Nikol'skii-Besov spaces and their approximation characteristics for Dunkl harmonic analysis (Q326916)

The space \(L_{p,\alpha}\) consists of functions \(f:\,{\mathbb R}\to{\mathbb R}\), for which the following norm NEWLINE\[NEWLINE \left\|f\right\|_{p,\alpha}:= \left(\int_{{\mathbb R}}\left|f\left(x\right)\right|^p\,d\mu_\alpha\left(x\right)\right)^{1/p}, \quad 1\leq p<\infty, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \left\|f\right\|_{\infty,\alpha}:=\mathop{\text{ess\,sup}}_{x\in{\mathbb R}}\left|f\left(x\right)\right| NEWLINE\]NEWLINE is finite. Here, \(\alpha>-1/2\), and NEWLINE\[NEWLINE d\mu_\alpha\left(x\right)=\frac{\left|x\right|^{2\alpha+1}}{2^{\alpha+1}\Gamma\left(\alpha+1\right)}\,dx. NEWLINE\]NEWLINENEWLINENEWLINENEWLINENEWLINEFor \(f\in C^1\left({\mathbb R}\right)\), the Dunkl differential-difference operator is defined by NEWLINE\[NEWLINE D_\alpha f\left(x\right)=\frac{df}{dx}\left(x\right)+\left(\alpha+\frac{1}{2}\right) \frac{f\left(x\right)-f\left(-x\right)}{x}. NEWLINE\]NEWLINENEWLINENEWLINENEWLINENEWLINEThe spaces \(L_{p,\alpha}\) are embedded into \({\mathcal S}'\), the dual of the Schwartz space \({\mathcal S}\). For \(u\in{\mathcal S}'\), \(D_\alpha u\) is defined by NEWLINE\[NEWLINE \left<D_\alpha u,\varphi\right>:=-\left<u,D_\alpha\varphi\right>,\quad\varphi\in{\mathcal S}, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \left<f,g\right>:=\int_{{\mathbb R}}f\left(x\right)g\left(x\right)\,d\mu_\alpha\left(x\right). NEWLINE\]NEWLINENEWLINENEWLINENEWLINENEWLINEUsing a generalized translation operator, the authors define a special modulus of smoothness \(\omega_k\left(f,\delta\right)_{p,\alpha}\) for a function \(f\in L_{p,\alpha}\). Now, let \(1\leq p<\infty\), and \(r>0\) be real numbers, and let \(k\), and \(m\) be non-negative integers with the property \(m<;r<k+m\). The spaces \(H_p^{r,\alpha}\) consist of all functions \(f\in L_{p,\alpha}\), such that \(D_\alpha f,D_\alpha^2f,\dots,D_\alpha^mf\in L_{p,\alpha}\), and NEWLINE\[NEWLINE \omega_k\left(D_\alpha^mf,\delta\right)_{p,\alpha}\leq A_f\delta^{r-m},\quad\delta>0, NEWLINE\]NEWLINE for some \(A_f>0\).NEWLINENEWLINENEWLINENEWLINEThe following expression defines a semi-norm on \(H_p^{r,\alpha}\): NEWLINE\[NEWLINE h_p^r\left(f\right):=\sup_{\delta>0}\frac{\omega_k\left(D_\alpha^mf,\delta\right)_{p,\alpha}}{\delta^{r-m}}. NEWLINE\]NEWLINE Furthermore, \(H_p^{r,\alpha}\) is a Banach space with the norm NEWLINE\[NEWLINE \left\|f\right\|_{H_p^{r,\alpha}}:=\left\|f\right\|_{p,\alpha}+h_p^r\left(f\right). NEWLINE\]NEWLINENEWLINENEWLINENEWLINENEWLINEThe authors show that the space \(H_p^{r,\alpha}\) does not depend on the actual choice of \(k\) and \(m\) (see the Theorem below).NEWLINENEWLINENEWLINENEWLINEThe space of all entire functions of exponential type at most \(\sigma\) whose restrictions to \({\mathbb R}\) belong to \(L_{p,\alpha}\) is denoted by \(E_{p,\alpha}^\sigma\). The best approximation of \(f\in L_{p,\alpha}\) by such functions is NEWLINE\[NEWLINE E_\sigma\left(f\right)_{p,\alpha}:=\inf\left\{\left\|f-g\right\|_{p,\alpha}\,:\,g\in E_{p,\alpha}^\sigma\right\}. NEWLINE\]NEWLINENEWLINENEWLINENEWLINENEWLINEOne of the main results of the article gives a characterization of the space \(H_p^{r,\alpha}\) by the order of approximation by functions from the \(E_{p,\alpha}^\sigma\) class:NEWLINENEWLINENEWLINENEWLINETheorem. If \(f\in H_p^{r,\alpha}\), then NEWLINE\[NEWLINE E_\sigma\left(f\right)_{p,\alpha}\leq c_2\frac{h_p^r\left(f\right)}{\sigma^r}, NEWLINE\]NEWLINE for \(\sigma\geq 1\). Conversely, if \(f\in L_{p,\alpha}\) and NEWLINE\[NEWLINE E_\sigma\left(f\right)_{p,\alpha}\leq \frac{A}{\sigma^r}, NEWLINE\]NEWLINE for \(\sigma\geq 1\), where \(A\) is a constant that does not depend on \(\sigma\) (but depends on \(f\)), then \(f\in H_p^{r,\alpha}\) for all non-negative integers \(k\), \(m\), such that \(m<r<m+k\), and NEWLINE\[NEWLINE \left\|f\right\|_{H_p^{r,\alpha}}\leq C\left(\left\|f\right\|_{p,\alpha}+A\right), NEWLINE\]NEWLINE where \(C=C\left(k,m,r,\alpha\right)\) is a constant.NEWLINENEWLINENEWLINENEWLINEThe other main result is a description of the space \(B_{p,q,\alpha}^r\) in terms of the best approximations.NEWLINENEWLINENEWLINENEWLINEThe article should be interesting for specialists in harmonic analysis, functional analysis, and approximation theory.