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This article is cited in 12 scientific papers (total in 12 papers)
Order estimates of derivatives of the multidimensional periodic Dirichlet $\alpha$-kernel in a mixed norm
È. M. Galeev
Abstract:
In this paper the author establishes a sharp order estimate, in the mixed norm of $L_p(\mathbf T^n)$ for $1<p<\infty$ and in $L_\infty(\mathbf T^n)$ ($\mathbf T^n =[-\pi,\pi]^n$ is the $n$-dimensional torus), of the derivatives of order $\beta \in \mathbf R^n$ of the multidimensional Dirichlet $\alpha$-kernel $D_{\alpha,\mu}$ and the function $F_{\alpha,\mu}$, $\alpha>0$, $\mu>0$, which are sums of exponentials $e^{i(k,t)}$ lying respectively inside and outside a “graduated hyperbolic cross”, i.e., the set $\{k\in\square_s\mid(\alpha,s)\leqslant \mu\}$, where $\square_s=\{k\in\mathbf Z^n\mid2^{s_{j-1}} \leqslant|k_j|<2^{s_j},\, j=1,\ldots,n\}$, $s>0$.
Bibliography: 11 titles.
Received: 12.12.1980
Citation:
È. M. Galeev, “Order estimates of derivatives of the multidimensional periodic Dirichlet $\alpha$-kernel in a mixed norm”, Math. USSR-Sb., 45:1 (1983), 31–43
Linking options:
https://www.mathnet.ru/eng/sm2179https://doi.org/10.1070/SM1983v045n01ABEH002584 https://www.mathnet.ru/eng/sm/v159/i1/p32
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