Algebraic Fourier reconstruction of piecewise smooth functions

Spectral Reconstruction of Piecewise Smooth Functions

Asymptotic Behavior of the Fourier Transform of Piecewise-smooth Functions

Let D be a bounded domain in Rn such that ∂D is a union of a finite number of smooth hypersurfaces in general position. The asymptotics of the Fourier transform of a piece-wise smooth function f(x) with discontinuities of the first kind on ∂D, in particular of the characteristic function of D, is found in almost all directions. The results are based on our recent work on the singularities of th...

Recovering piecewise smooth functions from nonuniform Fourier measurements

In this paper, we consider the problem of reconstructing piecewise smooth functions to high accuracy from nonuniform samples of their Fourier transform. We use the framework of nonuniform generalized sampling (NUGS) to do this, and to ensure high accuracy we employ reconstruction spaces consisting of splines or (piecewise) polynomials. We analyze the relation between the dimension of the recons...

Spectral Reconstruction of Piecewise Smooth Functions from Their Discrete Data

Abstract. This paper addresses the recovery of piecewise smooth functions from their discrete data. Reconstruction methods using both pseudo-spectral coefficients and physical space interpolants have been discussed extensively in the literature, and it is clear that an a priori knowledge of the jump discontinuity location is essential for any reconstruction technique to yield spectrally accurat...

A Hybrid Approach to Spectral Reconstruction of Piecewise Smooth Functions

Consider a piecewise smooth function for which the (pseudo-)spectral coefficients are given. It is well known that while spectral partial sums yield exponentially convergent approximations for smooth functions, the results for piecewise smooth functions are poor, with spurious oscillations developing near the discontinuities and a much reduced overall convergence rate. This behavior, known as t...