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

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**68**(1999), 749-765 Request permission

## 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.## References

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## Additional Information

**Karlheinz Gröchenig**- Affiliation: Department of Mathematics, The University of Connecticut, Storrs, CT. 06269-3009
- Email: groch@math.uconn.edu
- Received by editor(s): October 25, 1996
- Additional Notes: This work was partially supported by NSF grant DMS-9306430.
- © Copyright 1999 American Mathematical Society
- Journal: Math. Comp.
**68**(1999), 749-765 - MSC (1991): Primary 30E05, 30E10, 42A10, 94A12
- DOI: https://doi.org/10.1090/S0025-5718-99-01029-7
- MathSciNet review: 1613711