Generalized sampling and the stable and accurate reconstruction of piecewise analytic functions from their Fourier coefficients
@article{Adcock2014GeneralizedSA, title={Generalized sampling and the stable and accurate reconstruction of piecewise analytic functions from their Fourier coefficients}, author={Ben Adcock and Anders Christian Hansen}, journal={Math. Comput.}, year={2014}, volume={84}, pages={237-270} }
Suppose that the first m Fourier coefficients of a piecewise analytic function are given. Direct expansion in a Fourier series suffers from the Gibbs phenomenon and lacks uniform convergence. Nonetheless, in this paper we show that, under very broad conditions, it is always possible to recover an n-term expansion in a different system of functions using only these coefficients. Such an expansion can be made arbitrarily close to the best possible n-term expansion in the given system. Thus, if a…
16 Citations
A Stability Barrier for Reconstructions from Fourier Samples
- MathematicsSIAM J. Numer. Anal.
- 2014
It is proved that any stable method for resolving the Gibbs phenomenon can converge at best root-exponentially fast in $m$ and any method with faster convergence must also be unstable, and in particular, exponential convergence implies exponential ill-conditioning.
Beyond Consistent Reconstructions: Optimality and Sharp Bounds for Generalized Sampling, and Application to the Uniform Resampling Problem
- Mathematics, Computer ScienceSIAM J. Math. Anal.
- 2013
This paper gives a complete and formal analysis of generalized sampling, the main result of which being the derivation of new, sharp bounds for the accuracy and stability of this approach.
Recovering Piecewise Smooth Functions from Nonuniform Fourier Measurements
- Mathematics, Computer Science
- 2015
This paper analyzes the relation between the dimension of the reconstruction space and the bandwidth of the nonuniform samples, and shows that it is linear for splines and piecewise polynomials of fixed degree, and quadratic for piecewisePolynomial of varying degree.
Exclusive robustness of Gegenbauer method to truncated convolution errors
- MathematicsJ. Comput. Phys.
- 2022
Sampling and Reconstruction in Distinct Subspaces Using Oblique Projections
- MathematicsJournal of Fourier Analysis and Applications
- 2018
We study reconstruction operators on a Hilbert space that are exact on a given reconstruction subspace. Among those the reconstruction operator obtained by the least squares fit has the smallest…
Spectral reconstruction in Fourier modal methods: on exclusive robustness of Gegenbauer method
- Mathematics
- 2021
Fourier spectral methods constitute a rigorous memory-minimizing technology for the analysis of photonic structures. However, the discrete geometry of these artifacts can lead to the Gibbs…
Stable Extrapolation of Analytic Functions
- MathematicsFound. Comput. Math.
- 2019
An asymptotically best extrapolant e(x) is constructed as a least squares polynomial approximant of degree M∗≤12N, which is known to be needed from approximation theory.
Generalized sampling: stable reconstructions, inverse problems and compressed sensing over the continuum
- MathematicsArXiv
- 2013
A Generalized Prony Method for Filter Recovery in Evolutionary System via Spatiotemporal Trade Off
- MathematicsArXiv
- 2015
The accuracy analysis based on the spectral properties of the operator $A$ and the initial state $x$ is performed, and the idea is based on a nonlinear, generalized Prony method similar to AK14.
References
SHOWING 1-10 OF 61 REFERENCES
Algebraic Fourier reconstruction of piecewise smooth functions
- MathematicsMath. Comput.
- 2012
It is proved that the locations of the jumps (and subsequently the pointwise values of the function) can be reconstructed with at least "half the classical accuracy".
Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions
- Mathematics
- 1995
Kowledge of a truncated Fourier series expansion for a 2π-periodic function of finite regularity, which is assumed to be piecewise smooth in each period, is used to accurately reconstruct the…
A Stability Barrier for Reconstructions from Fourier Samples
- MathematicsSIAM J. Numer. Anal.
- 2014
It is proved that any stable method for resolving the Gibbs phenomenon can converge at best root-exponentially fast in $m$ and any method with faster convergence must also be unstable, and in particular, exponential convergence implies exponential ill-conditioning.
On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function
- Mathematics
- 1992
Generalization of the inverse polynomial reconstruction method in the resolution of the Gibbs phenomenon
- Mathematics
- 2004
Beyond Consistent Reconstructions: Optimality and Sharp Bounds for Generalized Sampling, and Application to the Uniform Resampling Problem
- Mathematics, Computer ScienceSIAM J. Math. Anal.
- 2013
This paper gives a complete and formal analysis of generalized sampling, the main result of which being the derivation of new, sharp bounds for the accuracy and stability of this approach.
Complete algebraic reconstruction of piecewise-smooth functions from Fourier data
- Mathematics, Computer ScienceMath. Comput.
- 2015
In this paper we provide a reconstruction algorithm for piecewise-smooth functions with a-priori known smoothness and number of discontinuities, from their Fourier coefficients, posessing the maximal…
The resolution of the Gibbs phenomenon for spherical harmonics
- MathematicsMath. Comput.
- 1997
The Gibbs phenomenon is entirely overcome by proving that knowledge of the first N spherical harmonic coefficients yield an exponentially convergent approximation to a spherical piecewise smooth function f(θ, Φ) in any subinterval in which the function is analytic.
Convergence acceleration of modified Fourier series in one or more dimensions
- MathematicsMath. Comput.
- 2011
This paper extends Eckhoff's method to the convergence acceleration of multivariate modified Fourier series by suitable augmentation of the approximation basis and demonstrates how to increase the convergence rate to an arbitrary algebraic order.