| Abstract: | Let L p (S), 0 < p < +∞, be a Lebesgue space of measurable functions on S with ordinary quasinorm ∥·∥ p . For a system of sets {B t |t ∈ [0, +∞) n } and a given function ψ: [0, +∞) n ↦ [ 0, +∞), we establish necessary and sufficient conditions for the existence of a function f ∈ L p (S) such that inf {∥f − g∥ p p g ∈ L p (S), g = 0 almost everywhere on SB t } = ψ (t), t ∈ [0, +∞) n . As a consequence, we obtain a generalization and improvement of the Dzhrbashyan theorem on the inverse problem of approximation by functions of the exponential type in L 2. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1116–1127, August, 2006. |
