Multiparameter inverse problem of approximation by functions with given supports

  • A. N. Nesterenko
  • G. V. Radzievskii

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 $L_2$.
Published
25.08.2006
How to Cite
Nesterenko, A. N., and G. V. Radzievskii. “Multiparameter Inverse Problem of Approximation by Functions With Given Supports”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, no. 8, Aug. 2006, pp. 1116–1127, https://umj.imath.kiev.ua/index.php/umj/article/view/3517.
Section
Research articles