In this paper, we study best approximation polynomials \(p_m^{(\text {abs})}\) and \(p_m^{(\text {rel})}\) for the reciprocal function \(f(x) = \frac{1}{x}\) defined on \([a,\,b] \subset (0,\,\infty )\) . Here the best approximation polynomials of degree m for f minimize the uniform absolute and relative error between p and f, respectively, over all polynomials p of degree at most m. We provide representations of the best approximation polynomials in the basis of shifted Chebyshev polynomials. Furthermore, we derive recursion formulas that compute best approximation polynomials of degree \(2m-1\) from corresponding polynomials of degree \(m-1\) , where the close relationship between these polynomials is exploited. Both, the Chebyshev basis representations and the dyadic recursion formulas are shown to lead to fast and numerically stable algorithms for the evaluation of \(p_m^{(\text {abs})}\) and \(p_m^{(\text {rel})}\) on fine grids. Numerical experiments illustrate these results.
