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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type
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by Karlheinz Gröchenig PDF
Math. Comp. 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.
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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