New methods for quasi-interpolation approximations: Resolution of odd-degree singularities Buhmann, Martin Jäger, Janin Jódar, Joaquín Rodríguez González, Miguel Luis Radial basis functions Quasi-interpolation Approximation orders 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 reproduction 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. 2024-07-16T13:55:58Z 2024-07-16T13:55:58Z 2024-04-03 journal article M. Buhmann et al. 223 (2024) 50–64. [https://doi.org/10.1016/j.matcom.2024.03.032] https://hdl.handle.net/10481/93160 10.1016/j.matcom.2024.03.032 eng http://creativecommons.org/licenses/by/4.0/ open access Atribución 4.0 Internacional Elsevier