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Nonlinear approximation of functions in two dimensions by sums of exponential functions

Andersson, Fredrik LU ; Carlsson, Marcus LU and de Hoop, Maarten V. (2010) In Applied and Computational Harmonic Analysis 29(2). p.156-181
Abstract
We consider the problem of approximating a given function in two dimensions by a sum of exponential functions, with complex-valued exponents and coefficients. In contrast to Fourier representations where the exponentials are fixed, we consider the nonlinear problem of choosing both the exponents and coefficients. In this way we obtain accurate approximations with only few terms. Our approach is built on recent work done by G. Beylkin and L Monzon in the one-dimensional case. We provide constructive methods for how to find the exponents and the coefficients, and provide error estimates. We also provide numerical simulations where the method produces sparse approximations with substantially fewer terms than what a Fourier representation... (More)
We consider the problem of approximating a given function in two dimensions by a sum of exponential functions, with complex-valued exponents and coefficients. In contrast to Fourier representations where the exponentials are fixed, we consider the nonlinear problem of choosing both the exponents and coefficients. In this way we obtain accurate approximations with only few terms. Our approach is built on recent work done by G. Beylkin and L Monzon in the one-dimensional case. We provide constructive methods for how to find the exponents and the coefficients, and provide error estimates. We also provide numerical simulations where the method produces sparse approximations with substantially fewer terms than what a Fourier representation produces for the same accuracy. (c) 2009 Elsevier Inc. All rights reserved. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Sparse representations, Systems of polynomial equations, Frequency estimation, Nonlinear approximation, Prony's method in, several variables, Hankel operators, AAK theory in several variables
in
Applied and Computational Harmonic Analysis
volume
29
issue
2
pages
156 - 181
publisher
Elsevier
external identifiers
  • wos:000278798300003
  • scopus:78549295512
ISSN
1096-603X
DOI
10.1016/j.acha.2009.08.009
language
English
LU publication?
yes
id
dffd71c8-9f52-4f21-846d-1a8fa0404640 (old id 1630775)
date added to LUP
2010-07-21 13:46:28
date last changed
2018-06-03 03:21:07
@article{dffd71c8-9f52-4f21-846d-1a8fa0404640,
  abstract     = {We consider the problem of approximating a given function in two dimensions by a sum of exponential functions, with complex-valued exponents and coefficients. In contrast to Fourier representations where the exponentials are fixed, we consider the nonlinear problem of choosing both the exponents and coefficients. In this way we obtain accurate approximations with only few terms. Our approach is built on recent work done by G. Beylkin and L Monzon in the one-dimensional case. We provide constructive methods for how to find the exponents and the coefficients, and provide error estimates. We also provide numerical simulations where the method produces sparse approximations with substantially fewer terms than what a Fourier representation produces for the same accuracy. (c) 2009 Elsevier Inc. All rights reserved.},
  author       = {Andersson, Fredrik and Carlsson, Marcus and de Hoop, Maarten V.},
  issn         = {1096-603X},
  keyword      = {Sparse representations,Systems of polynomial equations,Frequency estimation,Nonlinear approximation,Prony's method in,several variables,Hankel operators,AAK theory in several variables},
  language     = {eng},
  number       = {2},
  pages        = {156--181},
  publisher    = {Elsevier},
  series       = {Applied and Computational Harmonic Analysis},
  title        = {Nonlinear approximation of functions in two dimensions by sums of exponential functions},
  url          = {http://dx.doi.org/10.1016/j.acha.2009.08.009},
  volume       = {29},
  year         = {2010},
}