Best approximation and cyclic variation diminishing kernels (Q1359965)

Let \(\widetilde C\) be the set of continuous \(2\pi\)-periodic functions \(f:\mathbb{R} \to \mathbb{R}\) and let \(\widetilde C^2\) be the set of kernels \(K:\mathbb{R}^2 \to\mathbb{R}\) which are continuous and \(2\pi\)-periodic in each variable. In this paper the function \(K\) is a cyclic variation diminishing kernel, i.e. it verifies a property generalizing total positivity. The authors characterize the elements of best uniform approximation to \(f\in \widetilde C\) from the sets a) \({\mathcal M} =\int_0^{2\pi} K(x,y) h(y)dy\): \(|h(y) |\leq 1\), a.e., \(y\in [0,2\pi]\}\); b) \({\mathcal P}^+_{2n} (\xi)= \{\sum^{2n}_{j=1} (-1)^{j+1} \int_{\xi_j}^{\xi_{j+1}} K(x,y)dy\): \(\xi= \xi_1\leq \cdots \leq\xi_{2n+1} =\xi_1+2 \pi\}\); c) \({\mathcal P}_{2n} =\cup_\xi {\mathcal P}^+_{2n} (\xi)\); d) \({\mathcal M}_\infty =\{\int_0^{2\pi} K(x,y) d\mu(y)\): \(\mu\) is a finite Borel nonnegative measure on \([0,2\pi]\}\). The uniqueness of best approximation elements is also analysed. Nonlinear results are proved by using a characterization of best approximants from quasi-Chebyshev spaces. Other characterizations are given in terms of the properties of the error function.