A new generalization of multiquadric functions $\phi(x)=\sqrt{c^{2d}+||x||^{2d}}$, where $x\in\mathbb{R}^n$, $c\in \mathbb{R}$, $d\in \mathbb{N}$, is presented to increase the accuracy of quasi-interpolation further. With the restriction to Euclidean spaces of odd dimensionality, the generalization can be used to generate a quasi-Lagrange operator that reproduces all polynomials of degree $2d-1$. In contrast to the classical multiquadric, the convergence rate of the quasi-interpolation operator can be significantly improved by a factor $h^{2d-n-1}$, where $h>0$ represents the grid spacing. Among other things, we compute the generalized Fourier transform of this new multiquadric function. Finally, an infinite regular grid is employed to analyse the properties of the aforementioned generalization in detail.

翻译：本文提出一类新的广义多二次函数$\phi(x)=\sqrt{c^{2d}+||x||^{2d}}$（其中$x\in\mathbb{R}^n$，$c\in \mathbb{R}$，$d\in \mathbb{N}$），旨在进一步提升拟插值精度。该广义形式在奇数维欧氏空间限制下，可构造出能精确再现$2d-1$次多项式的拟拉格朗日算子。与传统多二次函数相比，其拟插值算子的收敛速度可通过因子$h^{2d-n-1}$显著提升（其中$h>0$表示网格间距）。此外，本文计算了该新型多二次函数的广义傅里叶变换，并采用无限正则网格对所提广义函数的性质进行详细分析。