On the Range of the Radon Transform on Z n and the Related Volberg ’ s Uncertainty Principle

We characterize the image of exponential type functions under the discrete Radon transform R on the lattice Zn of the Euclidean space Rn (n ≥ 2). We also establish the generalization of Volberg’s uncertainty principle on Z, which is proved by means of this characterization. The techniques of which we make use essentially in this paper are those of the Diophantine integral geometry as well as the Fourier analysis.


Introduction
First of all, we recall briefly that the uncertainty principle states, roughly speaking, that a nonzero function and its Fourier transform cannot both be sharply localized, which can be interpreted topologically by the fact that they cannot have simultaneously their supports in a same too small compact (see the Heisenberg uncertainty principle in [1]).Considerable attention has been devoted to discovering different forms of the uncertainty principle on many settings such as certain types of Lie groups and homogeneous trees.Several versions of the uncertainty principle have been established by many authors in the last few decades.Among the contributions dealing with this important topic, let us quote principally [1][2][3][4].On the other hand, we note that the uncertainty principle is one of the major themes of the classical Fourier analysis as well as its neighboring parts of the mathematical analysis.
In this paper, we are interested in developing the study of the restriction of the Radon transform  of  ∈  1 (Z  ) to G (1) .By means of Theorem 1 stated below, considered here as the first main result, we succeeded in proving the second main result concerning the generalization of Volberg's UP on Z  (see Theorem 2).We precise that G (1) is the most important subset of the discrete Grassmannian G for our study of both fundamental results (see Sections 3 and 4).
The purpose of this paper is to study the characterization of the image of exponential type functions under , as well as the generalization of Volberg's UP on Z  .
Our work is motivated by the fact that the uncertainty principle for the discrete Radon transform  on Z  plays a fundamental role in the field of physics, especially in quantum mechanics.
Our paper is organized as follows.
In Section 2, we fix, once and for all, some notation and also give certain properties of the discrete Radon transform  on Z  , which will be useful in the sequel of this paper.Moreover, we recall Volberg's theorem on Z in the same section.
Section 3 deals with the characterization of the image of exponential type functions under , which is given by the following main theorem (see Theorem 4).
Theorem 1 (characterization of the image of exponential type functions under ).Let  be a positive function of  1 (Z  ).Then (i) the following two conditions are equivalent: where  > 1 is an absolute constant; (ii) the following two equivalences hold: where  > 0 is an absolute constant, where  > 0 is an absolute constant.
Section 4 is devoted to establishing the generalization of Volberg's UP on the lattice Z  (see Theorem 2 below).We make here use of the discrete Fourier transform F, which maps a function  ∈  1 (Z  ) to a function F on T  defined by where T  = R  /Z  is the -dimensional torus.
Theorem 2 (generalization of Volberg's UP on the lattice Z  ).

Notations and Preliminaries
In this section, we fix some notation which will be useful in the sequel of this paper and recall certain properties of the discrete Radon transform on Z  ( ≥ 2).We also introduce various functional spaces.For 1 ≤  < +∞, let   (Z  ) (resp.,  ∞ (Z  )) be the space of all complex-valued functions  defined on Z  such that ∑ ∈Z  |()|  < +∞ (resp., sup ∈Z  |()| < +∞).Let us denote by  0 (Z  ) the space of all complex-valued functions  defined on Z  such that () → 0 as ‖‖ → +∞, with for all  = ( 1 , . . .,   ) ∈ Z  .It is clear that, for 1 ≤  <  < +∞, we have the following inclusions: For 1 ≤  < +∞, we denote by ‖ ⋅ ‖  the discrete norm on the space   (Z  ) defined by We define the discrete Radon transform  on Z  as follows: for all (, ) ∈ P × Z and  ∈  1 (Z  ), where (, ) is the hyperplane in Z  defined by and () being the greatest common divisor of the integers  1 , . . .,   (see [5] for more details), and  denotes the usual inner product of  and  regarded as two vectors of the Euclidean space R  .The set () = { ∈ Z  | /‖‖ 2 ∈ Z} can be written as follows: Because, for all (, ) ∈ Z 2 such that  ̸ = , we have We note that Indeed, for  ∈ Z  \ {0}, we can take  = /() ∈ P, then /‖‖ 2 = () ∈ Z.Moreover, 0 ∈ () for all  ∈ P.
For a function  ∈  1 (Z  ), we define its discrete Fourier transform F on the -dimensional torus T  as follows: We define the discrete one-dimensional Fourier transform F 1 by where T is the one-dimensional torus.
At the end of this section, we recall Volberg's theorem on Z.

Characterization of the Image of Exponential Type Functions under 𝑅
In this section, we study the characterization of the image of exponential type functions under the discrete Radon transform  on Z  .More precisely, we state the following main theorem which will be proved after introducing some intermediate lemmas.
Theorem 4 (characterization of the image of exponential type functions under ).Let  be a positive function of  1 (Z  ).Then (i) the following two conditions are equivalent: where  > 1 is an absolute constant; (ii) the following two equivalences hold: where  > 0 is an absolute constant, where  > 0 is an absolute constant.
In order to avoid any too long proof of Theorem 4 and then prove it clearly, we need the following useful lemmas.Lemma 5. Let  > 0 and  be the function defined on Z  by: () = exp(−‖‖ 2 ), for all  ∈ Z  .Then there exists a constant  > 0 (which depends only on  and ) such that, for all (, ) ∈ P × Z, we have Proof.Let (, ) = ∑ ∈Z  ,=‖‖ 2 exp(( 2 −‖‖ 2 )).The left inequality of ( 21) is trivial since  > 0 implies clearly that  > 0. To show the right inequality of (21), it suffices to prove the inequality (, ) < , for all (, ) ∈ P × Z.For this, we distinguish two cases.
Lemma 7. Let  ∈  1 (Z  ) verifying the following condition: where  is a strictly positive absolute constant.Then  = 0.
We now return to the proof of Theorem 4.
We now give the proof of Theorem 10.
Proof of Theorem 10.We prove this theorem by two steps as follows.