Operators for one-side approximation by algebraic polynomials
Combining ideas of Freud and Nevai with a special interpolation problem of Hermite type the author constructs operators \(\phi_ n\) and \(\Phi_ n\) for the one-sided approximation of functions \(f\in BV[0,1]\). These operators are nonlinear and polynomial. The main result of the paper is a complete characterization and saturation theorem for these operators in the \(L_ p\)-metric, \(1\leq p\leq \infty\). It turns out that the saturation order is \(n^{-1/2}\). The appropriate measure for the smoothness of functions is here Popov's first order \(\tau\)-modulus of f with respect to p. The paper closes with a sufficient condition for a prescribed weighted approximation order in terms of the integration properties of a modified maximal function of f.
scientific article; zbMATH DE number 4103670