On Exact Inequalities of Hardy]Littlewood]Polya Type

The following problem is investigated for certain Hilbert function spaces: for a given AG0 find the infimum of the set of BG0 such that the inequality 5 x k. 5 22FA5x522qB5x r . 5 22 , for k, rgNj 04, 0Fk-r, holds for all sufficiently smooth functions. An analogous problem is investigated under some restrictions on the spectrum of functions.