Linear versus Non-Linear Acquisition of Step-Functions

@article{Ettinger2007LinearVN,
  title={Linear versus Non-Linear Acquisition of Step-Functions},
  author={Boris Ettinger and Niv Sarig and Yosef Yomdin},
  journal={Journal of Geometric Analysis},
  year={2007},
  volume={18},
  pages={369-399}
}
We address in this article the following two closely related problems. 1. How to represent functions with singularities (up to a prescribed accuracy) in a compact way. 2. How to reconstruct such functions from a small number of measurements.The stress is on a comparison of linear and non-linear approaches. As a model case, we use piecewise-constant functions on [0,1], in particular, the Heaviside jump function ℋt=χ[0,t]. Considered as a curve in the Hilbert space L2([0,1]) it is completely… 

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References

SHOWING 1-10 OF 74 REFERENCES

Adaptive Edge Detectors for Piecewise Smooth Data Based on the minmod Limiter

An adaptive edge detection procedure based on a cross-breading between the local and global detectors is introduced, achieved by using the minmod limiter to suppress spurious oscillations near discontinuities while retaining high-order accuracy away from the jumps.

Nonlinear approximation

This is a survey of nonlinear approximation, especially that part of the subject which is important in numerical computation, and emphasis will be placed on approximation by piecewise polynomials and wavelets as well as their numerical implementation.

Semialgebraic complexity of functions

Connect the dots: how many random points can a regular curve pass through?

Given a class Γ of curves in [0, 1]2, we ask: in a cloud of n uniform random points, how many points can lie on some curve γ ∈ Γ? Classes studied here include curves of length less than or equal to

Spectral Reconstruction of Piecewise Smooth Functions from Their Discrete Data

This paper addresses the recovery of piecewise smooth functions from their discrete data. Reconstruction methods using both pseudo-spectral coecients and physical space interpolants have been

Exact sampling results for some classes of parametric nonbandlimited 2-D signals

We present sampling results for certain classes of two-dimensional (2-D) signals that are not bandlimited but have a parametric representation with a finite number of degrees of freedom. While there

Positivity, sums of squares and the multi-dimensional moment problem

Let K be the basic closed semi-algebraic set in R n defined by some finite set of polynomials S and T, the preordering generated by S. For K compact, f a polynomial in n variables nonnegative on K

Multivariate moment problems: Geometry and indeterminateness

The most accurate determinateness criteria for the multivariate mo- ment problem require the density of polynomials in a weighted Lebesgue space of a generic representing measure. We propose a

Positivity, sums of squares and the multi-dimensional moment problem II ⁄

The paper is a continuation of work initiated by the first two authors in [K– M]. Section 1 is introductory. In Section 2 we prove a basic lemma, Lemma 2.1, and use it to give new proofs of key

Detection of Edges in Spectral Data

We are interested in the detection of jump discontinuities in piecewise smooth functions which are realized by their spectral data. Specifically, given the Fourier coefficients, { fk 5 ak 1 ibk}k51 N
...