The Beurling Theorem in Space–Time Algebras

In this work, the space–time Fourier transform SFT introduced by E. Hitzer satisfies some uncertainty principles of the algebra for space–time Cl(3,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ Cl_{(3,1)} $$\end{document}-valued signals over the space–time vector space R(3,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb {R}}^{(3,1)} $$\end{document}. An analog of the Beurling theorem for the SFT is obtained. As a direct consequence of Beurling’s theorem, other versions of the uncertainty principle, such as Hardy’s, Gelfand–Shilov’s, Cowling–Price’s and Morgan’s theorems are also deduced.


Introduction
Let f : R n → C be a square summable function in Lebesgue's sense over R n with n ≥ 1 be the dimension, that is f ∈ L 2 (R n ). Let us denote by ·, · , the scalar product and by · the Euclidean norm on R n . Thus, if f ∈ L 2 (R n ), then the Fourier transform of f is defined by The uncertainty principle is a collection of statements in harmonic analysis, each of them quantifying in some form the fundamental duality between a function f and its Fourier transform f , which prevents both representations from being "simultaneously concentrated in small sets." This principle has several quantitative versions which were proved by Hardy, Miyachi, Morgan, Gelfand-Shilov, and Cowling-Price (see [3,8,14] and the references therein).
Beurling's theorem, given by Beurling [1] and proved by Hörmander [12], says that for any non-trivial function f in L 2 (R), the function (x, y) → f (x) f (y) is never integrable on R 2 with respect to the measure e |xy| dxdy, where f is the Fourier transform of f given by (1) for n = 1. It is stated as follows: Theorem 1.1 [12] Let f ∈ L 2 (R) be such that Then f = 0 almost everywhere. This theorem is generalized by Bonami, Demange and Jaming [2] by giving solutions in terms of Hermite functions, asserting that Theorem 1.2 [2] Let f ∈ L 2 (R n ) and N ≥ 0 such that Then f (x) = P(x)e −<Ax,x> , where A is a real positive definite symmetric matrix and P is a polynomial of degree < (N −n) 2 .
In particular, f is identically 0 when N ≤ n. We may call this a master theorem as the theorems of Hardy, Morgan, Cowling-Price and theorem of Gelfand-Shilov can also be obtained from this generalized version of Beurling's theorem. The mutual dependencies of these uncertainty theorems can be schematically displayed as follows: Beurling's theorem  [17], on symmetric spaces of noncompact type [16], on Heisenberg groups and two step nilpotent Lie groups [15]. Recently, these theorems are also considered in the context of the Dunkl transform by Gallardo and Trimèche [6], Kawazoe and Mejjaoli [13].
We mention to the readers that in [4], the authors proved Beurling's theorem for the quaternion Fourier transform QFT in the space of square integral functions in the quaternion algebra H over R where It is generalized also in [18] in the Clifford algebras. To the best of our knowledge, these theorems for the SFT given by Hitzer [9] have not been derived yet. In our paper, a generalization of the Beurling theorem for the SFT is proved. As an easy consequence of Beurling's theorem, other versions of the uncertainty principle, such as the Hardy, Gelfand-Shilov, Cowling-Price and Morgan's theorem, are also deduced.

Space-Time Algebra in Minkowski Space-Time
In mathematical physics, it is well-known that the theory of special relativity was originally proposed by Albert Einstein, formerly student of the German mathematician Hermann Minkowski, is ideally studied in the context of a four-dimensional space, known as the Minkowski space-time R (3,1) .
We introduce in R (3,1) , the following orthonormal vector basis, The space-time algebra over R (3,1) , denoted by Cl (3,1) , is defined as an associative, non-commutative algebra which has a 16-dimensional graded basis: {1, e t , e 1 , e 2 , e 3 , e 12 , e 13 , e 23 , e t1 , e t2 , e t3 , i 3 , e t12 , e t13 , e t23 , i st }, where e 12 = e 1 e 2 , e t12 = e t e 1 e 2 and so on, with i 3 and i st are, respectively, the spatial unit volume tri-vector, and the pseudo-scalar given by: The associative geometric multiplication of the basis vectors satisfies: e k e l + e l e k = 2 k δ k,l , k, l ∈ {t, 1, 2, 3}, where δ k,l is the Kronecker symbol, and Obviously: Additionally, we get the following multiplication property ( Table 1). The general elements of space-time algebra Cl (3,1) are called multivectors. A multivector M ∈ Cl (3.1) can be represented grade wise as where A ⊂ {t, 1, 2, 3} and M A ∈ R. We thus can express every space-time algebra Cl (3,1) -valued function f in the form where f A : R (3,1) → R are real-valued functions and e A are the basis elements of Cl (3,1) given by (2). Also by (3), every multivector f ∈ Cl (3,1) can be written as follows Table 1 Involutions of space-time algebra Cl (3,1) denotes the grade k-vector part of f .
The principal reverse f of f is given by where f means to change in the basis decomposition of f the sign of every vector of negative square e A = α 1 e α 1 α 2 e α 2 . . . α k e α k , where e A = e α 1 e α 2 . . . e α k is a grade k basis element of Cl (3,1) . Note that the principal reverse is linear, involutive and anti-automorphic, this means (3,1) .
We introduce the complexified Clifford algebra C ⊗ Cl (3,1) , where C denotes the complex-valued numbers, and every multi-vector f ∈ C ⊗ Cl (3,1) can be written as and we add the rule i † C = −i C with i C ∈ C is the complex imaginary unit. The scalar part of a space-time multi vector f is defined as The following cyclic multiplication symmetry holds true Sc( f gh) = Sc(h f g) for all f , g, h ∈ Cl (3,1) .
For all f , g ∈ Cl (3,1) , the scalar product f * g is defined by If f = g, we obtain the modulus of a multivector f ∈ Cl (3,1) as It is not difficult to see that for f , g ∈ Cl (3,1) , one can obtain Moreover, the Cauchy-Schwarz inequality is true (see Appendix of [25]): (3,1) .
The ± split for space-time algebra (space-time split) was introduced by E. Hitzer as follows (see [9]): for all f ∈ Cl (3,1) . The above split leads immediately to We define the inner product ( f , g) and the scalar product f , g of two functions f , g : R (3,1) → Cl (3,1) , respectively, by which induces the following L 2 (R (3,1) , Cl (3,1) )-norm: For 1 ≤ a < ∞, the linear spaces L a (R (3,1) , Cl (3,1) ) are introduced as: L a (R (3,1) , Cl (3,1) ) := f : For a = ∞, the L ∞ -norm is defined by Let μ ∈ Cl (3,1) , with μ 2 = −1, we have a natural generalization of Euler's formula for space-time algebra, as follows (see [5,Lemma 3]): for all θ ∈ R. Now, we provide the definition of the SFT and we give its important properties (for more details see [11]). (3,1) , Cl (3,1) ). Then, the space-time Fourier transform of f is defined by where Note that we usually omit the upper indexes showing the special square roots of −1 selected for the transform, as in We also observe that the SFT can be seen as a special case of the general steerable two-sided Clifford Fourier transform [19,20]. Furthermore, we emphasize that the exponential kernel factors in (9) do not commute with the space-time function f , something that must always be considered in the establishment of the main properties of the SFT.
A change of variable gives the following Definition 2.3 [10] Let f , g ∈ L 2 (R (3,1) , Cl (3,1) ). The convolution of two space-time signals f , g is defined as provided that the integral exists.

Auxiliary Results
In the remainder of this paper, we denote by · the Euclidean norm in R 3 . In order to obtain our results, we will need some auxiliary results. and Proof Let x = te t + − → x ∈ R (3,1) and z = z t e t + − → z ∈ C ⊗ R (3,1) . Then, we have It follows from (8) that with Taking into account (17) and the following inequality, Similarly, we obtain It follows from (4) and (18) that This leads to (15), by virtue of the Cauchy-Schwarz's inequality We proceed in the same way as for the proof of (15), we have (16). (3,1) , R), then we have Proof Let ω ∈ R (3,1) , by the definition of the SFT, the relation (13) and Fubini's theorem, we have by virtue of the ± split for space-time algebra (6). It follows from (7) that Then this lemma is proved. (3,1) , R) and ϕ(x) = e − 1 2 |x| 2 , x ∈ R (3,1) . Then Proof Using Lemma 3.2, we have On the other hand, a direct computation gives: by virtue of i 2 3 = −1 and the well-known Gaussian Integral with Complex Offset: The result (20) is due to (21) and (22).

Lemma 3.4 Let x ∈ R. Then
where P n (x) is polynomial of degree n.
Proof After having made a series of derivations, we can see that where H n (x) are the probabilist's Hermite polynomials. The proof of this lemma is done by induction. (3,1) , Cl (3,1) ). Then we have Proof It follows from the definition of SFT that for all l ∈ {1, 2, 3}, we have by virtue of i 2 3 = −1 in the second equality. Therefore, we deduce by induction on α l ∈ N that By using the same lines as the proof of (26), we have for all α t ∈ N So, by (26) and (27), we find that We then obtain the desired results. (3,1) ) with deg Q = deg P. (3,1) . In 4 dimensions, a polynomial of degree k is given as follows:

Proof
The proof follows immediately from the previous lemma and (12).

Beurling's theorem for the SFT
In this section, we provide Beurling's theorem for the SFT.
Proof Let f ∈ L 2 (R (3,1) , Cl (3,1) ). It follows by (30) that Then, if the theorem is proved for f A , we obtain by (3) the result for f .
Moreover, for any z = a + i C b, a, b ∈ R (3,1) , we get It follows from (15), (16) and (34) that by virtue to (23), where C is a positive constant. We deduce that g is entire of order 2.
Third step: The function g admits an holomorphic extension to C ⊗ R (3,1) that is of order 2. Moreover, there exists a polynomial R such that for all z ∈ C ⊗ R (3,1) , Indeed, for all x ∈ R (3,1) and θ ∈ R, we have We should prove that g(z)g(i C z) with z ∈ C ⊗ R (3,1) is a polynomial. To show that, we define a new function G on C ⊗ R (3,1) by: G is entire of order 2, because g is. Since to prove our claim it is enough to show that G is a polynomial. For this, we use (36) and the Phragmèn-Lindelhöf principle [7], we find that G is a polynomial on C ⊗ R (3,1) . This proves that g(z)g(i C z) is a polynomial.
Fourth step: Following the proof of [2, Proposition 2.2, page 32-33], we find that g(x) = P(x)e Ax,x , where x ∈ R (3,1) and A a is symmetric matrix and P(x) is a polynomial. A direct computation shows that the form of the matrix A imposed by condition (32) is a diagonal and A = δ I . Then for some polynomial P and for some δ > 0. Also it follows from [2, Proposition 2.1] that the degree of the polynomial P satisfies deg P < (d − 4)/2. Therefore, By taking the SFT of both sides, we obtain by virtue of Lemma 3.7, where R is a polynomial of degree = deg P. That is Taking again the SFT, we get for some polynomial P of degree = deg P ≤ (d − 4)/2 and 0 < δ < 1/2.
Proof The proof of the corollary follows from the previous theorem, from the fact that | x, ω | ≤ |x||ω| and that Then the proof is complete

Conclusion
The main focus of this paper is to prove a new uncertainty principle for the spacetime Fourier transform SFT, namely Beurling's theorem. It shows that it is not possible for both a non-zero space-time signal and its SFT to decay very rapidly. The beauty of this theorem lies in the fact that many other theorems, such as Hardy's theorem, Gelfand-Shilov's theorem, Cowling-Price's theorem and Morgan's theorem, can be derived from it. In future studies, we will consider other uncertainty principles for the SFT, such as Miyachi's theorem.

Author Contributions
Equal contributions by all authors. All authors read and approved the final manuscript.
Funding None.

Availability of Data and Materials
No data were used to support this study.

Conflict of interest
The authors declare that they have no competing interests.