Abstract
In some applications, one is interested in reconstructing a function f from its Fourier series coefficients. The problem is that the Fourier series is slowly convergent if the function is non-periodic, or is non-smooth. In this paper, we suggest a method for deriving high order approximation to f using a Padé-like method. Namely, we do this by fitting some Fourier coefficients of the approximant to the given Fourier coefficients of f. Given the Fourier series coefficients of a function on a rectangular domain in Rd, assuming the function is piecewise smooth, we approximate the function by piecewise high order spline functions. First, the singularity structure of the function is identified. For example in the 2D case, we find high accuracy approximation to the curves separating between smooth segments of f. Secondly, simultaneously we find the approximations of all the different segments of f. We start by developing and demonstrating a high accuracy algorithm for the 1D case, and we use this algorithm to step up to the multidimensional case.
Highlights
Fourier series expansion is a useful tool for representing and approximating functions, with applications in many areas of applied mathematics
We look for an approximation S to f which is a combination of two components, p1 ∈ Π1 in Ω1 and p2 ∈ Π1 in Ω2, separated by Γ0 ( p), p ∈ Π2, such that (2M + 1)2 Fourier coefficients of f and S are matched in the least-squares sense: M
The basic crucial assumption behind the presented Fourier acceleration strategy is that the underlying function is piecewise ‘nice’
Summary
Fourier series expansion is a useful tool for representing and approximating functions, with applications in many areas of applied mathematics. For functions that are not periodic, the convergence rate is slow near the boundaries and the approximation by partial sums exhibits the Gibbs phenomenon. One approach is to filter out the oscillations, as discussed in several papers [1,2] Another useful approach is to transform the Fourier series into an expansion in a different basis. Further improvement of this approach is presented in [3] using Freud polynomials, achieving very good results for univariate functions with singularities. Nersessian and Poghosyan [8] have used a rational Padé type approximation strategy for approximating univariate non-periodic smooth functions. For multiple Fourier series of smooth non-periodic functions, a convergence acceleration approach was suggested by Levin and Sidi [9]. We start by developing and demonstrating a high accuracy algorithm for the 1D case, and use this algorithm to step up to the multidimensional case
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
