Some inequalities of the large sieve type
The author introduces a new method for considering large sieve inequalities. Selberg gave a method based on \(L^2\) estimates, which has been extended to \(0 < p < \infty\). For separated points \(x_1 < x_2 < \ldots < x_r\), with \( \min \{ x_{j+1} - x_j, j = 1, \ldots, r; 2\pi -(x_r - x_1) \} = \delta >0\), the Selberg estimate is that for any \(s\), a trigonometric polynomial of degree \(N\), \[\sum_{j =1}^r \vert s(x_j)\vert ^2 \leq \left( \frac{N}{2 \pi} + \frac{1}{\delta} \right) \int_T \vert s(x) \vert ^2 \,dx.\] This has been extended for \(0 < p < \infty, - \pi < x_1 < x_2 < \ldots < x_r < \pi\), with \(\delta\) as above, \[ \sum_{j =1}^r \vert s(x_j)\vert ^p \leq \left( \frac{N}{ \pi} + \frac{1}{\delta} \right) \frac{(p +1)e}{2}\int_T \vert s(x) \vert ^p \,dx.\] The author proves that if the \(x_j, \delta\) are as above and \(1 \leq p < \infty\), \[\sum_{j =1}^r \vert s(x_j)\vert ^p \leq C\int_T \vert s(x) \vert ^p \,dx\] where \[C = \frac{N \sigma_p}{ \vert \vert \cos \ t \vert \vert ^p_{L^p([- \pi/2, \pi/2]}}\] with \(\sigma_p = \frac{\pi}{N \delta}\), if \(\frac{\pi}{N \delta}\in Z\), and equal to \(1 + \left[ \frac{\pi}{N \delta} \right]\), otherwise. This can be simplified to (i) If \(\frac{\pi}{N \delta} \in Z, \vert \vert s\vert \vert_p = 1, p \geq 1, \) and \(\{ x_j \}_{j=1}^r\) separated as above, \[\sum_{j =1}^r \vert s(x_j)\vert ^p \leq \frac{p +1}{\delta}. \] (ii) If \(\frac{\pi}{N \delta} \notin Z\) ,\[\sum_{j =1}^r \vert s(x_j)\vert ^p \leq (p + 1) \left(\frac{N}{\pi} + \frac{1}{\delta}\right).\]
scientific article; zbMATH DE number 7491534