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

Abstract

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 ( L i p 1 2 ) . Then using the modified Markov–Bernstein inequalities, we estimate the value of | p n ( r ) ( w ρ 2 , x ) / p n ′ ( w ρ 2 , x ) | for r = 1 , 2 , … at zeros of p n ( w ρ 2 ; x ) and we apply this to estimate the coefficients of Hermite–Fejér interpolation polynomials. Here, p n ( w ρ 2 , x ) denotes the n th orthogonal polynomial with respect to an exponential-type weight w ρ ( x ) = | x | ρ w ( x ) , x ∈ R , ρ > − 1 / 2 .