We consider shift-invariant multiresolution spaces generated by q-elliptic splines in \({\mathbb {R}^{d}},d\ge 2,\) which are tempered distributions characterized by a complex-valued elliptic homogeneous polynomial q of degree \(m>d\). To construct Riesz bases of \(L^{2} ({\mathbb {R}^{d}}),\) a family of non-separable basic smooth functions are obtained by localizing a fundamental solution of the operator q(D), properly. The construction provides a generalization of some known elliptic scaling functions, the most famous being polyharmonic B-splines. Here, we prove that real-valued q leads to r-regular multiresolution analysis, with \(r=m-d-1.\) In addition, we prove that there exist r-regular non-separable prewavelet systems associated with not necessarily regular multiresolution analyses. These prewavelets have \(m-1\) vanishing moments and the approximation order of the prewavelet decomposition can be established.

我们认为通过产生移不变多分辨率空间q在-elliptic花键\（{\ mathbb {R} ^ {d}}，d \的Ge 2，\）被回火分布，其特征在于复值椭圆均匀多项式q的度\（m> d \）。为了构造\（L ^ {2}（{\ mathbb {R} ^ {d}}）\）的Riesz基，通过对算子q（D），正确。该构造提供了一些已知的椭圆缩放函数的概括，其中最著名的是多谐B样条。在这里，我们证明实值q导致r常规多分辨率分析，其中\（r = md-1。\）此外，我们证明存在与不一定常规多分辨率分析相关的r常规不可分离的子波系统。这些预子波具有\（m-1 \）消失矩，可以建立预子波分解的近似阶。