  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.
