Function approximation via the subsampled Poincaré inequality

Function approximation and recovery via some sampled data have long been studied in a wide array of applied mathematics statistics fields. Analytic tools, such as the Poincaré inequality, handy for estimating errors different scales. The purpose this paper is to study generalized where measurement function subsampled type, with small but non-zero lengthscale that will be made precise. Our analysis identifies inequality basic tool problems. We discuss demonstrate optimality concerning lengthscale, connecting it existing results literature. In application problems, accuracy using basis functions under regularity assumptions established by inequality. observe error bound blows up approaches zero, due fact underlying not regular enough well-defined pointwise values. A weighted version proposed address problem; its also discussed.