High-frequency wave propagation problems, often modeled by the Helmholtz equation, are very challenging to solve numerically. For instance, when using the Finite Element Method (FEM), the number of degrees of freedom needed to approximate the solution correctly grows at least as k^d , where k is the wavenumber and d the dimension (see e.g. [3]). An alternative has been recently proposed in [1]. There, Gaussian coherent states are used as basis functions instead of the piecewise polynomials of the FEM. For a wide range of source terms (such as those arising in plane-wave scattering), it is proved in [1] that Gaussian coherent states provide an accurate solution with only about k^{d−1/2} degrees of freedom, provided they are combined with a least squares formulation. However, this approach is currently limited in two ways. On the one hand, for Galerkin formulations (instead of least squares), no similar result has been proved. On the other hand, it has been numerically observed that Gaussian coherent states can lead to ill-conditioned discretisation matrices. In this work, we improve on this. We use as basis functions particular linear combinations of Gaussian coherent states which form what is called a Wilson basis [2], and which lead to well-conditioned discretised problems. We will show that with these new shape functions, both the least squares and the Galerkin formulations provide an accurate representation of the solution with a number of degrees of freedom scaling roughly as k^{d−1/2} . Furthermore, we will illustrate these theoretical findings with one-dimensional numerical examples. REFERENCES [1] Théophile Chaumont-Frelet, Victorita Dolean, and Maxime Ingremeau. “Efficient approximation of high-frequency Helmholtz solutions by Gaussian coherent states”. In: arXiv preprint arXiv:2208.04851 (2022). [2] Karlheinz Gröchenig. Foundations of time-frequency analysis. Springer Science & Business Media, 2001. [3] J.M. Melenk and S. Sauter. “Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions”. In: Math. Comp. 79.272 (2010), pp. 1871–1914.