Fast Fourier Transform (FFT){based methods, introduced by Moulinec and Suquet [1], are an efficient class of methods to solve conductivity and elasticity problems in periodic media. The extension of this method to non-periodic boundary conditions is a trending issue, as it is required to, among others, non-periodic media, defect modeling and wave propagation. Adaptations of the original method, using mirror unit cells [2] or an embedding buffer zone surrounding the unit cell [3] have been proposed to treat this problem. Nevertheless, direct application of arbitrary Dirichlet and/or Neumann boundary conditions is still an open problem. In this work, we propose a FFT-based method for arbitrary boundary conditions using an approximation space spanned by sine and/or cosine series. The solution field is split between a known term verifying the boundary conditions and an unknown term described by an approprite sine and/or cosine series, which describes the fuctations of the solution field in the unit cell without interfering with the boundary conditions. The method is assessed in several conductivity and elasticity problems including composite materials and fibrous networks, showing correct application of the boundary conditions without numerical artefacts, with the usual convergence rate of the FFT-based numerical methods. [1] H. Moulinec and P. Suquet, A numerical method for computing the overall response of nonlinear composites with complex microstructure. Computer Methods in Applied Mechanics and Engineering, 157(1-2), 69-94, 1998. [2] H. Grimm-Strele and M. Kabel, FFT-based homogenization with mixed uniform boundary conditions. International Journal for Numerical Methods in Engineering, 122(23), 7241-7265, 2021. [3] L. Gélébart, A modified FFT-based solver for the mechanical simulation of heterogeneous materials with Dirichlet boundary conditions. Comptes Rendus. Mécanique, 348(8-9), 693-704, 2020.