Linear versus Non-Linear Acquisition of Step-Functions

  title={Linear versus Non-Linear Acquisition of Step-Functions},
  author={Boris Ettinger and Niv Sarig and Yosef Yomdin},
  journal={Journal of Geometric Analysis},
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… 

Moment inversion problem for piecewise D-finite functions

We consider the problem of exact reconstruction of univariate functions with jump discontinuities at unknown positions from their moments. These functions are assumed to satisfy an a priori unknown

Algebraic Fourier reconstruction of piecewise smooth functions

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".


This paper presents some results on a well-known problem in Algebraic Sig- nal Sampling and in other areas of applied mathematics: reconstruction of piecewise-smooth functions from their integral

Local and Global Geometry of Prony Systems and Fourier Reconstruction of Piecewise-Smooth Functions

Many reconstruction problems in signal processing require solution of a certain kind of nonlinear systems of algebraic equations, which we call Prony systems. We study these systems from a general

Local and global geometry of Prony systems and Fourier reconstruction of piecewise-smooth functionsThis research is supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities, ISF grant 264/09 and the Minerva Foundation

  • Mathematics
  • 2014
Many reconstruction problems in signal processing require solution of a certain kind of nonlinear systems of algebraic equations, which we call Prony systems. We study these systems from a general

Complete algebraic reconstruction of piecewise-smooth functions from Fourier data

  • D. Batenkov
  • Mathematics, Computer Science
    Math. 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

Sampling Piecewise Sinusoidal Signals With Finite Rate of Innovation Methods

It is shown that under certain hypotheses on the sampling kernel, it is possible to perfectly recover the parameters that define the piecewise sinusoidal signal from its sampled version and to recover piecewise sine waves with arbitrarily high frequencies and arbitrarily close switching points.

Stability and super-resolution of generalized spike recovery

  • D. Batenkov
  • Computer Science
    Applied and Computational Harmonic Analysis
  • 2018

Signal Acquisition from Measurements via Non-Linear Models

The aim of this paper is to review some recent results in this direction, stressing the algebraic structure of the arising systems and mathematical tools required for their solutions, and to provide a solution method for a wide class of reconstruction problems.

Generalized Functions And Calculus Operators Of Mathematica Applied To Evaluation Of Influence Lines And Envelopes Of Statically Indeterminate Beams

Abstract The paper presents an analytical method of finding functions of influence lines of statically indeterminate beams. There are presented solutions of a fourth order equation with a right hand



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