On Chebyshev–Markov–Krein inequalities

We review the topic of Chebyshev–Markov–Krein inequalities, i.e. estimates for infν∈V(μ)∫fdνandsupν∈V(μ)∫fdν where μμ is a non-negative finite measure, and V(μ)V(μ) is the set of all non-negative finite measures νν satisfying ∫udν=∫udμ for all u∈Uu∈U, where UU is a finite-dimensional subspace. For UU a finite-dimensional TT-space on [a,b][a,b], we prove correct necessary and sufficient conditions for when a given non-negative function f∈C[a,b]f∈C[a,b] satisfies ∫aξ−fdμξ≤∫aξ−fdν≤∫aξ+fdν≤∫aξ+fdμξ for every ν∈V(μ)ν∈V(μ) and all ξ∈(a,b)ξ∈(a,b), where μξμξ is the unique canonical representation in V(μ)V(μ) containing the point ξξ.