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 S\B 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.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1116–1127, August, 2006.
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Radzievskii, G.V., Nesterenko, A.N. Multiparameter inverse problem of approximation by functions with given supports. Ukr Math J 58, 1261–1274 (2006). https://doi.org/10.1007/s11253-006-0132-3
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DOI: https://doi.org/10.1007/s11253-006-0132-3