A Patchwise Local Fourier Extension Method for Function Approximation on General Two-Dimensional Domains

We propose a patchwise local Fourier extension method for approximating smooth functions on general two dimensional domains with curved boundaries. The domain is embedded into a Cartesian background grid and decomposed into rectangular interior patches and one-side curved trapezoidal boundary patches. After local data transfer, all patches are converted into fixed-size tensor-product arrays and approximated by a truncated-SVD stabilized local Fourier extension procedure. Unlike global Fourier frame approximations, the proposed method localizes both the geometry and the ill-conditioned extension process. For fixed local parameters, the local algebraic operations are performed on fixed-size systems, and the reference Fourier extension matrices and their singular value decompositions are reused across patches. Boundary patches require additional one-dimensional transfer or completion steps, but their costs remain uniformly bounded by the local resolution. Consequently, the online complexity is $O(N)$, where $N$ denotes the total number of retained output points for fixed local resolution. Numerical experiments on smooth curved domains and on a mildly rough boundary domain demonstrate that the method achieves high accuracy with a fixed set of local parameters. The smooth-cover correction reduces the boundary-induced error by several orders of magnitude in the full-domain rough-boundary test, without changing the underlying scan-based partition.