In this paper, we study functional approximations where we choose the so-called radial basis function method and more specifically, quasi-interpolation. From the various available approaches to the latter, we form new quasi-Lagrange functions when the orders of the singularities of the radial function's Fourier transforms at zero do not match the parity of the dimension of the space, and therefore new expansions and coefficients are needed to overcome this problem. We develop explicit constructions of infinite Fourier expansions that provide these coefficients and make an extensive comparison of the approximation qualities and - with a particular focus - polynomial precision and uniform approximation order of the various formulae. One of the interesting observations concerns the link between algebraic conditions of expansion coefficients and analytic properties of localness and convergence.

翻译：本文研究函数逼近问题，选用径向基函数方法，特别是拟插值方法。针对现有各种拟插值方法，我们构造了新的拟拉格朗日函数，以解决径向函数傅里叶变换在零点处的奇点阶数与空间维数奇偶性不匹配的问题，并为此需要新的展开式及系数。我们开发了提供这些系数的无穷傅里叶展开显式构造方法，并对各种逼近公式的逼近性能——尤其关注多项式精度与一致逼近阶——进行了广泛比较。一个有趣的发现涉及展开系数的代数条件与局部性和收敛性分析性质之间的关系。