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Identification problems in sparse sampling

Banff International Research Station for Mathematical Innovation and Discovery

We consider the interpolation of an $n$-variate exponential sum $$ F(x_1, \ldots, x_n) = \sum_{j=1}^t c_j e^{f_{j,1} x_1 + f_{j,2} x_2 + \cdots + f_{j,n} x_n}. $$ In the univariate case, $n=1$, there is an entire branch of algorithms, which can be traced back to Prony's method dated in the 18th century, devoted to the recovery of the $2t$ unknowns, $c_1, \ldots, c_t, f_{1}, \ldots, f_{t}$, in $$ F(x) = \sum_{j=1}^t c_j e^{f_{j} x}. $$ In the multivariate case, $n>1$, it remains an active research topic to identify and separate distinct multivariate parameters from results computed by a Prony-like method from samples along projections. On top of the above, if the $f_{j,k}$ are allowed to be complex, the evaluations of the imaginary parts of distinct $f_{j,k}$ can also collide. This aliasing phenomenon can occur in either the univariate or the multivariate case. Our method interpolates $F(x_1, \ldots, x_n)$ from $(n+1) \cdot t$ evaluations. Since the total number of parameters $c_j$ and $f_{j,k}$ is exactly $(n+1) \cdot t$, we interpolate $F(x_1, \ldots, x_n)$ from the minimum possible number of evaluations. The method can also be used to recover the correct frequencies from aliased results. Essentially, we offer a scheme that can be embedded in any Prony-like algorithm, such as the least squares Prony version, ESPRIT, the matrix pencil approach, etc., thus can be viewed as a new tool offering additional possibilities in exponential analysis. This is joint work with A. Cuyt (Antwerp)

Mathematics; Approximations and expansions; Commutative rings and algebras; Classical analysis

video/mp4

Author affiliation: University of Antwerp, Belgium

2017-06-11

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61912

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/