On localization of functions in the Bernstein space

For ϱ > 0, let be the closed subspace of L ¹(ℝ) consisting of functions ƒ having the Fourier transforms ƒ concentrated in [−ϱ, ϱ]. Let a > 0. In this paper, we consider the problem of maximal localization of the L ¹ norm on [−a, a] of functions from B _ϱ ¹ . More precisely, for a given a, we investigate the supremum of the quantity E _a (ƒ) = ∫ _−a ^a |ƒ(x)|dx over all ƒ from the unit ball of the space B _ϱ ¹ . __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 573–590, October–December, 2007.