%~Mouliné par MaN_auto v.0.29.1 2023-11-06 17:12:21 \documentclass[CRMATH, Unicode, XML, published]{cedram} \TopicFR{Analyse harmonique} \TopicEN{Harmonic analysis} \usepackage[noadjust]{cite} \newcommand*\ee{{\mathcal E}} \newcommand*\ff{{\mathcal F}} \newcommand*\pp{{\mathcal P}} \newcommand*\xx{{\mathcal X}} \newcommand*\N{{\mathbb{N}}} \newcommand*\R{{\mathbb{R}}} \newcommand*\Z{{\mathbb{Z}}} \newcommand*{\dd}{\mathrm{d}} \newcommand*{\dmu}{\dd \mu} \newcommand*{\relabel}{\renewcommand{\labelenumi}{(\theenumi)}} \newcommand*{\romanenumi}{\renewcommand*{\theenumi}{\roman{enumi}}\relabel} \graphicspath{{./figures/}} \newcommand*{\mk}{\mkern -1mu} \newcommand*{\Mk}{\mkern -2mu} \newcommand*{\mK}{\mkern 1mu} \newcommand*{\MK}{\mkern 2mu} \hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red} \makeatletter \def\@setafterauthor{% \vglue3mm% % \hspace*{0pt}% \begingroup\hsize=12cm\advance\hsize\abstractmarginL\raggedright \noindent %\hspace*{\abstractmarginL}\begin{minipage}[t]{10cm} \leftskip\abstractmarginL \normalfont\Small \@afterauthor\par \endgroup \vskip2pt plus 3pt minus 1pt } \makeatother \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}-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. \[ \omega_m(f,\delta)_{2,\alpha}:=\sup_{00,\quad f\in L^{2}\left((-1,1), \dmu_{\alpha}\right). \] 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. \bibliographystyle{crplain} \bibliography{CRMATH_Saadi_20221010} \end{document}