Asymmetric approximations in the space $L_{p(t)}$

Abstract

We introduce the notion of $(α,β)$-norm in the space $L_{p(t)}$ of functions $x(t)$ for which $$\int\limits_0^1 {\left| {x(t)} \right|^{p(t)}< \infty }$$ where $p(t)$ is a positive measurable function. We establish a criterion for the element of the best $(α,β)$-approximation in the space $L_{p(t)}$. We obtain inequalities of the type of duality relations.