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Mathematics of Computation

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Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type

Author: Karlheinz Gröchenig
Journal: Math. Comp. 68 (1999), 749-765
MSC (1991): Primary 30E05, 30E10, 42A10, 94A12
MathSciNet review: 1613711
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Abstract: In many applications one seeks to recover an entire function of exponential type from its non-uniformly spaced samples. Whereas the mathematical theory usually addresses the question of when such a function in $L^2(\mathbb {R})$ can be recovered, numerical methods operate with a finite-dimensional model. The numerical reconstruction or approximation of the original function amounts to the solution of a large linear system. We show that the solutions of a particularly efficient discrete model in which the data are fit by trigonometric polynomials converge to the solution of the original infinite-dimensional reconstruction problem. This legitimatizes the numerical computations and explains why the algorithms employed produce reasonable results. The main mathematical result is a new type of approximation theorem for entire functions of exponential type from a finite number of values. From another point of view our approach provides a new method for proving sampling theorems.

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Additional Information

Karlheinz Gröchenig
Affiliation: Department of Mathematics, The University of Connecticut, Storrs, CT. 06269-3009

Keywords: Entire functions of exponential type, irregular sampling, Toeplitz matrices, approximation by trigonometric polynomials
Received by editor(s): October 25, 1996
Additional Notes: This work was partially supported by NSF grant DMS-9306430.
Article copyright: © Copyright 1999 American Mathematical Society