2017
Том 69
№ 7

# Multiparameter inverse problem of approximation by functions with given supports

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 $L_2$.

English version (Springer): Ukrainian Mathematical Journal 58 (2006), no. 8, pp 1261-1274.

Citation Example: Nesterenko A. N., Radzievskii G. V. Multiparameter inverse problem of approximation by functions with given supports // Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1116–1127.

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