2020/04/12 by David C. Levin, Levin, David
Engineering · #42B05 #65D15 #Advanced Numerical Analysis Techniques #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2004.05579
openalex publication_date 2020/04/12 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
In some applications, one is interested in reconstructing a function f from\nits Fourier series coefficients. The problem is that the Fourier series is\nslowly convergent if the function is non-periodic, or is non-smooth. In this\npaper, we suggest a method for deriving high order approximation to f using a\nPad 'e-like method. Namely, by fitting some Fourier coefficients of the\napproximant to the given Fourier coefficients of f. Given the Fourier series\ncoefficients of a function on a rectangular domain in \ℝd, assuming\nthe function is piecewise smooth, we approximate the function by piecewise high\norder spline functions. First, the singularity structure of the function is\nidentified. For example in the 2-D case, we find high accuracy approximation to\nthe curves separating between smooth segments of f. Secondly, simultaneously\nwe find the approximations of all the different segments of f. We start by\ndeveloping and demonstrating a high accuracy algorithm for the 1-D case, and we\nuse this algorithm to step up to the multidimensional case.\n