×

Characterizations of function spaces by the discrete Radon transform. (English) Zbl 1270.44001

The paper studies a discrete Radon transform acting on discrete hyperplanes in \(\mathbb{Z}^n\). Let \(\mathcal{P}\) denote the set of points \(a= (a_1,\ldots, a_n) \in \mathbb{Z}^n \setminus \{0\}\) such that \(\gcd (a_1,\ldots, a_n)=1\). For \(a \in \mathcal{P}\) and \(k \in \mathbb{Z}\), let \(H(a,k) = \{ x \in \mathbb{Z}^n : a \cdot x = k\}\), so that \(H(a,k)\) is a discrete hyperplane in \(\mathbb{Z}^n\). Let \(\mathcal{G}\) be the set of all discrete hyperplanes in \(\mathbb{Z}^n\), which can be parametrized as \(\mathcal{P} \times \mathbb{Z} / \{ \pm 1 \}\). The present paper then defines a discrete Radon transform \(R\), which maps functions on \(\mathbb{Z}^n\) to \(\mathcal{G}\), by \[ Rf( H(a,k)) = \sum_{m \in H(a,k)} f(m). \]
Previous work by the first author and A. Ihsane [Mediterr. J. Math. 5, No. 1, 77–99 (2008; Zbl 1182.44002)] proved a number of basic properties for \(R\), including the Strichartz-type inversion formula and an analogue of the Helgason support theorem. That work also showed that \(R\) is continuous as a linear mapping of the Schwartz spaces \(\mathcal{S}(\mathbb{Z}^n)\) into \(\mathcal{S} (\mathcal{G})\), where these must be appropriately defined in this discrete setting. The authors then ask whether the map \(R : \mathcal{S}(\mathbb{Z}^n) \rightarrow \mathcal{S} (\mathcal{G})\) is bijective; in fact, they show by an example that this is not the case.
The main focus of the present paper is to characterize the Radon transform image of the Schwartz space \(\mathcal{S}(\mathbb{Z}^n)\). Moreover, the authors define discrete Hardy spaces on \(\mathbb{Z}^n\) and \(\mathcal{G}\), and characterize their images under the discrete Radon transform \(R\).

MSC:

44A12 Radon transform
46F12 Integral transforms in distribution spaces

Citations:

Zbl 1182.44002
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1007/s00009-008-0137-2 · Zbl 1182.44002 · doi:10.1007/s00009-008-0137-2
[2] Folland G. B., Hardy Spaces on Homogeneous Groups (1982) · Zbl 0508.42025
[3] Helgason S., The Radon Transform (1980) · Zbl 0453.43011 · doi:10.1007/978-1-4899-6765-7_1
[4] Helgason, S. 1982. ”Ranges of the Radon transforms”. 63–70. Providence, RI: American Mathematical Society. Computed tomography (Cincinnati, OH, 1982), Proceedings of Symposia in Applied Mathematics, Vol. 27
[5] DOI: 10.1007/BF01458472 · Zbl 0554.46016 · doi:10.1007/BF01458472
[6] DOI: 10.1002/cpa.3160230311 · Zbl 0189.14803 · doi:10.1002/cpa.3160230311
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.