Abstract
For an \(L^{2}(\mathbb{R})\) function, the famous theorem by Paley and Wiener gives a beautiful relation between extensibility to an entire function of exponential type and the line support of its Fourier transform. However, there is a huge class of entire functions of exponential type which are not square integrable on an axis, but do have integrability properties on certain half lines. In this chapter we investigate such functions, their growth behavior, and their integrability properties in L p-norms. We show generalizations of a theorem of J. Korevaar and the Paley-Wiener theorem.
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Acknowledgements
This work was partially supported by the grant MEXT-CT-2004-013477, Acronym MAMEBIA, of the European Commission and by the DFG grant FO 792/2-1 awarded to Brigitte Forster.
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Forster, B., Semmler, G. (2015). Entire Functions in Generalized Bernstein Spaces and Their Growth Behavior. In: Pfander, G. (eds) Sampling Theory, a Renaissance. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-19749-4_8
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DOI: https://doi.org/10.1007/978-3-319-19749-4_8
Publisher Name: Birkhäuser, Cham
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