Journal
APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS
Volume 29, Issue 2, Pages 156-181Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.acha.2009.08.009
Keywords
Nonlinear approximation; Frequency estimation; Prony's method in several variables; AAK theory in several variables; Hankel operators; Systems of polynomial equations; Sparse representations
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Funding
- NSF CMG [DMS 0724644]
- Swedish Research Council [VR 61232-34]
- BP
- ConocoPhillips
- ExxonMobil
- ExxonMobil, StatoilHydro
- Total
- Geo-Mathematical Imaging Group at Purdue University
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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.
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