Localized nonlinear functional equations and two sampling problems in signal processing

AbstractLet 1 ≤ p ≤ ∞. In this paper, we consider solving a nonlinear functional equation f (x) = y,where x, y belong to ℓpand f has continuous bounded gradient in an inverse-closed subalgebra of ℬ (ℓ2), the Banach algebra of all bounded linear operators on the Hilbert space ℓ2. We introduce strict monotonicity property for functions f on Banach spaces ℓpso that the above nonlinear functional equation is solvable and the solution x depends continuously on the given data y in ℓp. We show that the Van-Cittert iteration converges in ℓpwith exponential rate and hence it could be used to locate the true solution of the above nonlinear functional equation. We apply the above theory to handle two problems in signal processing: nonlinear sampling termed with instantaneous companding and subsequently average sampling; and local identification of innovation positions and qualification of amplitudes of signals with finite rate of innovation.

[1]  Radu Balan,et al.  A Noncommutative Wiener Lemma and A Faithful Tracial State on Banach Algebras of Time-Frequency Shift Operators , 2005, math/0510178.

[2]  Yonina C. Eldar,et al.  Nonlinear and Nonideal Sampling: Theory and Methods , 2008, IEEE Transactions on Signal Processing.

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

[4]  K. Gröchenig,et al.  Wiener's lemma for twisted convolution and Gabor frames , 2003 .

[5]  Qiyu Sun,et al.  Local reconstruction for sampling in shift-invariant spaces , 2010, Adv. Comput. Math..

[6]  O. Norwood.  Density , 1993, International Society of Hair Restoration Surgery.

[7]  M. Zuhair Nashed,et al.  Sampling and Reconstruction of Signals in a Reproducing Kernel Subspace of $L^p({\Bbb R}^d)$ , 2009, ArXiv.

[8]  Qiyu Sun,et al.  Stability of Localized Operators , 2008, 0811.1612.

[9]  G. Minty.  Monotone (nonlinear) operators in Hilbert space , 1962 .

[10]  Nobukazu Teranishi,et al.  Photo response analysis in CCD image sensors with a VOD structure , 1995 .

[11]  A. Baskakov,et al.  Wiener's theorem and the asymptotic estimates of the elements of inverse matrices , 1990 .

[12]  Qiyu Sun,et al.  Wiener's lemma for infinite matrices with polynomial off-diagonal decay , 2005 .

[13]  Jeffrey S. Morris,et al.  Understanding the characteristics of mass spectrometry data through the use of simulation , 2005, Cancer informatics.

[14]  Thierry Blu,et al.  Sampling and exact reconstruction of bandlimited signals with additive shot noise , 2006, IEEE Transactions on Information Theory.

[15]  T. Inouye,et al.  Nonlinear amplitude compression in magnetic resonance imaging: quantization noise reduction and data memory saving , 1990, IEEE Aerospace and Electronic Systems Magazine.

[16]  N. Nikolski.  In search of the invisible spectrum , 1999 .

[17]  A. Aldroubi,et al.  SLANTED MATRICES, BANACH FRAMES, AND SAMPLING , 2007, 0705.4304.

[18]  M. Rieffel.  Leibniz seminorms for "Matrix algebras converge to the sphere" , 2007, 0707.3229.

[19]  N. Wiener.  A Note on Tauberian Theorems , 1932 .

[20]  Symmetry of Matrix Algebras and Symbolic Calculus for Infinite Matrices , 2022 .

[21]  Martin Vetterli,et al.  Sampling with finite rate of innovation: channel and timing estimation for UWB and GPS , 2003, IEEE International Conference on Communications, 2003. ICC '03..

[22]  I. Krishtal.  WIENER'S LEMMA: PICTURES AT AN EXHIBITION , 2011 .

[23]  Karlheinz Gröchenig,et al.  Norm‐controlled inversion in smooth Banach algebras, I , 2012, J. Lond. Math. Soc..

[24]  Qiyu Sun,et al.  Reconstructing Signals with Finite Rate of Innovation from Noisy Samples , 2009 .

[25]  Tosio Kato,et al.  Nonlinear semigroups and evolution equations , 1967 .

[26]  Predrag Pejovic,et al.  A new algorithm for simulation of power electronic systems using piecewise-linear device models , 1995 .

[27]  Thierry Blu,et al.  Sampling signals with finite rate of innovation , 2002, IEEE Trans. Signal Process..

[28]  Karlheinz Gröchenig,et al.  Convergence Analysis of the Finite Section Method and Banach Algebras of Matrices , 2010 .

[29]  Pier Luigi Dragotti,et al.  Sampling Schemes for Multidimensional Signals With Finite Rate of Innovation , 2007, IEEE Transactions on Signal Processing.

[30]  Karlheinz Gröchenig,et al.  Wiener’s Lemma: Theme and Variations. An Introduction to Spectral Invariance and Its Applications , 2010 .

[31]  R. Balan,et al.  Density, overcompleteness, and localization of frames , 2006 .

[32]  Nader Motee,et al.  Optimal Control of Spatially Distributed Systems , 2008, 2007 American Control Conference.

[33]  Qiyu Sun.  Wiener’s Lemma for Infinite Matrices II , 2007, 1001.1457.

[34]  Thomas Strohmer,et al.  The finite section method and problems in frame theory , 2005, J. Approx. Theory.

[35]  Karlheinz Gröchenig,et al.  Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices , 2006 .

[36]  Hany Farid,et al.  Blind inverse gamma correction , 2001, IEEE Trans. Image Process..

[37]  Karlheinz Gröchenig,et al.  Time-Frequency Analysis of Sjöstrand's Class , 2004 .

[38]  G. Petersen.  A Tauberian theorem , 1962 .

[39]  E. Kissin,et al.  Differential properties of some dense subalgebras of C*-algebras , 1994, Proceedings of the Edinburgh Mathematical Society.

[40]  R. Balan,et al.  Density, Overcompleteness, and Localization of Frames. I. Theory , 2005, math/0510360.

[41]  I. W. Sandberg,et al.  Notes on PQ theorems , 1994 .

[42]  W. L. Miranker,et al.  The recovery of distorted band-limited signals , 1961 .

[43]  Qiyu Sun,et al.  WIENER’S LEMMA FOR INFINITE MATRICES , 2007 .

[44]  George G. Lorentz,et al.  Constructive Approximation , 1993, Grundlehren der mathematischen Wissenschaften.

[45]  Stéphane Jaffard.  Propriétés des matrices « bien localisées » près de leur diagonale et quelques applications , 1990 .

[46]  Karlheinz Gröchenig,et al.  Noncommutative Approximation: Inverse-Closed Subalgebras and Off-Diagonal Decay of Matrices , 2009, 0904.0386.

[47]  David J. Sakrison,et al.  The effects of a visual fidelity criterion of the encoding of images , 1974, IEEE Trans. Inf. Theory.

[48]  Moe Z. Win,et al.  Evaluation of an ultra-wide-band propagation channel , 2002 .

[49]  Alan C. Bovik,et al.  Mean squared error: Love it or leave it? A new look at Signal Fidelity Measures , 2009, IEEE Signal Processing Magazine.

[50]  Yonina C. Eldar,et al.  Xampling at the Rate of Innovation , 2011, IEEE Transactions on Signal Processing.

[51]  K. Grōchenig,et al.  Banach algebras of pseudodifferential operators and their almost diagonalization , 2007, 0710.1989.

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

[53]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[54]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[55]  Qiyu Sun,et al.  Frames in spaces with finite rate of innovation , 2008, Adv. Comput. Math..

[56]  Akram Aldroubi,et al.  Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces , 2001, SIAM Rev..

[57]  Qiyu Sun,et al.  Nonuniform Average Sampling and Reconstruction of Signals with Finite Rate of Innovation , 2006, SIAM J. Math. Anal..

[58]  P. Hall,et al.  Innovated Higher Criticism for Detecting Sparse Signals in Correlated Noise , 2009, 0902.3837.

[59]  Nader Motee,et al.  Distributed Multi-Parametric Quadratic Programming , 2009, IEEE Transactions on Automatic Control.

[60]  R. Balan.  The noncommutative Wiener lemma, linear independence, and spectral properties of the algebra of time-frequency shift operators , 2008 .

[61]  Kasso A. Okoudjou,et al.  Invertibility of the Gabor frame operator on the Wiener amalgam space , 2007, J. Approx. Theory.

[62]  Mubarak Shah,et al.  Estimation of the radiometric response functions of a color camera from differently illuminated images , 2004, 2004 International Conference on Image Processing, 2004. ICIP '04..

[63]  Minh N. Do,et al.  A Theory for Sampling Signals from a Union of Subspaces , 2022 .

[64]  Massimo Fornasier,et al.  Optimal adaptive computations in the Jaffard algebra and localized frames , 2010, J. Approx. Theory.

[65]  Martin Vetterli,et al.  Sampling and reconstruction of signals with finite rate of innovation in the presence of noise , 2005, IEEE Transactions on Signal Processing.

[66]  T. Strohmer,et al.  Pseudodifferential operators on locally compact abelian groups and Sjöstrand's symbol class , 2006, math/0604294.

[67]  Felix E. Browder,et al.  Nonlinear mappings of nonexpansive and accretive type in Banach spaces , 1967 .

[68]  M. Unser.  Sampling-50 years after Shannon , 2000, Proceedings of the IEEE.

[69]  Israel Gohberg,et al.  The band method for positive and strictly contractive extension problems: An alternative version and new applications , 1989 .

[70]  Yonina C. Eldar,et al.  Nonlinear and Nonideal Sampling Revisited , 2010, IEEE Signal Processing Letters.

[71]  A. Aldroubi,et al.  Convolution, Average Sampling, and a Calderon Resolution of the Identity for Shift-Invariant Spaces , 2005 .