<p>Wu functions are a class of positive definite compactly supported radial basis functions in terms of piecewise polynomials. They are similar to Wendland functions that came up shortly later and they still require further investigation. Here, we prove that the native space of a Wu function is a dense subspace of a Sobolev space and give an explicit characterization of the native spaces of Wu functions. In addition, three definitions of Wu functions are introduced and proven to be equivalent. Based on these new equivalent definitions and the so-called <i>f</i>-form, we can generalize the Wu functions for the even-dimensional spaces, while the original Wu functions are proven to have odd maximal possible dimensions on which they are positive definite. Such functions in even-dimensional spaces will be called the “missing Wu functions”. Furthermore, we can generalize the Wu functions to “fractional”-dimensional spaces. We call all these Wu functions the generalized Wu functions. The closed form of the generalized Wu functions is given in terms of hypergeometric functions. Finally, we prove that the Wu functions and the missing Wu functions can be written as linear combinations of the generalized Wendland functions.</p>