An Adaptive Algorithm for Weighted Approximation of Singular Functions over $\mathbb{R}$

Abstract

We study the $\omega$-weighted $L^p$ approximation ($1\le p\le\infty$) of piecewise $r$-smooth functions $f:\mathbb{R}\to\mathbb{R}$. Approximations $\mathcal{A}_nf$ are based on $n$ values of $f$ at points that can be chosen adaptively. Assuming that the weight $\omega$ is Riemann integrable on any compact interval and asymptotically decreasing, a necessary condition for the error of approximation to be of order $n^{-r}$ is that $\|\omega\|_{L^{1/\gamma}}<\infty$, where $\gamma=r+1/p$. For the class $W_r$ of globally $r$-smooth functions, this condition is also sufficient. Indeed, we show a nonadaptive algorithm $\mathcal{P}_n^*$ with the worst case error $\sup_{f\in W_r}\|f-\mathcal{P}_n^*f\|_{L_\omega^p}/\|f^{(r)}\|_{L^\infty} \asymp\|\omega\|_{L^{1/\gamma}}n^{-r}$. Such worst case result does not hold in general for the class of piecewise $r$-smooth functions. However, if $p<\infty$ and the class is restricted to $\widehat F_r^1$ of functions with at most one singularity and uniformly bounded singular...