sum }}\widehat{f}_{k}e^{ikx},\text{ }\left( n=0,1,...\right) . \end{equation*}

We denote by $\left( \sigma _{n}\left( f\right) \right) $ the sequence of Fejér means of $\left( \ref{FS}\right) ,$ i.e.,

\begin{equation*} \sigma _{n}\left( f\right) \left( x\right) =\tfrac {1}{n+1}\underset {k=0}{\overset {n}{\sum }}S_{k}\left( f\right) \left( x\right) . \end{equation*}

In 1912, S.N. Bernstein obtained the following estimate for the approximation order by Fejér means.

Theorem A. [ 2 ] . Let $f\in H_{2\pi }^{\alpha }$ $\left( 0{\lt}\alpha \leq 1\right) .$ Then the estimate

\begin{equation} \left\Vert f-\sigma _{n}\left( f\right) \right\Vert _{C_{2\pi }}\ll \left\{ \begin{array}{cc} \frac{1}{n^{\alpha }}, & \alpha <1 \\[2mm] \frac{\log n}{n}, & \alpha =1\end{array}\right. \label{B} \end{equation}

holds for $n\geq 2.$

S.B. Stechkin extended Bernstein’s result as follows.

Theorem B. [ 14 ] . Let $f\in C_{2\pi }.$ Then the estimate

\begin{equation} \left\Vert f-\sigma _{n}\left( f\right) \right\Vert _{C_{2\pi }}\ll \tfrac {1}{n+1}\underset {k=0}{\overset {n}{\sum }}\omega \big( f,\tfrac {1}{k+1}\big) \label{S} \end{equation}

holds for every natural number $n.$

Let $p=\left( p_{n}\right) _{n=0}^{\infty }$ be a sequence of positive real numbers and let $P_{n}=\underset {k=0}{\overset {n}{\sum }}p_{k}. $ Nörlund means of the series $\left( \ref{FS}\right) $ with respect to the sequence $p$ are defined by

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

It is known that Nörlund summability method is regular if and only if $p_{n}/P_{n}\rightarrow 0$ as $n\rightarrow \infty $ [ 8 , p. 64 ] . It is clear that $N_{n}\left( p;f\right) $ coincides with $\sigma _{n}\left( f\right) $ in the special case $p_{n}=1$ $\left( n=0,1,...\right) $.

In 1976, A.S.B. Holland, B. Sahney and J. Tzimbalario obtained a more general result than Theorem B.

Theorem C. [ 9 ] . Let $p=\left( p_{n}\right) _{n=0}^{\infty }$ be a sequence of positive real numbers such that $np_{n}\ll P_{n}.$ Then for every $f\in C_{2\pi },$ the inequality

\begin{equation} \left\Vert f-N_{n}\left( p;f\right) \right\Vert _{C_{2\pi }}\ll \tfrac {1}{P_{n}}\underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}P_{k}\omega \big( f,\tfrac 1k\big) \label{HST} \end{equation}

holds.

It is clear that in the case $p_{n}=1$ $\left( n=1,2,...\right) $ (4) reduces to (3). Theorem C also extends a result of B. Sahney and D.S. Goel [ 13 ] which states that

\begin{equation} \left\Vert f-N_{n}\left( p;f\right) \right\Vert _{C_{2\pi }}\ll \tfrac {1}{P_{n}}\underset {k=1}{\overset {n}{\sum }}\tfrac {P_{k}}{k^{1+\alpha }} \label{SG} \end{equation}

for $f\in H_{2\pi }^{\alpha }$, where $\left( p_{n}\right) $ is a non-increasing sequence of positive real numbers.

These theorems can be found in the survey [ 10 ] . Also, we refer to the monographs [ 1 ] , [ 3 ] , [ 4 ] , [ 16 ] and [ 18 ] for more information and results about trigonometric approximation theory.

Approximation problems on cubes of the $d-$dimensional Euclidean space $\mathbb {R}^{d}$ are studied by assuming that the functions are $2\pi $-periodic in each of their variables (see, for example [ 16 , Sections 5.3 and 6.3 ] and [ 18 , vol. II, ch. XVII ] ). But, in the case of non-tensor product domain, for example for hexagonal domains in the Euclidean plane $\mathbb {R}^{2},$ another definition of periodicity is needed. For such domains the most useful periodicity is the periodicity with respect to the lattices.

Let $A$ be a non-singular $d\times d$ matrix. The discrete subgroup $A\mathbb {Z}^{d}=\big\{ Ak:k\in \mathbb {Z}^{d}\big\} $ of the Euclidean space $\mathbb {R}^{d}$ is called the lattice generated by $A$, and the matrix $A$ is called the generator matrix of this lattice. The lattice $A^{-tr}\mathbb {Z}^{d}$, where $A^{-tr}$ is the transpose of the inverse matrix $A^{-1},$ is called the dual lattice of $A\mathbb {Z}^{d}.$ A bounded set $\Omega \subset \mathbb {R}^{d}$ is said to tile $\mathbb {R}^{d}$ with the lattice $A\mathbb {Z}^{d}$ if

\begin{equation*} \underset {\alpha \in A\mathbb {Z}^{d}}{\sum }\chi _{\Omega }\left( x+\alpha \right) =1 \end{equation*}

holds almost everwhere, that is, for almost every $x\in \mathbb {R}^{d}$ there exists exactly one $\alpha \in A\mathbb {Z}^{d}$ such that $x+\alpha \in \Omega .$ In this case the set $\Omega $ is called a spectral set for the lattice $A\mathbb {Z}^{d}.$ One suppose that the spectral set $\Omega $ contains $0$ as an interior point and tiles $\mathbb {R}^{d}$ with the lattice $A\mathbb {Z}^{d}$ without overlapping and without gap, i.e.,

\begin{equation*} \underset {k\in \mathbb {Z}^{d}}{\sum }\chi _{\Omega }\left( x+Ak\right) =1 \end{equation*}

for all $x\in \mathbb {R}^{d}$ and $\Omega +Ak$ and $\Omega +Aj$ are disjoint if $k\neq j.$ For example we can take $\Omega =\big[ -\frac{1}{2},\frac{1}{2}\big) ^{d}$ for the standard lattice $\mathbb {Z}^{d}$ (the lattice generated by the identity matrix)$.$

Let $\Omega $ be the spectral set of the lattice $A\mathbb {Z}^{d}.$ $L^{2}\left( \Omega \right) $ becomes a Hilbert space with respect to the inner product

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

where $\left\vert \Omega \right\vert $ is the $d-$dimensional Lebesgue measure of $\Omega .$ A theorem of Fuglede states that the set $\big\{ e^{2\pi i\left\langle \alpha ,x\right\rangle }:\alpha \in A^{-tr}\mathbb {Z}^{d}\big\} $ is an orthonormal basis of the Hilbert space $L^{2}\left( \Omega \right) $, where $\left\langle \alpha ,x\right\rangle $ is the usual Euclidean inner product of $\alpha $ and $x$ [ 5 ] . According to this theorem, Fourier series and approximation on the spectral set of the lattice $A\mathbb {Z}^{d}$ can be studied by using the exponentials $e^{2\pi i\left\langle \alpha ,x\right\rangle }$ $\big( \alpha \in A^{-tr}\mathbb {Z}^{d}\big) .$

A function $f$ is said to be periodic with respect to the lattice $A\mathbb {Z}^{d}$ if

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

for all $k\in \mathbb {Z}^{d}.$

If we consider the standard lattice $\mathbb {Z}^{d}$ and its spectral set $\big[ -\frac{1}{2},\frac{1}{2}\big) ^{d},$ Fourier series with respect to this lattice coincide with usual multiple Fourier series of functions of $d-$variables.

We refer to [ 11 ] for more detailed information about Fourier analysis on lattices.

2 Hexagonal Fourier series

In the Euclidean plane $\mathbb {R}^{2},$ besides the standard lattice $\mathbb {Z}^{2}$ and the rectangular domain $\big[ -\frac{1}{2},\frac{1}{2}\big) ^{2},$ the simplest lattice is the hexagon lattice and the simplest spectral set is the regular hexagon.

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 satisfy $t_{1}+t_{2}+t_{3}=0$. If we define

\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}, \label{homogeneous} \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*}

We use bold letters $\mathbf{t}$ for homogeneous coordinates and we denote by $\mathbb {R}_{H}^{3}$ the plane $t_{1}+t_{2}+t_{3}=0,$ that is

\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\} . \end{equation*}

Also we use the notation $\mathbb {Z}_{H}^{3}$ for the set of points in $\mathbb {R}_{H}^{3}$ with integer components, that is $\mathbb {Z}_{H}^{3}=\mathbb {Z}^{3}\cap \mathbb {R}_{H}^{3}.$

It follows from $(\ref{homogeneous})$ that the Jacobian determinant of the change of variables $x=\left( x_{1},x_{2}\right) \rightarrow \mathbf{t}=\left( t_{1},t_{2},t_{3}\right) $ is $dx_{1}dx_{2}=\frac{2\sqrt{3}}{3}dt_{1}dt_{2}.$

In the homogeneous coordinates, the inner product on $L^{2}\left( \Omega \right) $ becomes

\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 $\Omega ,$ and the orthonormal basis of $L^{2}\left( \Omega \right) $ becomes

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

Also, a function $f$ is periodic with respect to the hexagonal lattice (or $H$-periodic) if and only if $f\left( \mathbf{t}\right) =f\left( \mathbf{t}+\mathbf{s}\right) $ whenever $\mathbf{s}\equiv \mathbf{0}$ $\left( \operatorname {mod}3\right) ,$ where $\mathbf{t}\equiv \mathbf{s}$ $\left( \operatorname {mod}3\right) $ defined 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*}

It is clear that the functions $\phi _{\mathbf{j}}\left( \mathbf{t}\right) $ are $H-$periodic. If the function $f$ is $H-$periodic then

\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 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*}

Note that, $\mathbb {H}_{n}$ consists of all points with integer components inside the hexagon $n\overline{\Omega }.$ Members of the set

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

are called hexagonal trigonometric polynomials. It is clear that the dimension of $\mathcal{H}_{n}$ is $\# \mathbb {H}_{n}=3n^{2}+3n+1.$

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) e^{-\frac{2\pi i}{3}\left\langle \mathbf{j},\mathbf{t}\right\rangle }d\mathbf{t},\text{ }\left( \mathbf{j}\in \mathbb {Z}_{H}^{3}\right) . \end{equation*}

The $n$th 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) \text{, }\left( n\in \mathbb {N}\right) . \end{equation*}

The partial sums have the integral representation

\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{s}\right) D_{n}\left( \mathbf{s}\right) d\mathbf{s}, \label{partial} \end{equation}

where

\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*}

is the Dirichlet kernel of order $n$.

It is known that ( [ 15 ] , [ 11 ] ) 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) ,\text{ }\left( n\in \mathbb {N}\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 $\mathbf{t}=\left( t_{1},t_{2},t_{3}\right) \in \mathbb {R}_{H}^{3}.$

More general information about hexagonal Fourier series can be found in [ 11 ] and [ 17 ] .

3 Main result

We denote by $C_{H}\big( \overline{\Omega }\big) $ the set of complex valued $H-$periodic continuous functions defined on $\mathbb {R}_{H}^{3}$. $C_{H}\big( \overline{\Omega }\big) $ becomes a Banach space with respect to the uniform norm

\begin{equation*} \left\Vert f\right\Vert _{C_{H}\big( \overline{\Omega }\big) }=\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}\big( \overline{\Omega }\big) $ is defined by

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

where $T_{\mathbf{h}}\left( f\right) \left( \mathbf{t}\right) =f\left( \mathbf{t}+\mathbf{h}\right) $ and

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

for $\mathbf{h}=\left( h_{1},h_{2},h_{3}\right) \in \mathbb {R}_{H}^{3}.$ It is known that [ 17 ] the modulus of continuity is a non-decreasing 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 $\lambda {\gt}0.$

For $0{\lt}\alpha \leq 1,$ we define the Hölder class $H^{\alpha }( \overline{\Omega }) $ of $H-$periodic continuous functions as

\begin{equation*} H^{\alpha }(\overline{\Omega }) :=\left\{ f\in C_{H}( \overline{\Omega }) :\omega _{H}\left( f,\delta \right) \ll \delta ^{\alpha },\text{ }\delta {\gt}0\right\} . \end{equation*}

The Fejér means of the series $\left( \ref{fourier}\right) $ are defined by

\begin{equation*} \sigma _{n}\left( f\right) \left( \mathbf{t}\right) =\tfrac {1}{n+1}\overset {n}{\underset {k=0}{\sum }}S_{k}\left( f\right) \left( \mathbf{t}\right) . \end{equation*}

The following analogue of Theorem A for hexagonal Fourier series was proved in [ 6 ] .

Theorem D. Let $f\in H^{\alpha }\left( \overline{\Omega }\right) $ $\left( 0{\lt}\alpha \leq 1\right) .$ Then the estimate

\begin{equation} \left\Vert f-\sigma _{n}\left( f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\ll \left\{ \begin{array}{cc} \frac{1}{n^{\alpha }}, & \alpha <1 \\[2mm] \frac{\left( \log n\right) ^{2}}{n}, & \alpha =1\end{array}\right. \label{G} \end{equation}

holds for $n\geq 2.$

Let $p=\left( p_{n}\right) _{n=0}^{\infty }$ be a sequence of positive real numbers and $\left( N_{n}\left( p;f\right) \right) $ be the sequence of Nörlund means of the series (7) with respect to the sequence $p,$ that is

\begin{equation} N_{n}\left( p;f\right) \left( \mathbf{t}\right) =\tfrac {1}{P_{n}}\overset {n}{\underset {k=0}{\sum }}p_{n-k}S_{k}\left( f\right) \left( \mathbf{t}\right) ,\text{ }\left( n\in \mathbb {N}\right) . \label{NS} \end{equation}

By considering $\left( \ref{partial}\right) ,$ we get

\begin{equation} N_{n}\left( p;f\right) \left( \mathbf{t}\right) =\tfrac {1}{\left\vert \Omega \right\vert }\underset {\Omega }{\int }f\left( \mathbf{t}-\mathbf{s}\right) F_{n}\left( p;\mathbf{s}\right) d\mathbf{s}, \label{NINT} \end{equation}

where

\begin{equation*} F_{n}\left( p;\mathbf{t}\right) :=\tfrac {1}{P_{n}}\overset {n}{\underset {k=0}{\sum }}p_{n-k}D_{k}\left( \mathbf{t}\right) . \end{equation*}

The aim of this work is to prove an analogue of Theorem C for hexagonal Fourier series. The main result is the following.

Theorem 1. Let $p=\left( p_{n}\right) $ be a non-increasing sequence of positive real numbers. Then the estimate

\begin{equation} \left\Vert f-N_{n}\left( p;f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\ll \tfrac {1}{P_{n}}\log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) \label{Th1} \end{equation}

holds for every $f\in C_{H}\big( \overline{\Omega }\big) $ and for every natural number $n.$

Proof â–¼

By (14), definition of $\omega _{H}\left( f,\cdot \right) ,$ (13) and (10) we have

\begin{equation} \left\vert f\left( \mathbf{t}\right) -N_{n}\left( p;f\right) \left( \mathbf{t}\right) \right\vert \ll \tfrac {1}{P_{n}}\int _\Omega \omega _{H}\left( f,\left\Vert \mathbf{s}\right\Vert \right) \left\vert p_{n}\! +\! \overset {n}{\underset {k=1}{\sum }}p_{n-k}\left( \Theta _{k}\left( \mathbf{s}\right) \! -\! \Theta _{k-1}\left( \mathbf{s}\right) \right) \right\vert d\mathbf{s}. \label{fminusNn} \end{equation}

Since the function

\begin{equation*} \mathbf{t}\rightarrow \omega _{H}\left( f,\left\Vert \mathbf{t}\right\Vert \right) \left\vert p_{n}+\underset {k=1}{\overset {n}{\sum }}p_{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 $\mathbf{t}=\left( t_{1},t_{2},t_{3}\right) \in \Omega ,$ it is sufficient to estimate the integral

\begin{equation*} I_{n}:= \int _\Delta \omega _{H}\left( f,\left\Vert \mathbf{t}\right\Vert \right) \left\vert p_{n}+\underset {k=1}{\overset {n}{\sum }}p_{n-k}\left( \Theta _{k}\left( \mathbf{t}\right) -\Theta _{k-1}\left( \mathbf{t}\right) \right) \right\vert d\mathbf{t,} \end{equation*}

where

\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*} I_{n} =& \underset {\Delta }{\int }\omega _{H}\left( f,\left\Vert \mathbf{t}\right\Vert \right) \left\vert p_{n}+\underset {k=1}{\overset {n}{\sum }}p_{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 p_{n}+\underset {k=1}{\overset {n}{\sum }} p_{n-k}\Big( \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{5.5cm}-\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}}\Big) \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} \end{equation*}

as in [ 17 ] , we get

\begin{align*} I_{n}= & 3\underset {\widetilde{\Delta }}{\int }\omega _{H}\left( f,s_{1}\! +\! s_{2}\right) \bigg\vert p_{n}\! \! +\! \! \underset {k=1}{\overset {n}{\sum }}p_{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) } \\ & \hspace{5 cm}-\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 $s_{2},$ we have

\begin{align*} I_{n}=6\underset {\Delta ^{\ast }}{\int }\left( s_{1}+s_{2}\right) ^{\alpha }\bigg\vert p_{n}+\underset {k=1}{\overset {n}{\sum }}p_{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 $\Delta ^{\ast }$ is the half of $\widetilde{\Delta }:$

\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*}

The change of variables

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

hence we have

\begin{equation*} I_{n}=3\underset {\Gamma }{\int }\omega _{H}\left( f,u_{1}\right) \left\vert p_{n}+\underset {k=1}{\overset {n}{\sum }}p_{n-k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}, \end{equation*}

where

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

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( \big( k+\tfrac {1}{2}\big) 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( \big( k+\tfrac {1}{2}\big) \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{align*} D_{k,3}^{\ast }\left( u_{1},u_{2}\right) & :=2\cos \left( \big( k+\tfrac {1}{2}\big) \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{align*}

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) }, \\ 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 considering the fact $\left( n+1\right) p_{n}\ll P_{n}$ and by (11) we get

\begin{eqnarray*} \underset {\Gamma }{\int }p_{n}\omega _{H}\left( f,u_{1}\right) du_{1}du_{2} & \leq & p_{n}\omega _{H}\left( f,1\right) \ll \tfrac {P_{n}}{n}\omega _{H}\left( f,1\right) \\ & =& \tfrac {P_{n}}{n}\omega _{H}\big( f,n\tfrac {1}{n}\big) \ll \tfrac {P_{n}}{n}n\omega _{H}\big( f,\tfrac {1}{n}\big) \\ & =& \underset {k=1}{\overset {n}{\sum }}\tfrac {1}{n}P_{n}\omega _{H}\big( f,\tfrac {1}{n}\big) \leq \underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) , \end{eqnarray*}

since the sequence $\left( P_{n}/n\right) $ non-increasing and $\omega _{H}\left( f,\cdot \right) $ is non-decreasing. Hence,

\begin{equation} I_{n}\ll I_{n}^{\ast }+\underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f, \tfrac {1}{k}\big) , \label{A} \end{equation}

where

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

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

we have

\begin{equation*} I_{n}^{\ast }=I_{n,1}^{\ast }+I_{n,2}^{\ast }+I_{n,3}^{\ast }, \end{equation*}

where

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

We shall need the well known inequalities

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

and

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

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

By (17) and (20) we obtain

\begin{align*} I_{n,1}^{\ast }& =\underset {\Gamma _{1}}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \\ & \leq \underset {\Gamma _{1}}{\int }\omega _{H}\left( f,u_{1}\right) \left( \underset {k=1}{\overset {n}{\sum }}\left( k+1\right) ^{2}p_{n-k}\right) du_{1}du_{2} \\ & \leq \left( n+1\right) ^{2}P_{n}\underset {\Gamma _{1}}{\int }\omega _{H}\left( f,u_{1}\right) du_{1}du_{2} \\ & =\left( n+1\right) ^{2}P_{n}\underset {0}{\overset {1/(3(n+1)) }{\int }}\underset {3u_{2}}{\overset {1/(n+1)}{\int }}\omega _{H}\left( f,u_{1}\right) du_{1}du_{2} \\ & \leq \left( n+1\right) ^{2}P_{n}\omega _{H}\left( f,\tfrac {1}{n+1}\right) \underset {0}{\overset {1/(3\left( n+1\right)) }{\int }}\underset {3u_{2}}{\overset {1/(n+1)}{\int }}du_{1}du_{2} \\ & \leq P_{n}\omega _{H}\left( f,\tfrac {1}{n}\right) =\underset {k=1}{\overset {n}{\sum }}\tfrac {1}{n}P_{n}\omega _{H}\big( f,\tfrac {1}{n}\big) . \end{align*}

Since the sequence $\left( P_{n}/n\right) $ is non-increasing we get

\begin{equation} I_{n,1}^{\ast }\leq \underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \label{B2} \end{equation}

We write the rectangle $\Gamma _{2}$ as $\Gamma _{2}=\Gamma _{2}^{\prime }\cup \Gamma _{2}^{\prime \prime },$ where

\begin{equation*} \Gamma _{2}^{\prime }:=\left\{ \left( u_{1},u_{2}\right) \in \Gamma _{2}:u_{2}\leq \tfrac {p_{n}}{3\left( n+1\right) P_{n}}\right\} \end{equation*}

and

\begin{equation*} \Gamma _{2}^{\prime \prime }:=\left\{ \left( u_{1},u_{2}\right) \in \Gamma _{2}:u_{2}\geq \tfrac {p_{n}}{3\left( n+1\right) P_{n}}\right\} \end{equation*}

to estimate $I_{n,2}^{\ast }.$

By (21) we obtain

\begin{eqnarray*} & & \! \! \! \! \! \! \! \! \! \! \underset {\Gamma _{2}^{\prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}D_{k,1}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\leq \\ & \leq & \overset {\frac{p_{n}}{3\left( n+1\right) P_{n}}}{\underset {0}{\int }}\underset {\frac{1}{n+1}}{\overset {1}{\int }}\omega _{H}\left( f,u_{1}\right) \left( \underset {k=1}{\overset {n}{\sum }}p_{n-k}\left\vert D_{k,1}^{\ast }\left( u_{1},u_{2}\right) \right\vert \right) du_{1}du_{2} \\ & \ll & P_{n}\overset {\frac{p_{n}}{3\left( n+1\right) P_{n}}}{\underset {0}{\int }}\underset {\frac{1}{n+1}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}du_{1}du_{2}=\tfrac {p_{n}}{3\left( n+1\right) }\underset {\frac{1}{n+1}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}du_{1} \\ & =& \tfrac {p_{n}}{3\left( n+1\right) }\underset {1}{\overset {n+1}{\int }}\omega _{H}\big( f,\tfrac {1}{t}\big) dt=\tfrac {p_{n}}{3\left( n+1\right) }\underset {k=1}{\overset {n}{\sum }}\left( \underset {k}{\overset {k+1}{\int }}\omega _{H}\big( f,\tfrac {1}{t}\big) dt\right) \\ & \leq & \tfrac {p_{n}}{n+1}\underset {k=1}{\overset {n}{\sum }}\omega _{H}\big( f,\tfrac {1}{k}\big) \leq \underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \end{eqnarray*}

For $j=2,3,$ by (20) and (21),

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

\begin{eqnarray*} & \leq & nP_{n}\underset {\frac{1}{n+1}}{\overset {1}{\int }}\overset {\frac{p_{n}}{3\left( n+1\right) P_{n}}}{\underset {0}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}du_{2}du_{1}\leq p_{n}\underset {\frac{1}{n+1}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}du_{1} \\ & =& p_{n}\underset {1}{\overset {n+1}{\int }}\tfrac {\omega _{H}\left( f,1/t\right) }{t}dt=\underset {k=1}{\overset {n}{\sum }}\left( \underset {k}{\overset {k+1}{\int }}\tfrac {\omega _{H}\left( f,1/t\right) }{t}dt\right) \\ & \leq & p_{n}\underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}\omega _{H}\big( f,\tfrac {1}{k}\big) =\underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}p_{n}\omega _{H}\big( f,\tfrac {1}{k}\big) \leq \underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \end{eqnarray*}

Hence we get

\begin{equation} \underset {\Gamma _{2}^{\prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\ll \underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \label{C} \end{equation}

To estimate the integrals $I_{n,3}^{\ast }$ and

we shall use the expression $\left( \ref{expression2}\right) $ of $D_{k}^{\ast }\left( u_{1},u_{2}\right) .$

Lemma 5. 11 of [ 12 ] yields

\begin{equation*} \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}\cos \left( \left( 2k+1\right) u_{2}\pi \right) \right\vert \ll P\big( \tfrac {1}{2\pi u_{2}}\big) \end{equation*}

and

\begin{equation*} \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}\cos \left( \left( 2k+1\right) \tfrac {u_{1}-u_{2}}{2}\pi \right) \right\vert \ll P\left( \tfrac {1}{\left( u_{1}-u_{2}\right) \pi }\right) \end{equation*}

for $\left( u_{1},u_{2}\right) \in \Gamma _{2}^{\prime \prime }\cup \Gamma _{3}$, where $P\left( t\right) :=P_{\left[ t\right] }.$ By Lemmas 5. 11 and 5. 10 of [ 12 ] , the fact

\begin{equation*} \sin \tfrac {u_{1}\pi }{2}\leq \tfrac {2}{\sqrt{3}}\sin \left( \tfrac {u_{1}+u_{2}}{2}\pi \right) , \end{equation*}

and (21), we get

\begin{equation*} \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}\cos \left( \left( 2k+1\right) \tfrac {u_{1}+u_{2}}{2}\pi \right) \right\vert \ll P\big( \tfrac {1}{u_{1}\pi }\big) \end{equation*}

for $\left( u_{1},u_{2}\right) \in \Gamma _{2}^{\prime \prime }\cup \Gamma _{3}.$ Hence by considering these inequalities and (21) we obtain

\begin{equation} \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}H_{k,1}\left( u_{1},u_{2}\right) \right\vert \ll \tfrac {1}{u_{1}^{2}}P\big( \tfrac {1}{2\pi u_{2}}\big) \label{hk1} \end{equation}

and

\begin{equation} \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}H_{k,j}\left( u_{1},u_{2}\right) \right\vert \ll \tfrac {1}{u_{1}u_{2}}P\big( \tfrac {3}{2\pi u_{1}}\big) \text{ }\left( j=2,3\right) \label{hk23} \end{equation}

for $\left( u_{1},u_{2}\right) \in \Gamma _{2}^{\prime \prime }\cup \Gamma _{3}.$

By (21) we obtain

\begin{eqnarray*} & & \! \! \! \! \! \! \! \! \! \! \underset {\Gamma _{2}^{\prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}H_{k,1}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\leq \\ & \leq & \underset {\frac{1}{n+1}}{\overset {1}{\int }}\overset {\frac{1}{3\left( n+1\right) }}{\underset {\frac{p_{n}}{3\left( n+1\right) P_{n}}}{\int }}\omega _{H}\left( f,u_{1}\right) \left( \underset {k=1}{\overset {n}{\sum }}p_{n-k}\left\vert H_{k,1}\left( u_{1},u_{2}\right) \right\vert \right) du_{2}du_{1} \\ & \leq & P_{n}\underset {\frac{1}{n+1}}{\overset {1}{\int }}\overset {\frac{1}{3\left( n+1\right) }}{\underset {\frac{p_{n}}{3\left( n+1\right) P_{n}}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}du_{2}du_{1}\leq \tfrac {P_{n}}{n+1}\underset {\frac{1}{n+1}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}du_{1} \\ & =& \tfrac {P_{n}}{n+1}\underset {1}{\overset {n+1}{\int }}\omega _{H}\big( f,\tfrac {1}{t}\big) dt\leq \tfrac {P_{n}}{n+1}\underset {k=1}{\overset {n}{\sum }}\omega _{H}\big( f,\tfrac {1}{k}\big) \\ & \leq & \underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \end{eqnarray*}

For $j=2,3$ by (25) we get

\begin{eqnarray*} & & \! \! \! \! \! \! \! \! \! \! \underset {\Gamma _{2}^{\prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}H_{k,j}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\ll \\ & \ll & \underset {\frac{1}{n+1}}{\overset {1}{\int }}\overset {\frac{1}{3\left( n+1\right) }}{\underset {\frac{p_{n}}{3\left( n+1\right) P_{n}}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}u_{2}}P\left( \tfrac {3}{2\pi u_{1}}\right) du_{2}du_{1} \\ & =& \log \left( \tfrac {P_{n}}{p_{n}}\right) \underset {\frac{1}{n+1}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}P\left( \tfrac {3}{2\pi u_{1}}\right) du_{1} \\ & =& \log \left( \tfrac {P_{n}}{p_{n}}\right) \underset {\frac{3}{2\pi }}{\overset {\frac{3}{2\pi }\left( n+1\right) }{\int }}\omega _{H}\left( f,\tfrac {3}{2\pi t}\right) \tfrac {P\left( t\right) }{t}dt \\ & =& \log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\left( \underset {\frac{3}{2\pi }k}{\overset {\frac{3}{2\pi }\left( k+1\right) }{\int }}\omega _{H}\left( f,\tfrac {3}{2\pi t}\right) \tfrac {P\left( t\right) }{t}dt\right) \leq \end{eqnarray*}

\begin{eqnarray*} & \leq & \log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {\omega _{H}\left( f,\frac{1}{k}\right) }{k}P\left( \tfrac {3}{2\pi }\left( k+1\right) \right) \\ & \ll & \log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \end{eqnarray*}

Thus, (23) and this inequality give

\begin{equation*} \underset {\Gamma _{2}^{\prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\ll \log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) , \end{equation*}

and hence

\begin{equation} I_{n,2}^{\ast }\ll \log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {P_{k}\omega _{H}\left( f,1/k\right) }{k}. \label{D} \end{equation}

By (24) and by the inequality

\begin{equation*} \tfrac {\omega _{H}\left( f,\delta _{2}\right) }{\delta _{2}}\leq 2\tfrac {\omega _{H}\left( f,\delta _{1}\right) }{\delta _{1}}\text{ }\left( \delta _{1}{\lt}\delta _{2}\right) \end{equation*}

which is easily obtained from (11),

\begin{eqnarray*} & & \! \! \! \! \! \! \! \! \! \! \underset {\Gamma _{3}}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}H_{k,1}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\ll \\ & \ll & \underset {\frac{1}{3\left( n+1\right) }}{\overset {\frac{1}{3}}{\int }}\underset {3u_{2}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}P\big( \tfrac {1}{2\pi u_{2}}\big) du_{1}du_{2} \\ & \ll & \underset {\frac{1}{3\left( n+1\right) }}{\overset {\frac{1}{3}}{\int }}\underset {3u_{2}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{1}u_{2}}P\big( \tfrac {1}{2\pi u_{2}}\big) du_{1}du_{2} \\ & =& \underset {\frac{1}{3\left( n+1\right) }}{\overset {\frac{1}{3}}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{2}}P\big( \tfrac {1}{2\pi u_{2}}\big) \log \big( \tfrac {1}{3u_{2}}\big) du_{2} \\ & \leq & \log \left( n+1\right) \underset {\frac{1}{3\left( n+1\right) }}{\overset {\frac{1}{3}}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{2}}P\big( \tfrac {1}{2\pi u_{2}}\big) du_{2} \\ & =& \log \left( n+1\right) \underset {\frac{3}{2\pi }}{\overset {\frac{3}{2\pi }\left( n+1\right) }{\int }}\omega _{H}\big( f,\tfrac {3}{2\pi t}\big) \tfrac {P\left( t\right) }{t}dt \\ & \ll & \log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \end{eqnarray*}

By (25), for $j=2,3,$

Thus,

\begin{equation} I_{n,3}^{\ast }\ll \log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \label{E} \end{equation}

Combining (16), (19), (22), (26) and (27) give (15).

Proof â–¼

4 Conclusions

$\textsc{Conclusion 1}. $ For $f\in H^{\alpha }\big( \overline{\Omega }\big) $ $\left( 0{\lt}\alpha \leq 1\right) ,$ Theorem 1 yields the following analogue of (5):

\begin{equation*} \left\Vert f-N_{n}\left( p;f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\ll \tfrac {1}{P_{n}}\log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {P_{k}}{k^{1+\alpha }}. \end{equation*}

Note that this estimate was obtained directly in [ 7 ] .

$\textsc{Conclusion 2}. $ In the case $p_{n}=1,$ $\left( n=0,1,...\right) ,$ (15) reduces to

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

which is the analogue of (3) for hexagonal Fourier series.

$\textsc{Conclusion 3}. $ In the case $p_{n}=1,$ $\left( n=0,1,...\right) $ and $f\in H^{\alpha }\big( \overline{\Omega }\big) $ $\left( 0{\lt}\alpha \leq 1\right) ,$ (15) gives

\begin{equation*} \left\Vert f-\sigma _{n}\left( f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\ll \left\{ \begin{array}{cc} \frac{\log n}{n^{\alpha }}, & 0{\lt}\alpha {\lt}1 \\[2mm] \frac{\left( \log n\right) ^{2}}{n}, & \alpha =1.\end{array}\right. \end{equation*}

This estimate yields the same approximation order with (12) in the case $\alpha =1.$

Acknowledgements

The author wants to thank the referee for her/his valuable suggestions.

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