The Bernstein inequality for certain weighted algebraic and trigonometric polynomials (Q701429)

Let \(Q_{m,n}\) denote the space of polynomials of degree at most \(2m\) on \((-\infty, \infty)\) in the numerator with the polynomial \((1+x^2)^n\) , \(m \leq n\), in the denominator. The author has shown earlier that certain conditions characterize interpolation of minimal norm in the spaces \(Q_{m,n}\) [\textit{T. Kilgore}, J. Approximation Theory 24, 273-288 (1978; Zbl 0428.41023)]. Approximation properties of these spaces are further studied here. In particular, the author investigates here the derivatives of functions in these spaces and establishes inequalities of Markov-Bernstein type.