Helgason Gabor Fourier transform and uncertainty principles
aa r X i v : . [ m a t h . C A ] D ec HELGASON-GABOR FOURIER TRANSFORM AND UNCERTAINTYPRINCIPLES
M. EL KASSIMI, M.BOUJEDDAINE AND S. FAHLAOUI
Abstract.
Windowing a Fourier transform is a useful tool, which gives us the similaritybetween the signal and time frequency signal, and it allows to get sense when/where ceratinfrequencies occur in the input signal, this method is introduced by Dennis Gabor. In thispaper, we generalize the classical Gabor-Fourier transform(GFT) to the Riemannian symmet-ric space called the Helgason Gabor Fourier transform (HGFT). We continue with provingseveral important properties of HGFT, like the reconstruction formula, the Plancherel for-mula, and Parseval formula. Finally we establish some local uncertainty principle such asBenedicks-type uncertainty principle.
Introduction
The Fourier transform has been a useful tool for analyzing frequency properties of a signal,but this transform still insufficient to represent and compute location information for a givensignal. To solve this problem, in [5] Gabor formulated a fundamental method by multiplyingthe function to be transformed by a Gaussian function. This transform becomes a powerfulmethod for determining the sinusoidal frequency and phase content of signal local sectionsconsidering its changes over time. In addition, it used for filtering and modifying the signalin the limited region.In classical case, the Gabor transform is given by ([5]) G{ f } ( b, ω ) = Z + ∞−∞ f ( t ) e − π ( t − τ ) e − iωt dt In the general case, we take the windowed function ϕ as square integral function. The GaborFourier transform has other names used in the literature, like as short-time Fourier transformand windowed Fourier transform. Motivated by this concept, in this paper we study a gen-eralization of the classical Gabor Fourier transform to the Riemannian Symmetric spaces see[9, 11], which we call the Helgason Gabor Fourier transform(HGFT).Then, we derive impor-tant harmonic analysis properties of HGFT. This paper is organized as follows, in the first section we remind some results about the clas-sical Helgason-Fourier transform, in the second we define the HGFT, and we establish for it aseveral harmonic analysis properties, such as the inversion formula, Plancherel and Parsevalformulas, in the last one we demonstrate some local uncertainty principles for HGFT likeBenedick’s theorem.
In this section we describe the necessary preliminaries regardingsemi-simple Lie groups and harmonic analysis on associated Riemannian symmetric spaces.If X is a Riemannian symmetric space of noncompact type then X can be viewed as aquotient space G/K where G is a connected, noncompact, semi-simple Lie group with finitecenter and K a maximal compact subgroup of G .Let G = N AK be an Iwasawa decomposition of G and let a be the Lie algebra of A . Denot-ing by M the centralizer of A in K and putting B = K/M . By writing g = n.expA ( g ) .u , g ∈ G ,where u ∈ K, A ( g ) ∈ a , n ∈ N , and for x = gK ∈ X and b = kM ∈ B = K/M , we write A ( x, b ) = A ( k − g ). Let dx be a G -invariant measure on X , and let db and dk be the respectivenormed K -invariant measures on B and K .Let o = K be the origin in X and denote the action of G on X by ( g, x ) gx for g ∈ G, x ∈ X . The Lie algebras of G and K are respectively denoted by g and k .We denote by C ∞ c ( X ) the set of infinity differentiable compactly-supported functions on X .Let dg be the element of the Haar measure on G.We assume that the Haar measure on G is normed, so that Z X f ( x )d x = Z G f ( go )d g, f ∈ C ∞ c ( X ) (1.1)Let a ∗ be the real dual of a and a ∗ C be its complexification ; the finite Weyl group W actson a ∗ . Suppose that Σ is the set of bounded roots (Σ ⊂ a ∗ ), Σ + is the set of positive boundedroots, and a + is the positive Weyl chamber so that a + = { h ∈ a : α ( h ) > α ∈ Σ + } . Denote by ρ the half-sum of the positive bounded roots (counted with their multiplicities) ;then ρ ∈ a ∗ . Let h , i be the Killing form on the Lie algebra a . For λ ∈ a ∗ , let A λ be thevector in a such that λ ( A ) = h A λ , A i for all A ∈ a . Given λ, µ ∈ a ∗ , we set h λ, µ i := h A λ , A µ i . ELGASON-GABOR FOURIER TRANSFORM AND UNCERTAINTY PRINCIPLES 3
The correspondence λ A λ enables us to identify a ∗ with a . Using this identification, wecan translate the action of the Weyl group W to a . Let a ∗ + = { λ ∈ a ∗ : A λ ∈ a ∗ } The Helgason Fourier transform is a powerful tool in harmonic analysis on noncompact Rie-mannian symmetric spaces
G/K ([11]). This transform associates to any smooth compactlysupported right K -invariant function f on G .For integrable functions f on C ∞ c ( X ) , b ∈ K and λ ∈ a ∗ , Helgason-Fourier transform isdefined as in ([9])by: b f ( λ, b ) = Z X f ( x ) e ( − iλ + ρ )( A ( x,b )) d x, λ ∈ a ∗ , b ∈ B = K/M (1.2)We will assume throughout this paper X is of rank 1, and hence dim a ∗ = 1. In this casewe identify a ∗ C with C by identifying λ ρ with λ ∈ C . Under this identification, a ∗ = R bymeans of the correspondence λ λα, λ ∈ R .We norm the measure on X and we conclude this section with the following properties, dueto Helgason.The original function f ∈ C ∞ c ( X ) can then be reconstructed from b f by means of theinversion formula f ( x ) = 1 | W | Z a ∗ × B b f ( λ, b ) e ( iλ + ρ )( A ( x,b )) | c ( λ ) | − d λ d b, (1.3)where | W | is the order of the Weyl group of G/K , dλ is the element of the Euclidean measureon a ∗ and c ( λ ) is the Harish-Chandra function.We also state the Plancherel formula for the Fourier transform: Theorem 1.1.
The Fourier transform defined on C c ( X ) by (1) extends to an isometry of L ( X ) onto L ( a ∗ × B ) (with the measure | c ( λ ) | − d λ d b on a ∗ × B ). Moreover, Z X f ( x ) f ( x ) d x = 1 | W | Z a ∗ × B c f ( λ, b ) c f ( λ, b ) e ( iλ + ρ )( A ( x,b )) | c ( λ ) | − d λ d b, (1.4) for all f , f ∈ L ( X ) . Proof.
See [10, Theorem 2, page 227].It follows from the above arguments that for f ∈ L ( X ), we have M.EL KASSIMI M.BOUJEDDAINE AND S.FAHLAOUI Z X | f ( x ) | d x = 1 | W | Z a ∗ × B | b f ( λ, b ) | d µ ( λ )d b = Z a ∗ + × B | b f ( λ, b ) | d µ ( λ )d b, (1.5)where d µ ( λ ) := | c ( λ ) | − d λ .Given h ∈ G . For a function f ∈ C ( X ), the translation operator T h is given by the formula( T h f )( x ) := Z K f ( gkho )d k. We remind that a function ϕ ∈ C ( X ) is called a spherical function if ϕ is K -invariant, ϕ ( o ) = 1, and for each D ∈ C ∞ c ( X ), there exists λ D ∈ C such that Dϕ = λ D ϕ .We now list down some well known properties of the elementary spherical functions on X based on the Harish-Chandras result [9, Chapitre 4, Theorem 4.3].First, we give the following lemma proved in [9, Lemma 3]. Lemma 1.2.
For f ∈ L ( X ) , we have \ ( T h ( f ))( λ, b ) = ϕ λ ( h ) . b f ( λ, b ) , h ∈ G where b f ( λ, b ) is the Fourier transform of f The classical Gabor transform of a function f ∈ L ( R ) cannot possess a support of finiteLebesgue measure. In [15] the author showed that the portion of this transform lying outsidesome set M of finite Lebesgue measure cannot be arbitrarily small, either. For sufficientlysmall M , this can be seen immediately by estimating the Hilbert-Schmidt norm of a suitablydefined operator. In this section, we try to give some new harmonic analysis results relatedto Gabor transform in the case of Riemannian symmetric space X .We define first the Gabor-Helgason transform by: G ϕ { f } ( λ, b, h ) = Z X f ( x ) T h − ϕ ( x ) e ( − iλ + ρ )( A ( x,b )) dx with λ ∈ a ∗ , b ∈ B = K/M
ELGASON-GABOR FOURIER TRANSFORM AND UNCERTAINTY PRINCIPLES 5
Before to give the reconstruction formula for the HGFT, we needthe following lemma, which proves that, the translation T h is an isometric operator for thenorm of the space L . Lemma 2.1.
For every function f ∈ L ( X ) and h ∈ G : We have k T h f k L ( X ) = k f k L ( X ) Proof.
Applying the relation (1.1), we get, k T h f k L ( X ) = Z X | T h f ( x ) | dx = Z G | T h f ( go ) | dg = Z G T h f ( go ) T h f ( go ) dg = Z G Z K f ( gkho ) dk Z K f ( gk ′ ho ) dk ′ dg = Z K Z K ( Z G f ( gkho ) f ( gk ′ ho ) dg ) dkdk ′ = Z K Z K ( Z G f ( go ) f ( go ) dg ) dkdk ′ = Z G | f ( go ) | dg = k f k L ( X ) (cid:3) Theorem 2.2.
Let ϕ ∈ L ( X ) be a window function. Then every function f ∈ L ( X ) , canbe reconstructed by f ( x ) = 1 | W |k ϕ k Z X Z a ∗ × B G ϕ ( λ, b, h ) e ( iλ + ρ )( A ( x,b )) T h − ϕ ( x ) | c ( λ ) | − dλdbdh Proof.
We can obtain the inversion formula by using the fact that: G ϕ { f } ( λ, b, h ) = ( \ f T h − ϕ )( λ, b )So, f ( x ) T h − ϕ ( x ) = 1 | W | Z a ∗ × B G ϕ { f } ( λ, b, h ) e ( iλ + ρ )( A ( x,b )) | c ( λ ) | − dλdb, (2.1)We multiply the both sides of (2.1) by T h − ϕ , we obtain f ( x ) | T h − ϕ ( x ) | = 1 | W | Z a ∗ × B G ϕ { f } ( λ, b, h ) e ( iλ + ρ )( A ( x,b )) | c ( λ ) | − dλdbT h − ϕ ( x ) (2.2) M.EL KASSIMI M.BOUJEDDAINE AND S.FAHLAOUI on integer the inequality (2.2) with respect the measure dh , w get f ( x ) Z G | T h − ϕ ( x ) | dh = 1 | W | Z G Z a ∗ × B G ϕ { f } ( λ, b, h ) e ( iλ + ρ )( A ( x,b )) | c ( λ ) | − T h − ϕ ( x ) dλdbdh by using the first lemma 2.1, we obtain, f ( x ) k ϕ k L ( X ) = 1 | W | Z G Z a ∗ × B G ϕ { f } ( λ, b, h ) e ( iλ + ρ )( A ( x,b )) T h − ϕ ( x ) | c ( λ ) | − dλdbdh (2.3)Now, simplifying both sides of 2.3 by k ϕ k L ( X ) , we get our result. f ( x ) = 1 | W |k ϕ k L ( X ) Z G Z a ∗ × B G ϕ { f } ( λ, b, h ) e ( iλ + ρ )( A ( x,b )) T h − ϕ ( x ) | c ( λ ) | − dλdbdh (cid:3) Theorem 2.3 (Plancherel formula) . For f ∈ L ( X ) and ϕ ∈ L ( X ) a windowed function,we have kG ϕ { f }k L ( a ∗ × B × G ) = k f k L ( X ) k ϕ k L ( X ) Proof.
We have, kG ϕ { f }k L ( a ∗ × B × G ) = k \ f T h − ϕ k L ( a ∗ × B × G ) = k f T h − ϕ k L ( X × G ) = Z G Z X f ( x ) T h − ϕ ( x ) f ( x ) T h − ϕ ( x ) dxdh = Z G Z X f ( x ) T h − ϕ ( x ) T h − ϕ ( x ) f ( x ) dxdh = Z G Z X | f ( x ) | | T h − ϕ ( x ) | dxdh using the equation (1.1), lemma 2.1 and the Fubini’s theorem, we have,= Z G Z G | f ( go ) | Z K ϕ ( gkh − o ) dk Z K ϕ ( gk ′ h − o ) dk ′ dhdg = Z G | f ( go ) | Z G Z K Z K ϕ ( gkh − o ) ϕ ( gk ′ h − o ) dkdk ′ dhdg using the invariance of the Haar measure dg by K, we get= Z G | f ( go ) | Z G ϕ ( h − o ) Z K Z K ϕ ( h − o ) dkdk ′ dhdg = Z G | f ( go ) | Z K dk Z K dk ′ Z G | ϕ ( h − o ) | dhdg = Z X | f ( x ) | dx Z X | ϕ ( y ) | dy = k f k L ( X ) k ϕ k L ( X )ELGASON-GABOR FOURIER TRANSFORM AND UNCERTAINTY PRINCIPLES 7 (cid:3) Theorem 2.4 (Parseval’s identity ) . Let ϕ ∈ L ( X ) be a window function and f, g ∈ L ( X ) arbitrary. Then we have Z G × a ∗ × B G ϕ { f } ( λ, b, h ) G ϕ g ( λ, b, h ) | c ( λ ) | − dλdbdh = k ϕ k L ( X ) Z X f ( x ) g ( x ) dx Proof. we have by the lemma 2.1 Z G × a ∗ × B G ϕ { f } ( λ, b, h ) G ϕ g ( λ, b, h ) | c ( λ ) | − dλdbdh = Z G × a ∗ × B \ ( f ( . ) T h − ϕ ( . ))( λ, b, h ) \ ( g ( . ) T h − ϕ ( . ))( λ, b, h ) | c ( λ ) | − dλdbdh = Z G Z X f ( x ) T h − ϕ ( x ) g ( x ) T h − ϕ ( x ) dxdh = Z G Z X f ( x ) T h − ϕ ( x ) T h − ϕ ( x ) g ( x ) dxdh = Z G Z X f ( x ) g ( x ) | T h − ϕ ( x ) | dxdh = Z X f ( x ) g ( x ) dx Z G | T h − ϕ ( y ) | dh Such as the proof of the Plancherel’s theorem 2.3, Applying the equation (1.1), lemma 2.1and the Fubini’s theorem, we get,= Z G Z G f ( go ) g ( go ) Z K ϕ ( gkh − o ) dk Z K ϕ ( gk ′ h − o ) dk ′ dhdg = Z G f ( go ) g ( go ) Z G Z K Z K ϕ ( gkh − o ) ϕ ( gk ′ h − o ) dkdk ′ dhdg using the invariance of the Haar measure dg by K, we obtain= Z G f ( go ) g ( go ) Z G ϕ ( h − o ) Z K Z K ϕ ( h − o ) dkdk ′ dhdg = Z G f ( go ) g ( go ) Z K dk Z K dk ′ Z G | ϕ ( h − o ) | dhdg = Z X f ( x ) g ( x ) dx Z X | ϕ ( y ) | dy = k ϕ k L ( X ) Z X f ( x ) g ( x ) dx (cid:3) M.EL KASSIMI M.BOUJEDDAINE AND S.FAHLAOUI
We shall now discuss the validity of some uncertainty principles in the case of Gabor-Helgason transform.
In quantum physics, the uncertainty principles state that, we cannot give simultaneouslythe position and moment time of particle with high precision. The formulation mathematics ofthis concept is that, the function and its Fourier transform cannot both be sharply localized.Many formulations are given, the first one is proved by Heinseberg in 1927 [8], after, manyauthors give some generations, such as, Hardy’s theorem [7], Morgan’s theorem [12]. Yearsafter, the locally uncertainty principles arise, those theorems asset that, when the uncertaintyof the momentum is small, the probability of being localized at any point is very small [2, 3, 14].Our first result will be the following local uncertainty principle,
Lemma 3.1.
Let ϕ, f ∈ L ( X ) , we have kG ϕ { f }{ f } ( ω, y ) k L ∞ ( a ∗ × B × X ) ≤ k f k L ( X ) k ϕ k L ( X ) (3.1) Proof.
We have |G ϕ { f }{ f } ( λ, b, h )) | = | Z X f ( x ) T h − ϕ ( x ) e ( − iλ + ρ )( A ( x,b )) dx |≤ Z X | f ( x ) || T h − ϕ ( x ) | dx Using Hölder inequality we get our result kG ϕ { f }{ f } ( λ, b, h ) k L ∞ ( a ∗ × B × X ) ≤ k f k L ( X ) k ϕ k L ( X ) (cid:3) Theorem 3.2.
Let ϕ a windowed function and let Σ a subset of a ∗ × B × X such that < m (Σ) < + ∞ , for all f ∈ L ( X ) we have, k f k L ( X ) k ϕ k L ( X ) ≤ p − m (Σ) kG ϕ { f } χ Σ c k L ( a ∗ × B × X ) (3.2) Proof.
For every f ∈ L ( X ); we have kG ϕ { f }k L ( X ) = kG ϕ { f } χ Σ k L ( a ∗ × B × X ) + kG ϕ { f } χ Σ c k L ( a ∗ × B × X ) (3.3) ELGASON-GABOR FOURIER TRANSFORM AND UNCERTAINTY PRINCIPLES 9
Applying the (3.1) and the Plancherel formula 2.3, we get kG ϕ { f } χ Σ k L ( a ∗ × B × X ) ≤ m (Σ) kG ϕ { f }k L ∞ ( a ∗ × B × X ) (3.4) ≤ m (Σ) k ϕ k L ( X ) k f k L ( X ) , (3.5)Thus, by the equation (3.3) kG ϕ { f } χ Σ k L ( a ∗ × B × X ) ≥ (1 − m (Σ) ) k ϕ k L ( X ) k f k L ( X ) k f k L ( X ) k ϕ k L ( X ) ≤ p − m (Σ) kG ϕ { f }{ f } χ Σ c k L ( a ∗ × B × X ) (cid:3) Theorem 3.3 (Concentration of G ϕ { f } in small sets ) . Let ϕ be a window function and Σ ⊂ a ∗ × B × X with m (Σ) < .Then, for f ∈ L ( X ) we have kG ϕ { f } − χ Σ G ϕ { f }k L ( a ∗ × B × X ) ≥ k ϕ k L ( X ) k f k L ( X ) (1 − m (Σ) k L ( X ) (3.6) Proof.
We have kG ϕ { f } − χ Σ G ϕ { f }k L ( a ∗ × B × X ) = kG ϕ { f } (1 − χ Σ ) k L ( a ∗ × B × X ) From theorem 2.3 kG ϕ { f } − χ Σ G ϕ { f }k L ( a ∗ × B × X ) ≥ kG ϕ { f }k L ( a ∗ × B × X ) (1 − m (Σ)) ≥ k ϕ k L ( X ) k f k L ( X ) (1 − m (Σ))Hence, kG ϕ { f } − χ Σ G ϕ { f }k L ( a ∗ × B × X ) ≥ k ϕ k L ( X ) k f k L ( X ) (1 − m (Σ)) (cid:3) Theorem 3.4.
Let s > . Then there exists a constant C s > such that, for all f, ϕ ∈ L ( X ) k f k L ( X ) k ϕ k L ( X ) ≤ C s (cid:18)Z a ∗ × B × X | ( λ, b, h ) | s |G ϕ { f } ( λ, b, h ) | | c ( λ ) | − dλdbdh (cid:19) (3.7) Proof.
Let 0 < r ≤ B r = { ( λ, b, h ) ∈ a ∗ × B × X : | ( λ, b, h ) | < r } theball of center 0 and radius r in a ∗ × B × X . Fix 0 < t ≤ m ( B t ) < k f k L ( X ) k ϕ k L ( X ) ≤ t s (1 − m ( B t )) Z | ( λ,b,h ) | >t t s |G ϕ { f } ( λ, b, h ) | | c ( λ ) | − dλdbdh ≤ t s (1 − m ( B t )) Z Z | ( λ,b,h ) | >t | ( λ, b, h ) | s |G ϕ { f } ( λ, b, h ) | | c ( λ ) | − dλdbdh ≤ t s (1 − m ( B t )) Z Z a ∗ × B × X | ( λ, b, h ) | s |G ϕ { f } ( λ, b, h ) | | c ( λ ) | − dλdbdh then, k f k L ( X ) k ϕ k L ( X ) ≤ t s (1 − m ( B t )) Z Z a ∗ × B × X | ( λ, b, h ) | s |G ϕ { f } ( λ, b, h ) | | c ( λ ) | − dλdbdh (3.8)we take the square of both sides of inequality (3.8), we get, k f k L ( X ) k ϕ k L ( X ) ≤ t s p − m ( B t ) (cid:18)Z Z a ∗ × B × X | ( λ, b, h ) | s |G ϕ { f } ( λ, b, h ) | | c ( λ ) | − dλdbdh (cid:19) We obtain the desired result by taking C s = t s p − m ( B t ). (cid:3) Now, before to give a version of Benedicks-type theorem for the Gabor Helgason Fouriertransform, we start by giving the following notations and definition.Let P ϕ : L ( X ) → L ( X ) be orthogonal projection of L ( X ) on the space G ϕ ( L ( X )).Let Σ be a measurable subset of a ∗ × B × X such that 0 < m (Σ) < + ∞ , where m is the Haarmeasure of a ∗ × B × X , we consider the operator P Σ defined on L ( X ) by P Σ F = χ Σ F where F ∈ L ( a ∗ × B × X )The usual norm of the operator P Σ P ϕ is defined by k P Σ P ϕ k = sup {k P Σ P ϕ ( F ) k L ( a ∗ × B × X ) , k F k L ( X ) ≤ } . Definition 3.5.
Let Σ a measurable subset of a ∗ × B × G and ϕ ∈ L ( X ) a nonzero windowfunction. We say that Σ is weakly annihilating, if any function f ∈ L ( X ) vanishes when itsHelgason Gabor Fourier transform with respect to the window ϕ is supported in Σ . Theorem 3.6 (Benedicks-type uncertainty principle for G ϕ ) . Let r, R > . Let ϕ ∈ L ( X ) ∩ L ∞ ( X ) be a non zero window function such that suppϕ ⊂ B r and let Σ = S × B R ⊂ a ∗ × B × X , be a subset of finite measure. Then Im { P ϕ } ∩ Im { P Σ } = { } ELGASON-GABOR FOURIER TRANSFORM AND UNCERTAINTY PRINCIPLES 11 i.e, Σ is weakly annihilating.Proof. Let F ∈ Im { P ϕ } ∩ Im { P Σ } , then, there exists a function f ∈ L ( X ) such that, F = G ϕ ( f ) and supp { F } ⊂ Σ.Then for all ( λ, h, b ) ∈ Σ F ( λ, h, b ) = \ ( f T h − ϕ )( λ, b )Thus supp { \ ( f T h − ϕ ) } ⊂ S , with m ( S ) < + ∞ . On other hand suppϕ ⊂ B r , we have supp { f T h − ϕ } ⊂ B r + R Hence, by the Benedicks theorem of the Helgason transform(see theorem 6.1 in [11]).We deduce that f T h − ϕ ≡ F = 0. (cid:3) References [1] M.J. Bastiaans,
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Department of Mathematics and Computer Sciences, Faculty of Sciences,Equipe d’Analyse Harmonique et Probabilités, University Moulay Ismaïl,BP 11201 Zitoune, Meknes, Morocco
Mohammed El kassimi
E-mail address : [email protected] Mustapha Boujeddaine
E-mail address : [email protected] Saïd Fahlaoui
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