Fourier reconstruction of univariate piecewise-smooth functions from non-uniform spectral data with exponential convergence rates

作者：Platte Rodrigo B^*; Gutierrez Alexander J; Gelb Anne

来源：Applied and Computational Harmonic Analysis, 2015, 39(3): 427-449.

DOI：10.1016/j.acha.2014.10.002

• Resampling
• Non-uniform Fourier data
• Edge detection
• Chebyshev polynomials
• Spectral convergence

Reconstruction of piecewise smooth functions from non-uniform Fourier data arises in sensing applications such as magnetic resonance imaging (MRI). This paper presents a new method that uses edge information to recover the Fourier transform of a piecewise smooth function from data that is sparsely sampled at high frequencies. The approximation is based on a combination of polynomials multiplied by complex exponentials. We obtain super-algebraic convergence rates for a large class of functions with one jump discontinuity, and geometric convergence rates for functions that decay exponentially fast in the physical domain when the derivatives satisfy a certain bound. Exponential convergence is also proved for piecewise analytic functions of compact support. Our method can also be used to improve initial jump location estimates, which are calculated from the available Fourier data, through an iterative process Finally, if the Fourier transform is approximated at integer values, then the IFFT can be used to reconstruct the underlying function. Post-processing techniques, such as spectral reprojection, can then be used to reduce Gibbs oscillations.

• 出版日期2015-11