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