Abstract
In this paper, we extend the theory of optimal approximations of functions f:ℝ→ℝ in the L 1(ℝ)-metric by entire functions of prescribed exponential type (bandlimited functions). We solve this problem for the truncated and the odd Gaussians using explicit integral representations and properties of truncated theta functions obtained via the maximum principle for the heat operator. As applications, we recover most of the previously known examples in the literature and further extend the class of truncated and odd functions for which this extremal problem can be solved, by integration on the free parameter and the use of tempered distribution arguments. This is the counterpart of the work (Carneiro et al. in Trans. Am. Math. Soc., 2012), where the case of even functions is treated.
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Acknowledgements
The authors are thankful to Marian Bocea for helpful discussions regarding the maximum principle of the heat operator and to Jeffrey D. Vaaler for the discussions on the extremal problem. E. Carneiro acknowledges support from the Institute for Advanced Study via the National Science Foundation agreement No. DMS-0635607 and support from the CNPq-Brazil grants 473152/2011-8 and 302809/2011-2.
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Communicated by Doron S. Lubinsky.
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Carneiro, E., Littmann, F. Bandlimited Approximations to the Truncated Gaussian and Applications. Constr Approx 38, 19–57 (2013). https://doi.org/10.1007/s00365-012-9177-8
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DOI: https://doi.org/10.1007/s00365-012-9177-8
Keywords
- Truncated Gaussian
- Exponential type
- Extremal functions
- Onesided best approximation
- Tempered distributions