  We obtain new effective results in best approximation theory, specifically
moduli of uniqueness and constants of strong unicity, for the problem of best
uniform approximation with bounded coefficients, as first considered by Roulier
and Taylor. We make use of techniques from the field of proof mining, as
introduced by Kohlenbach in the 1990s. In addition, some bounds are obtained
via the Lagrangian interpolation formula as extended through the use of Schur
polynomials to cover the case when certain coefficients are restricted to be
zero.
