Inequalities of Bernstein and Jackson-Nikol'skii type and estimates of the norms of derivatives of Dirichlet kernels
Let the function \(\Lambda (t) = \Lambda (t_1 ,\dots,t_s )\) be continuous on the \(s\)-dimensional cube \([0,1]^s\) and nondecreasing in each of its variable \(t_k ,\;k = 1,\dots,s\). Moreover we assume that \(\Lambda (t) > 0\) if \(\prod_{j = 1}^s {t_j > 0} \) and \(\Lambda (t) = 0\) if \(\prod_{j = 1}^s {t_j = 0} \). For \(N>0\) we set \(\Gamma(\Lambda, N)=\{m\in Z^s_+:\Lambda(2^{-m})\geq 1/N\}\), where \(2^{-m}=(2^{-m_1},\dots, 2^{-m_s})\). Let us introduce the set \(Q(\Lambda, N)=\cup_{n\in\Gamma (\Lambda,N)}\rho(n)\), where \[ \rho(n)=\{m\in Z^s: 2^{n_j-1}\leq |m_j|<2^{n_j},\;j=1,\dots,s\}. \] The author considers the \(\Lambda\)-Dirichlet kernel \(D_{Q(\Lambda, N)}=\sum_{n\in\Gamma(\Lambda, N)}\delta_n(x)\), where \[ \delta_n(x)=\sum_{m\in\rho(n)}e^{i(m,x)},\quad (m,x)=\sum^s_{k=1}m_kx_k. \] In Theorem 1 two-sided estimates for the \(L^p(T^s)\) norms of the Weil derivative \(D^{(\beta)}_{Q(\Lambda,N)}\) of order \(\beta\in R^s\) of the Dirichlet kernel \(D_{Q(\Lambda,N)}\) is given. Theorem 2 gives a criterion for the inclusion \(F^{(\beta)}_{Q(\Lambda,N)}\in L^p(T^s)\), where \(F_{Q(\Lambda, N)}=\sum_{n\in Z^s_+\setminus \Gamma(\Lambda,N)}\delta_n(x)\). Also Bernstein and Nikolskii type inequalities for trigonometric polynomials with spectrum generated by the level surfaces of the function \(\Lambda (t)\) in \(L^p(T^s)\) are established.
scientific article; zbMATH DE number 5155500