Abstract

We characterize the image of exponential type functions under the discrete Radon transform R on the lattice of the Euclidean space . We also establish the generalization of Volberg's uncertainty principle on , 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.

1. 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 [1–4]. 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.

We consider here the lattice of the Euclidean space (). For and , the linear Diophantine equation has an infinity of solutions in if and only if is an integral multiple of the greatest common divisor of the integers , where denotes the usual inner product of and regarded as two vectors of the Euclidean space (see [5] for more details). Therefore, for , the set of all its solutions in is infinite and forms a discrete hyperplane in . Let be the set consisting of all hyperplanes in , where . We note that plays a role as discrete Grassmannian and can be parametrized as (see [5, Section 2]). Moreover, can be written as the following disjoint union: where with , for all .

As an analogue of the Euclidean case, the discrete Radon transform on , which maps a function to a function on , is defined by for all , where is the space of all complex-valued functions defined on such that (see [5] for more details on this Radon transform).

In this paper, we are interested in developing the study of the restriction of the Radon transform of to . 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 (see Theorem 2). We precise that is the most important subset of the discrete Grassmannian 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 .

Our work is motivated by the fact that the uncertainty principle for the discrete Radon transform on 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 , which will be useful in the sequel of this paper. Moreover, we recall Volberg’s theorem on 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 . Then  the following two conditions are equivalent:(1), (2),  where is an absolute constant;  the following two equivalences hold:(3),  where is an absolute constant,(4),  where is an absolute constant.

Section 4 is devoted to establishing the generalization of Volberg’s on the lattice (see Theorem 2 below). We make here use of the discrete Fourier transform , which maps a function to a function on defined by where is the -dimensional torus.

For and , we denote by    the set (resp., the restriction of to ), where by putting and .

In this section, we consider the function defined on by and also (where ) and , respectively, given by for all , with , respectively, being the one-dimensional torus. Now, we state the following main theorem (see Theorem 10).

Theorem 2 (generalization of Volberg’s on the lattice ). Let satisfying the following three conditions(1), for all , where , are strictly positive absolute constants.(2)The function belongs to .(3). Then .

2. 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 . We also introduce various functional spaces. For , let be the space of all complex-valued functions defined on such that . Let us denote by the space of all complex-valued functions defined on such that as , with for all . It is clear that, for , we have the following inclusions:

For , we denote by the discrete norm on the space defined by

We define the discrete Radon transform on as follows: for all and , where is the hyperplane in defined by and being the greatest common divisor of the integers (see [5] for more details), and denotes the usual inner product of and regarded as two vectors of the Euclidean space .

The set can be written as follows: Because, for all such that , we have We note that Indeed, for , we can take , then . Moreover, for all .

For a function , we define its discrete Fourier transform on the -dimensional torus as follows:

We define the discrete one-dimensional Fourier transform by where is the one-dimensional torus.

For , the Fourier coefficients of are denoted by () and defined by

Let , with . The inversion formula for the discrete Radon transform is given by for all and (see [5, Theorem 4.1] and also [6]).

At the end of this section, we recall Volberg’s theorem on .

Theorem 3 (Volberg’s theorem, see [1]). Let and suppose that is a nontrivial function on such that as . Moreover, suppose that its Fourier transform is integrable on the one-dimensional torus . Then .

3. Characterization of the Image of Exponential Type Functions under

In this section, we study the characterization of the image of exponential type functions under the discrete Radon transform on . 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 . Then  the following two conditions are equivalent:(1), (2),  where is an absolute constant;   the following two equivalences hold:(3),  where is an absolute constant,(4),  where 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 and be the function defined on by: , for all . Then there exists a constant (which depends only on and ) such that, for all , we have

Proof. Let . The left inequality of (21) is trivial since implies clearly that . To show the right inequality of   (21), it suffices to prove the inequality , for all . For this, we distinguish two cases.
(1) The Case When . We have and ; then, by the Cauchy-Schwarz inequality, . It follows that ; thus since by hypothesis. The above inequality can be transformed as follows: which implies (21).
(2) The Case When , That Is, . Since , we can change the order of the coordinates of , then we can take , and thus , with . It follows that Let . From (23) and (24), we have for all . Consequently, where . Then by the above inequality, we obtain And Lemma 5 is proved.

Lemma 6. Let . We assume that there exists a function such that as , satisfying the following condition: Then is supported at the origin; that is, for all .

Proof. Let . Permuting the coordinates of , we can assume that . Applying hypothesis (27) to and , where , we obtain Since , this implies that as ; therefore, the right hand side of inequality (28) tends to zero as . By the inversion formula (see [5, Theorem 4.1]), the left hand side of the same inequality converges to as . Then . And Lemma 6 is proved.

We cannot hope to obtain more than this result: given a function on such that for and , we have if , and . Inequality (27) is verified for every function such that .

Lemma 7. Let verifying the following condition: where is a strictly positive absolute constant. Then .

Proof. By applying Lemma 6, we obtain , for all . Now, we show that . Condition (29) implies that , for all . Then, since we have with , we infer that . Thus, , for all . And this proves Lemma 7.

We now return to the proof of Theorem 4.

Proof of Theorem 4. The implication of Theorem 4 follows from Lemma 5. On the other hand, we deduce equivalence (resp., (4)) of Theorem 4 from Lemma 6 (resp., Lemma 7). Consequently, to complete the proof of Theorem 4, it remains to prove the implication of this theorem. For this, suppose that is satisfied and prove . Under our assumption, we have where is an absolute constant. Multiplying the members of (31) by , with and , (31) becomes but Then (32) can be transformed as follows: Since , there exists a constant (which does not depend on ) such that but For , equality (36) becomes where is the canonical orthonormal basis of . Moreover, (37) implies that the series is convergent; then there exists a constant (which does not depend on ) such that for all and such that . Therefore, for all and and thus which gives by summing with respect to Inequality (41) can be transformed as follows: Hence, Since , inequality (43) implies for all . It follows from inequality (44) that for all such that . Thus, for all such that , which proves the implication . And this completes the proof of Theorem 4.