We prove that the native space of a Wu function is a dense subspace of a Sobolev space. An explicit characterization of the native spaces of Wu functions is given. Three definitions of Wu functions are introduced and proven to be equivalent. Based on these new equivalent definitions and the so called $f$-form tricks, we can generalize the Wu functions into the even-dimensional spaces $\R^{2k}$, while the original Wu functions are only defined in the odd-dimensional spaces $\R^{2k+1}$. Such functions in even-dimensional spaces are referred to as the `missing Wu functions'. Furthermore we can generalize the Wu functions into `fractional'-dimensional spaces. We call all these Wu functions the generalized Wu functions. The closed form of the generalized Wu functions are 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.

翻译：我们证明了Wu函数的原生空间是Sobolev空间的稠密子空间。给出了Wu函数原生空间的显式刻画。引入了三种Wu函数的定义并证明了其等价性。基于这些新的等价定义和所谓的$f$-形式技巧，我们将Wu函数推广到偶数维空间$\R^{2k}$，而原始Wu函数仅定义于奇数维空间$\R^{2k+1}$。这些偶数维空间中的函数被称为"缺失的Wu函数"。进一步地，我们将Wu函数推广到"分数"维空间。我们将所有这些Wu函数统称为广义Wu函数。给出了广义Wu函数以超几何函数表示的闭式形式。最后，我们证明了Wu函数和缺失的Wu函数可以表示为广义Wendland函数的线性组合。