Jackson's theorem in the space \(L_{2}(\mathbb R^{d})\) with power weight
Suppose that \(d\in\mathbb{N}\), \(\mathbb{R}^d\) is \(d\)-dimensional Euclidean space with inner product \((x,y)=\sum_{i=1}^dx_iy_i\) and modulus \(|x|=\sqrt{(x,x)}\) for all \(x,\,y\in\mathbb{R}^d\), \(\lambda=(\lambda_1,\ldots,\lambda_d)\in \mathbb{R}^d\), \(\lambda_i\in [-1/2,\infty)\), \(i\in\{1,\ldots,d\}\), \(\sigma(\lambda)=\lambda_1+\cdots+\lambda_d\), \[ v_{\lambda}(x)=\frac1{2^{\sigma(\lambda)+d}\prod_{i=1}^d\Gamma(\lambda_i+1)} \prod_{i=1}^d|x_i|^{2\lambda_i+1} \] for all \(x\in \mathbb{R}^d\) is the power weight in \(\mathbb{R}^d\), \(d\mu_{\lambda}(x)=v_{\lambda}(x)dx\), and \(L_{2,\lambda}(\mathbb{R}^d)\) is the space of complex Lebesgue measurable functions \(f\) on \(\mathbb{R}^d\) with finite norm \[ \|f\|_{2,\lambda}=\left[\int_{\mathbb{R}^d}|f(x)|^2 \,d\mu_{\lambda}(x)\right]^{1/2}<\infty. \] For \(\lambda_i=-1/2\), \(i\in\{1,\ldots,d\}\), \(L_{2,\lambda}(\mathbb{R}^d)\) is simply denoted by \(L_2(\mathbb{R}^d)\). Assume that \(E_R(f)_2\) is the value of the best approximation of a function \(f\) by entire functions of exponential spherical type \(R\in (0,\infty)\) in the space \(L_2(\mathbb{R}^d)\), \[ \omega(\delta,f)_2=\sup_{|t|\leq\delta}\|f(x+t)-f(x)\|_{L_2(\mathbb{R}^d)} \] is its modulus of continuity, and \(q_{\alpha}\) is the least positive zero of the Bessel function \(J_{\alpha}\) of order \(\alpha\in[-1/2,\infty)\). The sharp Jackson inequality in the space \(L_2(\mathbb{R}^d)\) \[ E_R(f)_2\leq\frac1{\sqrt{2}}\omega\left(\frac{2q_{d/2-1}}R,f\right)_2 \] is well known. Let \(R\in (0,\infty)\), \(B_R=\{x\in\mathbb{R}^d\,:\,|x|\leq R\}\), \(E^R_{2,\lambda}\) be the set of entire functions \(g\in L_{2,\lambda}(\mathbb{R}^d)\) of exponential spherical type \(R\) for which the support \({\displaystyle\text{supp}}\, \widehat{g}\subset B_R\), and \(E_R(f)_{2,\lambda} =\inf\{\|f-g\|_{2,\lambda}\,:\,g\in E^R_{2,\lambda}\}\) be the value of the best approximation of the function \(f\in L_{2,\lambda}(\mathbb{R}^d)\) and \[ \omega_1(\delta,f)_{2,\lambda}=\sup_{|t|\leq\delta} \left[\int_{\mathbb{R}^d}|(\tau^t_{\lambda}|f(y)-f(x)|^2)|_{y=x} \,d\mu_{\lambda}(x)\right]^{1/2},\quad \delta\in(0,\infty) \] its modulus of continuity. In this paper, for \(\lambda_i\in [-1/2,\infty)\) with \(i\in \{1,\ldots,d\}\) and any \(f\in L_{2,\lambda}(\mathbb{R}^d)\), the authors prove the following sharp inequality: \[ E_R(f)_{2,\lambda}\leq \frac1{\sqrt{2}}\omega_1 \left(\frac{2q_{\sigma(\lambda)+d-1}}R,f\right)_{2,\lambda}, \] where the quantity \(2q_{\sigma(\lambda)+d-1}/R\) in the above inequality is the least for which holds.
scientific article; zbMATH DE number 5980656