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\title{Equivalence of K-functionals and modulus of smoothness generated by a Dunkl type operator on the interval $(-1, 1)$}

\author{\firstname{Faouaz} \lastname{Saadi}\IsCorresp}
\address{Department of Mathematics, Laboratory of Topology, Algebra, Geometry, and Discrete Mathematics, Faculty of Sciences Aïn Chock University Hassan II, Casablanca, Morocco}
\email{sadfaouaz@gmail.com}

\author{\firstname{Radouan} \lastname{Daher}}
\address[1]{Department of Mathematics, Laboratory of Topology, Algebra, Geometry, and Discrete Mathematics, Faculty of Sciences Aïn Chock University Hassan II, Casablanca, Morocco}
\email{rjdaher024@gmail.com}

\begin{abstract}
Our aim in this paper is to show that the modulus of smoothness and the $K$-functionals constructed from the Sobolev-type space corresponding to the Dunkl operator are equivalent on the interval $(-1,1)$.
\end{abstract}

\subjclass{43A30, 46E35, 33D60}
\keywords{\kwd{Fourier--Dunkl series}
\kwd{Dunkl transform}
\kwd{generalized translation operator}
\kwd{$K$-functionals}
\kwd{modulus of smoothness}}
\dateposted{2023-11-17}

\begin{document}
\maketitle

\section{Introduction}

As it is well-known, the modulus of smoothness generated by the standard translation is equivalent with the Peetre’s K-functional, see e.g.~\cite[p.~171]{devore}. This property is extended to Dunkl translation by E. S. Belkina and S. S. Platonov (see~\cite{belkina}) and Bessel translation in~\cite{platonov}. In this paper, we prove the counterparts of results obtained in~\cite{belkina}, i.e., we establish the equivalence between $K$-functionals and modulus of smoothness in the Dunkl context (on $(-1,1)$) by using Fourier--Dunkl expansions introduced in~\cite{ciaurri}, instead of Dunkl transform. The orthonormal system associated with this kind of series is a generalization of the trigonometric one (in particular, the periodicity is lost).

Hereinafter the symbol $\alpha$ stands for a real value such that $\alpha>-1$. We consider the Dunkl operator $\Lambda_{\alpha}$ associated with the reflection group $\mathbb{Z}_{2}$ on $\mathbb{R}$ given by
\[
\Lambda_{\alpha} f(x)=\frac{\dd}{\dd x} f(x)+\frac{2 \alpha+1}{x}\left(\frac{f(x)-f(-x)}{2}\right).
\]
The initial value problem
\[
\begin{cases}
\Lambda_{\alpha} f(x)=i\lambda f(x), &\lambda \in \mathbb{R} \\
f(0)=1
\end{cases}
\]
has a unique solution $E_{\alpha}(i\lambda.)$ (called the \emph{Dunkl kernel}) given by:
\begin{equation}\label{expoo}
E_{\alpha}(i\lambda x)=j_{\alpha}(\lambda x)+\frac{i\lambda x}{2(\alpha+1)}j_{\alpha+1}(\lambda x), \; \ x\in\mathbb{R},
\end{equation}
where $j_{\alpha}$ is the normalized Bessel function of the first kind defined by
\begin{equation}\label{defj}
j_{\alpha}\left(x\right)=2^{\alpha} \Gamma(\alpha+1) \frac{J_{\alpha}\left(x\right)}{x^{\alpha}}
\end{equation}
and $J_{\alpha}$ is the Bessel functions of the first kind of order $\alpha$
\[
J_{\alpha}(x)=\left(\dfrac{x}{2}\right)^{\alpha} \sum_{n=0}^{\infty} \frac{\left(\frac{i x}{2}\right)^{2 n}}{n ! \Gamma(n+\alpha+1)}.
\]
From~\cite{rosler1}, for all $x\in \R$, we have
\begin{equation}\label{e<1}
|E_\alpha(ix)|\leq 1 \quad \text{and}\quad |E^\prime_\alpha(ix)|\leq 1.
\end{equation}


Let $L^{p}\left((-1,1), \dmu_{\alpha}\right), p\geq 1$, denote the Lebesgue spaces on the interval $(-1,1)$ endowed with the norm
\[
\|f\|_{\alpha,p}=\left(\int_{-1}^{1}|f(t)|^{p} \dmu_{\alpha}(x)\right)^\frac{1}{p}.
\]
where $\dmu_{\alpha}(x)=\left(2^{\alpha+1} \Gamma(\alpha+1)\right)^{-1}|x|^{2 \alpha+1} \dd x$. The Dunkl transform is a generalization of the Fourier transform. It is defined for $f\in L^{1}\left((-1,1), \dmu_{\alpha}\right)$ by the identity (see~\cite{Dunkl, dejeu})
\[
\ff_{\alpha} f(y)=\int_{\mathbb{R}} f(x) E_{\alpha}(-iy x) \dmu_{\alpha}(x), \quad y\in \R.
\]
The Fourier transform corresponds with the case $\alpha=-1 / 2$ because $E_{-1 / 2}(ix)=$ $\mathrm{e}^{ix}$ and $\dmu_{-1 / 2}$ is, up to a multiplicative factor, the Lebesgue measure on $\mathbb{R}$.



\section{Equivalence of K-functionals and modulus of smoothness generated by a Dunkl type operator on the interval \texorpdfstring{$(-1, 1)$}{(-1, 1)}.}

Let $\left\{\lambda_{n}:=\lambda_{\alpha+1,n}, n \in \mathbb{N}\right\}$ be the increasing sequence of positive zeros of $J_{\alpha+1}$. It is proved in~\cite{laforgia} that
\begin{equation}\label{lamda2}
\lambda_{n} \leqslant n \pi+\alpha \pi / 2+\pi / 4 \quad\text {for } \alpha > -1/2.
\end{equation}
In~\cite{ifantis} we find the following inequality
\[
\lambda_{n}>\alpha+n \pi-\frac{\pi}{2}+\frac{3}{2}, \quad \alpha>-1, \quad n=1,2, \dots
\]
then
\begin{equation}\label{lamda4}
\lambda_{n}>n, \quad \alpha>-1, \quad n=1,2, \dots.
\end{equation}
When the range of $\alpha$ is fixed, like $-1 < \alpha \leqslant -\frac{1}{2}$ (see Schafheitlin in~\cite[p.~490]{Watson}) and no (essential) restriction on $n$ :
\begin{equation}\label{lamda1}
n \pi+\alpha \pi / 2+\pi / 4 <\lambda_{n}<n \pi \quad\left(-1 < \alpha \leqslant -\frac{1}{2}\right).
\end{equation}
Let $c_\alpha=\pi+\max\{0,\alpha \pi / 2+\pi / 4\}$. Combining~\eqref{lamda2}, \eqref{lamda4}, and~\eqref{lamda1} gives
\begin{equation}\label{lamda3}
n<\lambda_{n} < c_\alpha n \quad \text{for all } \alpha>-1.
\end{equation}
The real-valued function $\Im E_{\alpha}(ix)=\frac{x}{2(\alpha+1)}j_{\alpha+1}(x)$ is odd and its zeros are $\left\{\lambda_{n}, n \in \mathbb{Z}\right\}$ where $\lambda_{-n}=-\lambda_{n}$ and $\lambda_{0}=0$.



Theorem~1 in~\cite{ciaurri} establishes that $\left\{E_{\alpha}\left(i \lambda_{n} x\right)\right\}_{n \in \mathbb{Z}}$ is a complete orthogonal system in $L^{2}\left((-1,1), \dmu_{\alpha}\right)$. That is to say
\[
\int_{-1}^{1}E_{\alpha}\left(i \lambda_{n} x\right) \overline{E_{\alpha}\left(i \lambda_{m} x\right)} \dmu_{\alpha}(x)=\left\|E_{\alpha}\left(i \lambda_{n} \,\cdot\,\right) \right\|^2 _{2,\alpha}\delta_{n m}.
\]
For each appropriate function $f$ on $(-1,1)$, we define its Fourier series related to the system $\left\{E_{\alpha}\left(i \lambda_{n} x\right)\right\}_{n\in \Z}$, which are called Fourier--Dunkl series, as
\[
f \sim \sum_{n \in \mathbb{Z}} c_{n}(f) E_{\alpha}\left(i \lambda_{n} x\right)\theta_n, \quad c_{n}(f)=\int_{-1}^{1} f(y) \overline{E_{\alpha}\left(i \lambda_{n} y\right)} \dmu_{\alpha}(y).
\]
and
\[
\theta_n=\left\|E_{\alpha}\left(i\lambda_{n} \cdot\right)\right\|^{-2}_{2,\alpha}.
\]
We notice that Ó. Ciaurri and his collaborators have studied in~\cite{ciaurri2} the weighted norm convergence of the Fourier--Dunkl series and proved in~\cite{ciaurri3} an uncertainty inequality associated to this system. From~\cite[Lemma~1]{ciaurri} we have
\begin{equation}\label{valeur du poids}
\theta_{n}=\dfrac{2^\alpha\Gamma(\alpha+1)}{|j_\alpha(\lambda_{n})|^2},\quad n\in \Z\setminus \{0\} \quad (\text{we recall that } \lambda_{n}:=\lambda_{\alpha+1,n})
\end{equation}
and $\theta_{0}=2^{\alpha+1}\Gamma(\alpha+2)$. The following asymptotic formulas hold for the Bessel function $J_{\alpha}(u)$ (\cite[p.~490]{Watson}):
\begin{equation}\label{asymptotic1}
J_{\alpha}(u)=\sqrt{\frac{2}{\pi u}}\left[\cos \left(u-\frac{\alpha \pi}{2}-\frac{\pi}{4}\right)+O\left(\frac{1}{u}\right)\right], u \rightarrow \infty.
\end{equation}
Combining~\eqref{valeur du poids} and~\eqref{asymptotic1} gives
\begin{equation}\label{equivalence}
\theta_n\sim \pi|\lambda_{n}|^{2\alpha+1}, |n| \rightarrow \infty.
\end{equation}
The sequence $\left\{c_{n}(f), n \in \mathbb{Z}\right\}$ is called the discrete Fourier--Dunkl transform of $f$. We define the weighted spaces $l^{p}\left(\mathbb{Z}, (\theta_{n})_{n\in \Z}\right)$ by
\[
l^{p}\left(\mathbb{Z}, (\theta_{n})_{n\in \Z}\right)=\left\{\left(x_{n}\right)_{n \in \mathbb{Z}}: \mathbb{Z} \longrightarrow \mathbb{C}:\left(\sum_{n\in \Z}\left|x_{n}\right|^{p} \theta_{n}\right)^{1 / p}<+\infty\right\}.
\]
If $f \in L^{2}\left((-1,1), \dmu_{\alpha}\right)$, then the sequence $\left\{c_{n}(f), n \in \mathbb{Z}\right\}$ belongs to $l^{2}\left(\mathbb{Z}, (\theta_{n})_{n\in \Z}\right)$ and we have
\begin{equation}\label{L2}
\|f\|_{2,\alpha}=\sqrt{\sum_{n=-\infty}^{\infty}\left|c_{n}(f)\right|^{2}\theta_n}.
\end{equation}


The Dunkl translation operator of a function $f$ is defined for all $h\in \R$ by (see~\cite{Appell-Dunkl,rosenblum})
\begin{equation}\label{translation}
\tau_y^{\alpha } f(x)=\sum_{n=0}^{\infty} \Lambda_\alpha^n f(x) \frac{y^n}{\gamma_{n, \alpha}}, \quad \alpha>-1
\end{equation}
where $\Lambda_\alpha^0$ is the identity operator, $\Lambda_\alpha^{n+1}=\Lambda_\alpha\left(\Lambda_\alpha^n\right)$, and
\[
\gamma_{n, \alpha}=
\begin{cases}2^{2 k} k !(\alpha+1)_k & \text {if } n=2 k, \\ 2^{2 k+1} k !(\alpha+1)_{k+1} & \text {if } n=2 k+1.
\end{cases}
\]
The definition~\eqref{translation} is valid only for $C^\infty$-functions, and assuming also that the series on the right is convergent. In particular, this can be guaranteed when $f$ is a polynomial, because the operator $\Lambda_\alpha$ applied to a polynomial of degree $k$ generates a polynomial of degree $k-1$, so the series~\eqref{translation} has only a finite number of nonzero summands. In the case $\alpha=-1 / 2$, the translation $\tau_y^\alpha f$ is just the Taylor expansion of a function $f$ around a fixed point $x$, that is,
\[
f(x+y)=\sum_{n=0}^{\infty} f^{(n)}(x) \frac{y^n}{n !}.
\]
Some properties of the translation operator, including an integral expression, can be found in~\cite{23,31,34, rosler}. For our purposes, we only need the identity~\cite[formula (4.2.2)]{rosenblum}
\begin{equation}\label{trans}
\tau_{h}^\alpha(E_{\alpha}\left(i \lambda x\right))=E_{\alpha}\left(i \lambda h\right)E_{\alpha}\left(i \lambda x\right) \text{ for all } x,h\in \R.
\end{equation}
That resembles the classical
\[
e^{\lambda(h+x)}=e^{\lambda h}e^{\lambda x}.
\]


The scalar product in the Hilbert space $ L^{2}\left((-1,1), \dmu_{\alpha}\right)$ obeys the formula
\[
(f,g)=\int_{-1}^1f(x)\overline{g(x)}\dmu_{\alpha}(x).
\]
We denote by $ \mathcal{E}=\mathcal{E}((-1,1)) $, the set of all infinitely differentiable with compact support included in the interval $(-1,1)$. By the partial integration one can verify the relation
\begin{equation}\label{intparts}
\int_{-1}^1\Lambda_{\alpha}f(x)\overline{g(x)}\dd\mu_{\alpha}(x)=\dfrac{f(1)\overline{g(1)}-f(-1)\overline{g(-1)}}{2^{\alpha+1}\Gamma(\alpha+1)}-\int_{-1}^1f(x) \overline{\Lambda_\alpha g(x)}\dd\mu_{\alpha}(x).
\end{equation}
Then for any functions $f,g\in \ee$, we have
\begin{equation}\label{p1}
(\Lambda_{\alpha}f,g)=-(f,\Lambda_{\alpha}g).
\end{equation}
As usual, we endow the space $\ee$ with a topology; this turns it into a topological vector space. Let $\ee^\prime$ stand for the set of generalized functions, i.e., linear continuous functionals on the space $\ee$. We denote the value of a functional $f\in \ee^\prime$ on a function $\varphi \in \ee$ by $\langle f, \varphi \rangle$.The space $ L^{2}\left((-1,1), \dmu_{\alpha}\right)$ is embedded into $\ee^\prime$, provided that for $ f\in L^{2}\left((-1,1), \dmu_{\alpha}\right)$ and $\varphi \in \ee^\prime$ we put
\[
\langle f, \varphi \rangle=\int_{-1}^1f(x)\varphi(x)\dmu_{\alpha}(x)
\]
We can extend the action of Dunkl operator $\Lambda_\alpha$ onto the space of generalized functions $\ee^\prime$, putting
\[
\langle \Lambda_\alpha f, \varphi \rangle=-\langle f, \Lambda_\alpha\varphi \rangle, f\in \ee^\prime, \varphi\in \ee.
\]
In particular, the action of the operator $\Lambda_{\alpha} f$ is defined for any function $ f\in L^{2}\left((-1,1), \dmu_{\alpha}\right)$, but, generally speaking, $\Lambda_{\alpha} f$ is a generalized function.

Analogously to $\Lambda_{\alpha}$, we can extend the operator $\tau_{h}^\alpha$ by continuity on the whole space $L^{2}\left((-1,1), \dmu_{\alpha}\right)$. Indeed, Let $P$ be a vector space generated by the system $\left\{E_{\alpha}\left(i \lambda_{n} x\right)\right\}_{n \in \mathbb{Z}}$ and $f\in P$, then $f$ can be written as $f(x)=\sum_{n=-m}^{m}c_{n}E_{\alpha}\left(i \lambda_{n} x\right)\theta_n$. Using~\eqref{e<1} and~\eqref{trans} we check easily that
\begin{equation}\label{bounded}
\|\tau_{h}^\alpha f\|_{2,\alpha}\leq \|f\|_{2,\alpha}.
\end{equation}
As $P$ is a dense subspace of $L^{2}\left((-1,1), \dmu_{\alpha}\right)$, it follows from~\eqref{bounded} that $\tau_{h}^\alpha$ can be extended by continuity to a bounded operator in $L^{2}\left((-1,1), \dmu_{\alpha}\right)$. The extended operator is also denoted by $\tau_{h}^\alpha$; inequality~\eqref{bounded} remains valid for it.

For every function $f\in L^{2}\left((-1,1), \dmu_{\alpha}\right)$ we define the differences $\Delta_{h}^{m}f$ of order, $m\in \mathbb{N}=\{1,2,3,\dots\}$, with step $h\in \R$ by the formula $\Delta_{h}^{1}f(t)=\Delta_{h}f(t)=(\tau_{h}^\alpha-I)f(t)$, where $I$ is the identity operator in $L^{2}\left((-1,1), \dmu_{\alpha}\right)$ and for $m>1$
\[
\Delta_{h}^{m}f(t)=\Delta_{h}(\Delta_{h}^{m-1}f(t))=(\tau_{h}^\alpha-I)^{m}f(t)=\sum_{\substack{i=0}}^{\substack{m}}(-1)^{m-1}\binom{m}{i}(\tau_{h}^\alpha)^{i}f(t),
\]
where
\[
(\tau_{h}^\alpha)^{0}f(t)=f(t),\quad (\tau_{h}^\alpha)^{i}f(t)=\tau_{h}^\alpha((\tau_{h}^\alpha)^{i-1}f(t)), \quad i=1,2,\dots,m.
\]
The moduli of smoothness generated by general translations are defined as follows.
\[
\omega_m(f,\delta)_{2,\alpha}:=\sup_{0<h\leq \delta} \left\| \Delta_{h}^m f \right\|_{2,\alpha},\quad \delta >0,\quad f\in L^{2}\left((-1,1), \dmu_{\alpha}\right).
\]

Let $W^m_{2,\alpha}$ be the Sobolev space constructed by the operator $\Lambda_{\alpha}$, i.e.,
\[
W^m_{2,\alpha}:=\left\lbrace f\in L^{2}\left((-1,1), \dmu_{\alpha}\right): \Lambda_{\alpha}^j f \in L^{2}\left((-1,1), \dmu_{\alpha}\right), j=1,2,\dots,m\right\rbrace.
\]
Then the corresponding K-functional is
\[
K(f,t,W^m_{2,\alpha}):=\inf\left\lbrace \left\|f-g \right\|_{2,\alpha} + t\left\|\Lambda_{\alpha}^m g \right\|_{2,\alpha} : g\in W^m_{2,\alpha}\right\rbrace
\]
where $f \in L^{2}\left((-1,1), \dmu_{\alpha}\right)$ and $t>0$. For brevity, we denote
\[
K_m(f,t)_{2,\alpha}:= K(f,t,W^m_{2,\alpha}).
\]
The following theorem establishes an equivalence between the modulus of smoothness and the K-functional. It is analogous to the theorem on the equivalence between the modulus of smoothness and the K-functional in classical approximation theory.

\begin{theo}\label{th1}
One can find positive numbers $C_1=C_1(m,\alpha)$ and $C_2=C_2(m,\alpha)$ which satisfy the
inequality
\[
C_1\omega_m(f,\delta)_{2,\alpha}\leq K_m(f,\delta^m)_{2,\alpha}\leq C_2\omega_m(f,\delta)_{2,\alpha}
\]
where $f \in L^{2}\left((-1,1), \dmu_{\alpha}\right)$ and $\delta>0$.
\end{theo}

Using that $E_{-1 / 2}(ix)=\mathrm{e}^{i x}$, it is easy to check that Theorem~1 reduces to an equivalence between the modulus of smoothness and the K-functional on the basis of Fourier series and usual translation.

When we consider real even and odd functions the Fourier--Dunkl series can be seen as Fourier--Dini and Fourier--Bessel series respectively. From this fact, applying Theorem~1 to even or odd functions, we can deduce analogs of Theorem~1 for these kinds of series.


\section{Proof}

Some technical facts are needed to prove our result. They are included in the next lemmas.

\begin{lemm}\label{noyaudedunkl}\leavevmode
\begin{enumerate}\romanenumi
\item \label{2i} For $x \in \mathbb{R}$ the following inequality is fulfilled
\[
\left|1-E_{\alpha}(ix)\right| \leq |x|.
\]
\item \label{2ii} For $|x| \geq 1$, there exists a certain constant $c > 0$ which depends only on $\alpha$ such that
\[
\left|1-E_{\alpha}(ix)\right| \geq c.
\]
\end{enumerate}
\end{lemm}

\begin{proof}

\begin{proof}[(\ref{2i})]
It follows by the estimates provided in~\eqref{e<1} together with standard application of Lagrange’s mean value theorem.
\let\qed\relax
\end{proof}

\begin{proof}[(\ref{2ii})]
The asymptotic formulas~\eqref{asymptotic1} imply that $j_\alpha(x) \rightarrow 0 $ as $|x| \rightarrow \infty$. Consequently, a number $\eta >0$ exists such that with $|x| \geq \eta $ the inequality $\left|j_\alpha(x)\right| \leq 1 / 2$ is true. Let
\[
m=\min _{1\leq |x|\leq\eta}\left|1-j_\alpha(x)\right|.
\]
With $|x| \geq 1$ we get the inequality $\left|1-j_\alpha(x)\right| \geq c$, where $c=\min \{m, 1 / 2\}$.
Taking only the reel part of $1-E_{\alpha}\left(i \lambda_{n} x\right)$ gives
\[
c\leq \left|1-j_\alpha(x)\right| \leq \left| 1-E_{\alpha}\left(i \lambda_{n} x\right)\right|.
\]
\end{proof}
\let\qed\relax
\end{proof}

\begin{prop}\label{lemth}
Let $ x \in (-1,1) $ and $h\in \R$. If $f\in L^{2}\left((-1,1), \dmu_{\alpha}\right)$ with
\begin{equation*}
f(x)=\sum_{n \in \mathbb{Z}} c_{n}(f) E_{\alpha}\left(i \lambda_{n} x\right)\theta_n,
\end{equation*}
then
\begin{equation}\label{transp}
\tau^{\alpha}_hf(x)=\sum_{n \in \mathbb{Z}} c_{n}(f)E_{\alpha}\left(i \lambda_{n} h\right) E_{\alpha}\left(i \lambda_{n} x\right)\theta_n.
\end{equation}
\end{prop}
\begin{proof}
By product formula~\eqref{trans} of $\tau^{\alpha}_h$, we have
\begin{equation*}
\tau^{\alpha}_hE_{\alpha}\left(i \lambda_{n} x\right)=E_{\alpha}\left(i \lambda_{n} h\right) E_{\alpha}\left(i \lambda_{n} x\right).
\end{equation*}
So, for any Dunkl-polynomial function
\begin{equation*}
\mathcal{Q}_{N}(x)=\sum_{n=-N}^{N}c_{n}(f) E_{\alpha}\left(i \lambda_{n} x\right)\theta_n,
\end{equation*}
(this can be considered as a generalization of the trigonometric polynomial in the classical case $\alpha=-1/2$) since $\tau^{\alpha}_h$ is linear, we have
\begin{equation}\label{tranpoly}
\tau^{\alpha}_h\mathcal{Q}_{N}(x)=\sum_{n=-N}^{N}c_{n}(f)E_{\alpha}\left(i \lambda_{n} h\right) E_{\alpha}\left(i \lambda_{n} x\right)\theta_n.
\end{equation}
By using the fact that $\tau^{\alpha}_h$ is extended to a continuous linear operator in $L^{2}\left((-1,1), \dmu_{\alpha}\right)$ and the set of all polynomials $\mathcal{Q}_{N}(x)$ is everywhere dense in $L^{2}\left((-1,1), \dmu_{\alpha}\right)$, passage to the limit in~\eqref{tranpoly} gives the desired equality.
\end{proof}
\begin{coro}
Let $f\in L^{2}\left((-1,1), \dmu_{\alpha}\right)$ and $h\in \R$, then
\begin{equation}\label{2.10}
\left\|\Delta_{h}^{m} f \right\|_{2,\alpha}\leq 2^m \left\| f \right\|_{2,\alpha}.
\end{equation}
\end{coro}
\begin{proof}
Let $h\in\R$. According to the formula~\eqref{transp}, we obtain
\[
\begin{aligned}
c_n\left(\Delta_{h}^1 f\right) &=c_n\left(\tau_{h}^\alpha f\right)-c_n(f)\\
&=\left(E_{\alpha}\left(i\lambda_{n} h\right) -1\right) c_{n}(f).
\end{aligned}
\]
Using induction with respect to $m$, we have
\begin{equation}\label{induction}
c_n\left(\Delta_{h}^{m} f\right) =\left(E_{\alpha}\left(i\lambda_{n} h\right) -1\right)^{m} c_{n}(f).
\end{equation}
Then
\[
c_n\left(\Delta_{h}^{m} f\right)
\leq2^{m} c_{n}(f).
\]
\end{proof}

\begin{lemm}\label{fourrrr}
If $f \in \ee$, we get
\begin{equation}\label{fourrr}
c_n(\Lambda_\alpha f)=i\lambda_{n}c_n(f)
\end{equation}
for all $ n \in \mathbb{Z}.$
\end{lemm}
\begin{proof}
Let $f \in \ee$, we put $c_{n}(f)=\int_{-1}^{1} f(y) \overline{E_{\alpha}\left(i \lambda_{n} y\right)} \dmu_{\alpha}(y)$. It follows from~\eqref{p1} that
\begin{align*}
c_n(\Lambda_\alpha f)&=\int_{-1}^{1} \Lambda_\alpha f(y)E_{\alpha}\left(-i \lambda_{n} y\right)\dmu_{\alpha}(y),\\
&=-\int_{-1}^{1}f(y)\Lambda_\alpha E_{\alpha}\left(-i \lambda_{n} y\right)\dmu_{\alpha}(y),\\
&=i \lambda_{n}\int_{-1}^{1}f(y) E_{\alpha}\left(-i \lambda_{n} y\right)\dmu_{\alpha}(y),\\
&=i \lambda_{n}c_{n}(f).
\end{align*}
Then the equality~\eqref{fourrr} is valid in $\mathcal{E}$.
\end{proof}

\begin{remark}\label{remark}
Using induction with respect to $m$ and Lemma~\ref{fourrrr}, we can see that for all $f \in W^m_{2,\alpha} $
\begin{equation}\label{fourderr}
c_n(\Lambda_\alpha^m f)=(i\lambda_{n})^mc_n(f)
\end{equation}
for all $n\in \mathbb{Z}$ and $m=0, 1, 2, \dots.$
\end{remark}

\begin{lemm}\label{3.1}
Assume that $\delta>0$ and $f \in W^m_{2,\alpha} $. The following inequality is true:
\[
\omega_m(f,\delta)_{2,\alpha}\leq \delta^{m}\left\|\Lambda^{m} f \right\|_{2,\alpha}.
\]
\end{lemm}
\begin{proof}
Let $h\in (0,\delta]$. According to the formula~\eqref{induction}, we have
\begin{equation*}
c_n\left(\Delta_{h}^{m} f\right) =\left(E_{\alpha}\left(i\lambda_{n} h\right) -1\right)^{m} c_{n}(f).
\end{equation*}
It follows from the Parseval identity~\eqref{L2} and Lemma~\ref{noyaudedunkl} that
\begin{align*}
\left\|\Delta_{h}^{m} f \right\|^2_{2,\alpha}=
\sum_{n\in\Z}\left(1-E_{\alpha}\left(i\lambda_{n} h\right)\right)^{2m}\left|c_{n}(f)\right|^{2}\theta_n
&\leq h^{2m}\sum_{n\in\Z}\left(\dfrac{1-E_{\alpha}\left(i\lambda_{n} h\right)}{\lambda_{n} h}\right)^{2m}\left|\lambda_{n}^mc_{n}(f)\right|^{2}\theta_n
\\
&\leq h^{2m}\left\|\Lambda^{m}_\alpha f \right\|^2_{2,\alpha}\leq \delta^{2m}\left\|\Lambda^{m}_\alpha f \right\|^2_{2,\alpha}.
\end{align*}
Calculating the supremum with respect to all $h\in (0,\delta]$, we obtain $\omega_m(f,\delta)_{2,\alpha}\leq \delta^{m}\left\|\Lambda^{m}_\alpha f \right\|_{2,\alpha}$.
\end{proof}

\begin{definition}
For any function $f \in L^{2}\left((-1,1), \dmu_{\alpha}\right)$ and any number $\sigma >0$, we define the function
\[
\pp_\sigma(f)(t) :=\sum_{n\in \Z}\xx_\sigma(\lambda_{n})c_n(f)E_{\alpha}\left(i\lambda_{n} t\right)\theta_{n}
\]
where $\xx_\sigma(n)$ is the characteristic function defined by
\[
\xx_\sigma(\lambda_n):=
\begin{cases}
1 & \text{if\/ } |\lambda_{n}|\leq \sigma,\\
0 & \text{if\/ } |\lambda_{n}|> \sigma.
\end{cases}
\]
\end{definition}

\begin{prop}
Let $\sigma >0$.
For any function $f \in L^{2}\left((-1,1), \dmu_{\alpha}\right)$ the following inequality is true:
\[
\|f-\pp_\sigma(f)\|_{2,\alpha}\leq C\|\Delta^m_{\frac{1}{\sigma}} f\|_{2,\alpha}.
\]
\end{prop}
\begin{proof}
Using the Parseval equality, we obtain
\begin{align}
\|f-\pp_\sigma(f)\|^2_{2,\alpha}&=\sum_{n\in \Z}(1-\xx_\sigma(\lambda_n))|c_n(f)|^2\theta_{n},\\
&=\sum_{n\in \Z}\dfrac{(1-\xx_\sigma(\lambda_n))}{\left(1-E_{\alpha}\left(i\lambda_{n} \sigma^{-1}\right) \right)^{2m} }\left(1-E_{\alpha}\left(i\lambda_{n} \sigma^{-1}\right) \right)^{2m} |c_n(f)|^2\theta_{n}\label{3.2}
.
\end{align}
Note that $C_1 \leq \left| 1-E_{\alpha}\left(i x\right)\right|$ with $|x|\geq 1$ (see Lemma~\ref{noyaudedunkl}). Hence
\begin{equation}\label{3.3}
\sup_{n\in \Z} \dfrac{1-\xx_\sigma(\lambda_n)}{1-E_{\alpha}\left(i\lambda_{n} \sigma^{-1}\right)}\leq \sup_{|x|\geq 1} \dfrac{1}{1-E_{\alpha}\left(ix\right)}\leq \dfrac{1}{C_1}.
\end{equation}
Relations~\eqref{3.2} and~\eqref{3.3} give
\[
\|f-\pp_\sigma(f)\|_{2,\alpha}\leq C\|\Delta^m_{\frac{1}{\sigma}} f\|_{2,\alpha}
\]
where $C=\dfrac{1}{C_1^m}$.
\end{proof}

\begin{coro}\label{c3.1}
For any function $f \in L^{2}\left((-1,1), \dmu_{\alpha}\right)$ the following inequality is true:
\[
\|f-\pp_\sigma(f)\|_{2,\alpha}\leq C\omega_m\left(f,\frac{1}{\sigma}\right) _{2,\alpha}.
\]
\end{coro}

\begin{prop}
Suppose that $f \in L^{2}\left((-1,1), \dmu_{\alpha}\right)$, $m \in\N$, and $\sigma>0$. Then we have
\[
\|\Lambda_{\alpha}^m\pp_\sigma(f)\|_{2,\alpha}\leq C_3\sigma^m\|\Delta^m_{\frac{1}{\sigma}} f\|_{2,\alpha}.
\]
\end{prop}
\begin{proof}
Using the Parseval equality, we obtain
\begin{align}
\|\Lambda_{\alpha}^m\pp_\sigma(f)\|^2_{2,\alpha}&=\sum_{n\in \Z} \lambda_{n}^{2m}\xx_\sigma(\lambda_{n})|c_n(f)|^2\theta_{n},\\
&=\sigma^{2m}\sum_{n\in \Z}\dfrac{\xx_\sigma(\lambda_{n})(\lambda_{n}\sigma^{-1})^{2m}}{\left(1-E_{\alpha}\left(i\lambda_{n} \sigma^{-1}\right) \right)^{2m} }\left(1-E_{\alpha}\left(i\lambda_{n} \sigma^{-1}\right) \right)^{2m} |c_n(f)|^2\theta_{n}.\label{3.5}
\end{align}
Note that
\begin{equation}\label{3.4}
\sup_{n\in \Z} \left| \dfrac{\xx_\sigma(\lambda_n) \lambda_{n}\sigma^{-1}}{1-E_{\alpha}\left(i\lambda_{n} \sigma^{-1}\right)}\right| \leq\sup_{|x|\leq 1} \left| \dfrac{x}{1-E_{\alpha}\left(ix\right)}\right|=C_2.
\end{equation}
Then formula~\eqref{3.5} yields
\[
\|\Lambda_{\alpha}^m\pp_\sigma(f)\|_{2,\alpha}\leq C_3\sigma^m\|\Delta^m_{\frac{1}{\sigma}} f\|_{2,\alpha}
\]
where $C_3=C_2^m$.
\end{proof}

\begin{coro}\label{c3.2}
For any function $f \in L^{2}\left((-1,1), \dmu_{\alpha}\right)$ the following inequality is true:
\[
\|\Lambda_{\alpha}^m\pp_\sigma(f)\|_{2,\alpha}\leq C_3\sigma^m\omega_m\left(f,\frac{1}{\sigma}\right) _{2,\alpha}.
\]
\end{coro}

\goodbreak
\begin{proof}[Proof of Theorem~\ref{th1}]\ 
\begin{proof}[Proof of the inequality $2^{-m}\omega_m(f,\delta)_{2,\alpha}\leq K_m(f,\delta^m)_{2,\alpha}$]\ 

Let $h\in (0,\delta]$, $g\in W_{2,\alpha}^m$. Using Lemma~\ref{3.1} and inequality~\eqref{2.10}, we obtain
\begin{align*}
\left\|\Delta_{h}^{m} f \right\|_{2,\alpha}\leq\left\|\Delta_{h}^{m} f-g \right\|_{2,\alpha}+\left\|\Delta_{h}^{m} g \right\|_{2,\alpha}&\leq 2^m \left\| f-g \right\|_{2,\alpha}+h^m\left\|\Lambda_\alpha^{m} f \right\|_{2,\alpha}
\\
&\leq 2^m\left(\left\| f-g \right\|_{2,\alpha} + h^m\left\|\Lambda_\alpha^{m} f \right\|_{2,\alpha}\right).
\end{align*}
Calculating the supremum with respect to $h\in (0,\delta]$ and the infimum with respect to all possible functions $g\in W_{2,\alpha}^m$ we obtain
\[
2^{-m}\omega_m(f,\delta)_{2,\alpha}\leq K_m(f,\delta^m)_{2,\alpha}.
\]
\let\qed\relax
\end{proof}

\begin{proof}[Proof of the inequality $K_m(f,\delta^m)_{2,\alpha}\leq C_1\omega_m(f,\delta)_{2,\alpha}$]\ 

Since $\pp_\sigma(f) \in W_{2,\alpha}^m$ by the definition of a $K$-functional we have
\[
K_m(f,\delta^m)_{2,\alpha}\leq \left\| f-\pp_\sigma(f) \right\|_{2,\alpha}+ \delta^m\left\|\Lambda_\alpha^{m} \pp_\sigma(f) \right\|_{2,\alpha}.
\]
Using Corollaries~\ref{c3.1} and~\ref{c3.2}, this gives
\[
K_m(f,\delta^m)_{2,\alpha}\leq \omega_m\left(f,\frac{1}{\sigma}\right) _{2,\alpha} + C_3(\delta \sigma)^n\omega_m\left(f,\frac{1}{\sigma}\right) _{2,\alpha}.
\]
Since $\sigma$ is an arbitrary positive value, choosing $\sigma = 1/\delta$, we obtain the inequality.
\end{proof}
\let\qed\relax
\end{proof}

\subsection*{Acknowledgements} The authors would like to thank the anonymous referee and J. Faraut, who both found misprints as well as suggested valuable improvements of the text.


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