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

DOI:
https://doi.org/10.1090/S0025-5718-99-01029-7

MathSciNet review:
1613711

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

Email:
groch@math.uconn.edu

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