AN "ALGEBRAIC" RECONSTRUCTION OF PIECEWISE-SMOOTH FUNCTIONS FROM INTEGRAL MEASUREMENTS

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 measurements (like moments, Fourier coefficients, Radon trans- form, etc.). Our results concern reconstruction (from the moments or Fourier coefficients) of signals in two specific classes: linear combinations of shifts of a given function, and "piecewise D-finite functions" which satisfy on each continuity interval a linear differential equation with polynomial coefficients. In each case the problem is reduced to a solution of a certain type of non-linear algebraic system of equations ("Prony-type system"). We recall some known methods for explicitly solving such systems in one variable, and provide extensions to some multi-dimensional cases. Finally, we investigate the local stability of solving the Prony-type systems.

[1]  Walter Galitschi,et al.  On inverses of Vandermonde and confluent Vandermonde matrices , 1962 .

[2]  Walter Gautschi,et al.  On inverses of Vandermonde and confluent Vandermonde matrices. II , 1963 .

[3]  I. P. Natanson.  Constructive function theory , 1964 .

[4]  Å. Björck,et al.  Algorithms for confluent Vandermonde systems , 1973 .

[5]  Dan Kalman,et al.  The Generalized Vandermonde Matrix , 1984 .

[6]  V. N. Sorokin,et al.  Rational Approximations and Orthogonality , 1991 .

[7]  Bhaskar D. Rao,et al.  Model based processing of signals: a state space approach , 1992, Proc. IEEE.

[8]  A. Aptekarev,et al.  Rational Approximations and Orthogonality, Translations of Mathematical Monographs, Volume 92, E. M. Nikishin and V. N. Sorokin, American Mathematical Society, 1991, viii + 221 pp , 1993 .

[9]  Knut S. Eckhoff.  Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions , 1995 .

[10]  M. Vetterli,et al.  Approximation and compression of piecewise smooth functions , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[11]  Thomas Hagstrom,et al.  Locating Discontinuities of a Bounded Function by the Partial Sums of Its Fourier Series , 1999, J. Sci. Comput..

[12]  P. Milanfar,et al.  Reconstructing planar domains from their moments , 2000 .

[13]  Anne Gelb,et al.  Detection of Edges in Spectral Data II. Nonlinear Enhancement , 2000, SIAM J. Numer. Anal..

[14]  Eitan Tadmor,et al.  High Resolution Methods for Time Dependent Problems with Piecewise Smooth Solutions , 2003, math/0305008.

[15]  Yu. I. Lyubich,et al.  The Sylvester-Ramanujan System of Equations and The Complex Power Moment Problem , 2004 .

[16]  Michael Elad,et al.  Shape from moments - an estimation theory perspective , 2004, IEEE Transactions on Signal Processing.

[17]  Martin Vetterli,et al.  Exact sampling results for some classes of parametric nonbandlimited 2-D signals , 2004, IEEE Transactions on Signal Processing.

[18]  E.J. Candes.  Compressive Sampling , 2022 .

[19]  Yosef Yomdin,et al.  Linear versus Non-Linear Acquisition of Step-Functions , 2007, ArXiv.

[20]  Thierry Blu,et al.  Extrapolation and Interpolation) , 2022 .

[21]  Yosef Yomdin,et al.  Signal Acquisition from Measurements via Non-Linear Models , 2008 .

[22]  Dmitry Batenkov,et al.  Moment inversion problem for piecewise D-finite functions , 2009, 0901.4665.

[23]  Dmitry Batenkov,et al.  Algebraic Fourier reconstruction of piecewise smooth functions , 2010, Math. Comput..