Polynomials of constant signs, deviating least from zero in \(L_{p,\sigma}(E)\) \((1\leq p\leq +\infty)\) (Q1317589)

The author investigates the problem to find \[ \mu_ n^ \pm (E)= \min_{P_ n\in {\mathcal P}_ n^ \pm (E)} \| P_ n\|_{p,\sigma}, \] where \[ {\mathcal P}_ n(E)= \{P_ n(x)= x^ n+\dots;\;P_ n(x)\geq 0,\;x\in E\} \] are sets of non-negative and non- positive polynomials, respectively; the polynomials \(P_ n^ \pm(x)\in {\mathcal P}_ n^ \pm(E)\) are deviating least from zero in \(L_{p,\sigma}(E)\). Detailed discussion is provided in terms of the existence of non-uniqueness are considered at length. The power and usefulness of polynomials of constant signs are also clearly exhibited.