Divergence-free interpolation has been extensively studied and widely used in approximating vector-valued functions that are divergence-free. However, so far the literature contains no treatment of divergence-free quasi-interpolation. The aims of this paper are two-fold: to construct an analytically divergence-free quasi-interpolation scheme and to derive its simultaneous approximation orders to both the approximated function and its derivatives. To this end, we first explicitly construct a divergence-free matrix kernel based on polyharmonic splines and study its properties both in the spatial domain and Fourier domain. Then, with this divergence-free matrix kernel, we construct a divergence-free quasi-interpolation scheme defined in the whole space ${\mathbb{R}}^d$ for some positive integer d. We also derive corresponding approximation orders of quasi-interpolation to both the approximated divergence-free function and its derivatives. Finally, by coupling divergence-free interpolation together with our divergence-free quasi-interpolation, we extend our construction to a divergence-free quasi-interpolation scheme defined over a bounded domain. Numerical simulations are presented at the end of the paper to demonstrate the validity of divergence-free quasi-interpolation.

无散插值已被广泛研究并广泛用于逼近无散的向量值函数。然而，到目前为止，文献中没有包含对无散准插值的处理。本文的目的有两个：构建一个解析无散度的准插值方案，并推导出其对逼近函数及其导数的同时逼近阶数。为此，我们首先明确地构建了一个基于多调和样条的无散矩阵核，并研究了它在空间域和傅里叶域中的性质。然后，利用这个无散矩阵核，我们构造了一个在整个空间中定义的无散准插值方案${\mathbb{R}}^d$对于一些正整数d。我们还推导出近似的无散函数及其导数的准插值的相应近似阶数。最后，通过将无散插值与我们的无散准插值相结合，我们将我们的构造扩展到定义在有界域上的无散准插值方案。本文最后给出了数值模拟，以证明无散准插值的有效性。