Estimates for the best asymmetric approximations of asymmetric classes of functions (Q1759956)

Let \(\pi_n\) denote the set of all algebraic polynomials of degree at most \(n\), let \(\alpha\) and \(\beta\) be positive numbers, let \(\sigma \geq 1\), and let \(\rho(x)\) satisfy the inequality \[ \left ( \sqrt{1-x^2} \right )^{\sigma} \leq \rho(x) \leq {{1} \over {\sqrt{1-x^2}}}, \quad x \in [-1,1]. \] For some classes \(W\) of functions \(f\) defined on \([-1,1]\) such that \(f^{(r-1)}\) is absolutely continuous and the \(r\)-th derivative satisfies a certain boundedness condition, the authors find asymptotically sharp estimates for \[ \sup_{f \in W}\inf_{p \in \pi_n} \int_{-1}^1 [ \alpha (f(x) - p(x))_+ + \beta (f(x) - p(x))_- ] \rho(x) \, dx. \]