On the quaternionic short-time Fourier and Segal-Bargmann transforms

In this paper, we study a special one dimensional quaternion short-time Fourier transform (QSTFT). Its construction is based on the slice hyperholomorphic Segal-Bargmann transform. We discuss some basic properties and prove different results on the QSTFT such as Moyal formula, reconstruction formula and Lieb's uncertainty principle. We provide also the reproducing kernel associated to the Gabor space considered in this setting.

Recently there has been an increased interest in the generalization of integral transforms to the quaternionic and Clifford se ings. Such kind of transforms are widely studied, since they help in the analysis of vector-valued signals and images. In the survey [7] it is explained that some hypercomplex signals are useful tools for extracting intrinsically 1D-features from images. e reader can find other motivations for studying the extension of time frequency-analysis to quaternions in [7] and the references therein. In the survey [15] the author states that this research topic is based on three main approaches: the eigenfunction approach, the generalized roots of −1 approach and the spin group approach. Using the second one a quaternionic short-time Fourier transform in dimension 2 is studied in [5]. In the paper [16] the same transform is defined in a Clifford se ing for even dimension more than two. In this paper we introduce an extension of the short-time Fourier transform in a quaternionic se ing in dimension one. To this end, we fix a property that relates the complex short-time Fourier transform and the complex Segal-Bargmann transform: where V ϕ is the complex short-time Fourier transform with respect to the Gaussian window ϕ (see [21,Def. 3.1]) and Gf (z) denotes the complex version of the Segal-Bargmann transform according to [21]. To achieve our aim we use the quaternionc analogue of the Segal-Bargmann transform studied in [18]. is integral transform is used also in [17] to study some quaternionic Hilbert spaces of Cauchy-Fueter regular functions. In [10] and [25] the authors introduce some special modules of monogenic functions of Bargmann-type in Clifford analysis.
In order to present our results, we adopt the following structure: in section 2 we collect some basic definitions and preliminaries. In section 3, we prove some new properties of the quaternionic Segal-Bargmann transform. In particular we deal with an unitary property and give a characterization of the range of the Schwartz space. Moreover, we provide some calculations related to the position and the momentum operators.
In section 4, we give a brief overview of the 1D Fourier transform [19] and show a Plancherel theorem in this framework.
In section 5, we define the 1D QSTFT in the following way where B S H is the quaternionic Segal-Bargmann transform. Using some properties of B S H we prove an isometric relation for the 1D QSTFT and a Moyal formula. ese implies the following reconstruction formula From this follows that the adjoint operator defines a le inverse. Furthermore, it gives the possibility to write the 1D QSTFT using the reproducing kernel associated to the Gabor space Finally, we show that the 1D QSTFT follows a Lieb's uncertainty principle, some classical uncertainty principles for quaternionic linear operators in quaternionic Hilbert spaces were considered in [27].

P
In 2006 a new approach to quaternionic regular functions was introduced and then extensively studied in several directions, and it is nowadays widely developed [3,13,14,20]. is new theory contains polynomials and power series with quaternionic coefficients in the right, contrary to the Fueter theory of regular functions defined by means of the Cauchy-Riemann Fueter differential operator. e meeting point between the two function theories comes from an idea of Fueter in the thirties and next developed later by Sce [26] and by Qian [23]. is connection holds in any odd dimension (and in quaternionic case) and has been explained in [11] in the language of slice regular functions with values in the quaternions and slice monogenic functions with values in a Clifford algebra. e inverse map has been studied in [12] and still holds in any odd dimension. Moreover, the theory of slice regular functions have several applications in operator theory and in Mathematical Physics. e spectral theory of the S-spectrum is a natural tool for the formulation of quaternionic quantum mechanics and for the study of new classes of fractional diffusion problems, see [8,9], and the references therein. To make the paper self-contained, we briefly revise here the basics of the slice regular functions. Let H denote the quaternion algebra with its standard basis be the unit sphere of imaginary units in H. Note that any q ∈ H \ R can be wri en in a unique way as q = x + Iy for some real numbers x and y > 0, and imaginary unit I ∈ S. For every given I ∈ S we define C I = R + RI. It is isomorphic to the complex plane C so that it can be considered as a complex plane in H passing through 0,1 and I. eir union is the whole space of quaternions A real differentiable function f : Ω → H, on a given domain Ω ⊂ H, is said to be a (le ) slice regular function if, for every I ∈ S, the restriction f I to C I , with variable q = x + Iy, is holomorphic on Ω I := Ω ∩ C I , that is, it has continuous partial derivatives with respect to x and y and the function ∂ I f : Ω I → H defined by vanishes identically on Ω I . e set of slice regular functions will be denoted by SR(Ω).
Characterization of slice regular functions on a ball B = B(0, R) centred at the origin is given in [20]. Namely we have on Ω I . e slice derivative of f is the function ∂ S f : Ω → H defined by ∂ I f on Ω I , for all I ∈ S.
In all the paper we will make use of the Hilbert space L 2 (R, dx) = L 2 (R, H), consisting of all the square integrable H-valued functions with respect to In [2] the authors introduce the slice hyperholomorphic quaternionic Fock space F 2,ν Slice (H), defined for a given I ∈ S to be where ν > 0, f I = f | C I and dλ I (p) = dxdy for p = x + yI. e right H-vector space F 2,ν Slice (H) is endowed with the inner product e associated norm is given by is quaternionic Hilbert space does not depend on the choice of the imaginary unit I. An associated Segal-Bargmann transform was studied in [18] by considering the kernel function obtained by means of generating function related to the normalized weighted Hermite functions where ψ ν k denote the normalized weighted Hermite functions: are the normalized quaternionic monomials which constitute an orthonormal basis of F 2,ν Slice (H). en, for any quaternionic valued function ϕ in L 2 (R, H) the slice hyperholomorphic Segal-Bargmann transform is defined by In particular, most of our calculations later will be with a fixed parameter even ν = 1 or ν = 2π.

F S B
In this section we prove some new properties of the quaternionic Segal-Bargmann transform. We start from an unitary property which is not found in literature in the following explicit form.
Proposition 3.1. Let f, g ∈ L 2 (R, H). en, we have Proof. Any f, g ∈ L 2 (R, H) can be expanded as On the other way, since We have by [18] B Using the same calculus we obtain

By pu ing together (3.3) and (3.4) we obtain
Finally, since (3.2) and (3.5) are equal we obtain the thesis. Range of the Schwartz space and some operators. We characterize the range of the Schwartz space under the Segal-Bargmann transform with parameter ν = 1 in the slice hyperholomorphic se ing of quaternions. We consider also some equivalence relations related to the position and momentum operators in this se ing. e quaternionic Schwartz space on the real line that we are considering in this framework is defined by For I ∈ S, the classical Schwartz space is given by Clearly, we have that . Moreover, we prove the following Proof. Let ψ ∈ S H (R). en, we can write where ϕ 1 and ϕ 2 are C I −valued functions. Note that for all α, β ∈ N we have In particular, this implies that ψ ∈ S H (R) if and only if ϕ 1 , ϕ 2 ∈ S C I (R). i.e, Proof. Let f ∈ SF (H), then by definition f = B S H ψ where ψ ∈ S H (R). Let I, J ∈ S, be such that I ⊥ J. us, Lemma 3.3 implies that . en, we take the restriction to the complex plane C I and get: where the complex Bargmann transform (see [6]) is given by In particular, we set f I := B S H (ψ), f 1 := B C I (ϕ 1 ) and f 2 := B C I (ϕ 2 ). en, we have f 1 , f 2 ∈ SF(C I ). us, by applying the classical result in complex analysis, see [24] we have f 1 (z) = ∞ n=0 a n z n and f 2 (z) = ∞ n=0 b n z n , ∀z ∈ C I . 7 Moreover, for all p > 0 the following conditions hold sup n∈N |a n |n p √ n! < ∞ and sup n∈N |b n |n p √ n! < ∞.
In particular, we have then f I (z) = ∞ n=0 a n z n + ( ∞ n=0 a n z n )J, ∀z ∈ C I . erefore, z n c n with c n = a n + b n J, for all z ∈ C I .
us, by taking the slice hyperholomorphic extension we get Moreover, note that c n = a n + b n J, n ∈ N. en, |c n | ≤ |a n | + |b n |, ∀n ∈ N. us, for all p > 0, we have Finally, we conclude that en, by definition of the quaternionic Segal-Bargmann kernel we can write 8 In this case, we can apply the Leibnitz rule with respect to the slice derivative and get However, using the series expansion of the exponential function and applying the slice derivative we know that erefore, we obtain eorem 3.6. Let ϕ ∈ D(X). en, we have Proof. Let ϕ ∈ D(X) and q ∈ H. en, we have erefore, using Lemma 3.5 we obtain As a quick consequence, we have On the other hand, we have also the following eorem 3.8. We denote by M q : ϕ −→ M q ϕ(q) = qϕ(q) the creation operator on F 2,1 Slice (H). en, we have

Proof. Let ϕ ∈ D(X) ∩ D(D). en, we have
However, note that for all (q, Finally, we just need to apply (B S H ) −1 to complete the proof.

1D F
In this section, we study the one dimensional quaternion Fourier transforms (QFT). Namely, we are considering here the 1D le sided QFT studied in chapter 3 of the book [19]. In order to have less problems with computations we add −2π to the exponential. for a given I ∈ S. Its inverse is defined by Let J ∈ S be such that J ⊥ I. We can split the signal ψ via symplectic decomposition into simplex and perplex parts with respect to I such that we have: where ψ 1 (t), ψ 2 (t) ∈ C I . e le sided 1D QFT of ψ becomes According to [19], most of the properties may be inherited from the classical complex case thanks to the equivalence between C I and the standard complex plane and the fact that QFT can be decomposed into a sum of complex subfield functions.

Now, we define two fundamental operators for time-frequency analysis.
Translation As in the classical case we have a commutative relation between the two operators.
Lemma 4.2. Let ψ be a function in L 2 (R, H) then we have Proof. It is just a ma er of computations From [19, Table 3.2] we have the following properties

From (4.2) and (4.3) follow easily that
en, we prove a version of the Plancherel theorem for 1D QFT. H). en, we have In particular, for any φ ∈ L 2 (R, H) we have Proof. Let φ, ψ ∈ L 2 (R, H). By inversion formula for the 1D QFT, see [19], we have φ(ω) = ∼ F I (F I (φ))(ω), ∀ω ∈ R. 11 us, direct computations using Fubini's theorem lead to As a direct consequence, we have for any φ ∈ L 2 (R, H) e following remark may be of interest in some other contexts.
Remark 4.4. e formal convolution of two given signals φ, ψ : R −→ H when it exists is defined by In particular, if the window function φ is real valued the 1D QFT satisfies the classical property F I (φ * ψ) = F I (φ)F I (ψ).

5.
F G e idea of the short-time Fourier transform is to obtain information about local properties of the signal f . In order to achieve this aim the signal f is restricted to an interval and a er its Fourier transform is evaluated. However, since a sharp cut-off can introduce artificial discontinuities and can create problems, it is usually chosen a smooth cut-off function ϕ called "window function". e aim of this section is to propose a quaternionic analogue of the short-time Fourier transform in dimension one with a Gaussian window function ϕ(t) = 2 1/4 e −πt 2 . For this, we consider the following formula [21,Prop. 3.4.1] where the variables (x, ω) ∈ R 2 have been converted into a complex vector z = x + iω, and Gf (z) is the complex version of the Segal-Bargmann transform according to [21]. erefore, we want to extend (5.1) to the quaternionic se ing.
To this end, we use the quaternionic analogue of the Segal-Bargmann transform [18] and the slicing representation of the quaternions q = x + Iω, where I ∈ S.
If the signal is complex we denote the short-time Fourier transform as V ϕ , while if the signal is H-valued we identify the short-time Fourier transform as V ϕ .
Definition 5.1. Let f : R → H be a function in L 2 (R, H). We define the 1D quaternion short time Fourier transform of f with respect to ϕ(t) = 2 1/4 e −πt 2 as where q = x + Iω and B S H (f )(q) is the quaternionic Segal-Bargmann transform defined in (2.2).
Using (2.2) with ν = 2π, we can write (5.2) in the following way From this formula we are able to put in relation the 1D quaternion short-time Fourier transform and the 1D quaternion Fourier transform defined in section 3.
Lemma 5.2. Let f be a function in L 2 (R, H) and ϕ(t) = 2 1/4 e −πt 2 , recalling the 1D quaternion Fourier transform we have Proof. By pu ing q = x + Iω in (5.3) we have Now, we prove a formula which relates the 1D quaternion Fourier transform and its signal through the 1D short-time Fourier transform.
Proposition 5.3. If ϕ is a Gaussian function ϕ(t) = 2 1/4 e −πt 2 and f ∈ L 2 (R, H) then Proof. Recalling the definition of modulation and of inner product on L 2 (R, H), by Lemma 5.2 we have

13
Using the Plancherel theorem for the 1D quaternion Fourier transform, the property (4.4) and the fact that F I (ϕ) = ϕ we have Finally, from (4.1) and (5.6) we get Proof. We use the slicing representation of the quaternions q = x + Iω and formula (5.2) to get Now, using the change of variable p =q √ 2 we have that dA(p) = 1 2 dω dx, hence by [18, m. 4.6] us, the 1D quaternionic short-time Fourier transform is an isometry from L 2 (R, H) into L 2 (R 2 , H).
Proposition 5.5 (Moyal formula). Let f, g be functions in L 2 (R, H). en we have Using the same change of variables as before p =q √ 2 and from (3.1) we obtain Slice (H) = 2 f, g L 2 (R,H) .
in (5.2) by [18,Lemma 4.4] we get Remark 5.7. From (5.4) we can prove (5.8) in another way. is proof may be of interest in some other contexts.
Let us assume f, g ∈ L 2 (R, H) and recall ϕ(t) = 2 1/4 e −πt 2 , by Lemma 5.2 and Plancherel theorem for the 1D quaternion Fourier transform we have Now, by Fubini's theorem and the fact that ϕ 2 2 = 1 we get If we put f = g in (5.9) we obtain (5.7).

5.2.
Inversion formula and adjoint of QSTFT. e 1D QSTFT with Gaussian window ϕ satisfies a reconstruction formula that we prove in the following.
Proof. For all y ∈ R, we set Let h ∈ L 2 (R, H). Fubini's theorem combined with Moyal formula for QSTFT leads to g, h L 2 (R,H) = R h(y)g(y)dy Hence, we have is ends the proof.
We note that the QSTFT admits a le side inverse that we can compute as follows eorem 5.9. Let ϕ denote the Gaussian window ϕ(t) = 2 1/4 e −πt 2 and let us consider the operator A ϕ : L 2 (R 2 , H) −→ L 2 (R, H) defined for any F ∈ L 2 (R 2 , H) by en, A ϕ is the adjoint of V ϕ . Moreover, the following identity holds H) and h ∈ L 2 (R, H). We use some calculations similar to the previous result and get H) .
In particular, this shows that H). From reconstruction formula we obtain (5.10).
Remark 5.10. We note that the identity V * ϕ V ϕ = 2Id provides another proof for the fact that QSTFT is an isometric operator and the adjoint V * ϕ defines a le inverse.

5.3.
e eigenfunctions of the 1D quaternion Fourier transform. rough the 1D QSTFT we can prove in another way that the eigenfunctions of the 1D quaternion Fourier transform are given by the Hermite functions.
Combining with (5.11) From (5.10) we know that V ϕ is injective, hence we have the thesis.

5.4.
Reproducing kernel property. e inversion formula gives us the possibility to write the 1D QSTFT using the reproducing kernel associated to the quaternion Gabor space, introduced in [1], with a Gaussian window that is defined by then K ϕ (ω, x; ω ′ , x ′ ) is the reproducing kernel i.e.
Using Fubini's theorem we have

5.5.
Lieb's uncertainty principle for QSTFT. e QSTFT follows the Lieb's uncertainty principle with some weak differences comparing to the classical complex case. Indeed, we first study the weak uncertainty principle which is the subject of this result eorem 5.13 (Weak uncertainty principle). Let f ∈ L 2 (R, H) be a unit vector (i.e ||f || = 1), U an open set of R 2 and ε ≥ 0 such that