Multiparameter inverse problem of approximation by functions with given supports - Nesterenko, A. N.; Radzievskii, G. V. (Ukrains’kyi Matematychnyi Zhurnal, 2006)
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 {∥f − g∥^p_p| g ∈ L_p(S), g = 0 almost everywhere on St = 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.