On Hilbert's inequality and its applications
The following Hilbert's inequality is well-known: \[ \sum_{m=0}^\infty\sum_{n=0}^\infty a_mb_n/(m+n+1)\leq\pi \Biggl(\sum_{m=0}^\infty a_m^2\Biggr)^{1/2} \Biggl(\sum_{n=0}^\infty b_n^2\Biggr)^{1/2} \] [see, for example, \textit{G. H. Hardy, J. E. Littlewood} and \textit{G. Pólya}: ``Inequalities'', 2nd ed. (1952; Zbl 0047.05302, reprint 1988; Zbl 0634.26008, 1st ed. 1934; Zbl 0010.10703)]. The author converts the above \(\ell_2\) norms into weighted \(\ell_2\) norms and gives a corresponding integral version. In both cases, the weights are explicitly given and the inequalities are sharp.
scientific article; zbMATH DE number 1053566