The Markov-Bernstein inequality and Hermite-Fejér interpolation for exponential-type weights

We investigate the coefficients of Hermite-Fejér interpolation polynomials based at zeros of orthogonal polynomials with respect to exponential-type weights. First, we obtain the modified Markov-Bernstein inequalities with respect to w∈F(Lip12). Then using the modified Markov-Bernstein inequalities, we estimate the value of |pn(r)(wρ2,x)/pn′(wρ2,x)| for r=1,2,… at zeros of pn(wρ2;x) and we apply this to estimate the coefficients of Hermite-Fejér interpolation polynomials. Here, pn(wρ2,x) denotes the nth orthogonal polynomial with respect to an exponential-type weight wρ(x)=|x|ρw(x), x∈R, ρ>−1/2.