Let $\XX$ be a compact, smooth, connected, Riemannian manifold without
boundary, $G:\XX\times\XX\to \RR$ be a kernel. Analogous to a radial basis
function network, an eignet is an expression of the form $\sum_{j=1}^M
a_jG(\circ,y_j)$, where $a_j\in\RR$, $y_j\in\XX$, $1\le j\le M$. We describe a
deterministic, universal algorithm for constructing an eignet for approximating
functions in $L^p(\mu;\XX)$ for a general class of measures $\mu$ and kernels
$G$. Our algorithm yields linear operators. Using the minimal separation
amongst the centers $y_j$ as the cost of approximation, we give modulus of
smoothness estimates for the degree of approximation by our eignets, and show
by means of a converse theorem that these are the best possible for every
\emph{individual function}. We also give estimates on the coefficients $a_j$ in
terms of the norm of the eignet. Finally, we demonstrate that if any sequence
of eignets satisfies the optimal estimates for the degree of approximation of a
smooth function, measured in terms of the minimal separation, then the
derivatives of the eignets also approximate the corresponding derivatives of
the target function in an optimal manner.

使用一种确定性的通用算法，可以构建一个 eignet 以逼近 Lp（μ; XX）中的函数，我们给出我们的 eignet 的光滑度估计，并在最小分离成本中给出尽可能好的近似度，我们还展示了导数的 eignet 如何最优地近似目标函数的相应导数。