Restricted Range Approximation of k-Convex Functions in Monotone Norms

摘要: Restricted range approximation of k-convex functions by k-concave in monotone norms (norms for which $|f(x)| \leqq |g(x)|,a x b$, implies $||f|| ||g||$) is treated. It proved that the class best approximants contains a polynomial degree $ k - 1$. The results generalize those Kimchi and Richter-Dyn [J. Math. Anal. Appl.,1976] where special case sup-norm without restrictions on ranges discussed. general approach used here includes study some properties polynomials such norms. indicate restricted derivative norms, it impossible to approximate arbitrarily closely function with one outside range.