Benedicks-Amrein-Berthier type theorem related to the two-sided Quaternion Fourier transform
aa r X i v : . [ m a t h . C A ] S e p Benedicks-Amrein-Berthier’s Theorem Relatedto the Quaternion Fourier Transform
Youssef El Haoui , * , Said Fahlaoui Department of Mathematics and Computer Sciences, Faculty of Sciences, University Moulay Ismail,Meknes 11201, MoroccoE-MAIL: [email protected], [email protected]
Abstract
The main objective of the present paper is to establish a new uncertainty principle (UP) for the two-sided quaternion Fourier transform (QFT). This result is an extension of a result of Benedicks, Amreinand Berthier, which states that a nonzero function in L ( R , H ) and its two-sided QFT cannot bothhave support of finite measure. Key words:
Quaternion Fourier transform, BenedicksAmreinBerthier’s theorem.
In harmonic analysis, It is known that if the supports of a function f ∈ L ( R d ) and its Fourier transform F { f } are contained in bounded rectangles, then f vanishes almost every where.Benedicks [2] relaxed the requirements for this conclusion by sharing that the supports of f and F { f } need only have finite measure, whose proof, first circulated as a preprint in 1974 but not formally publishedfor another decade, Amrein and Berthier [1] later gave a different proof.The classical Fourier transform, defined over the real line, is given by F { f ( x )} ( y ) = ∫ +∞−∞ e − πixy f ( x ) dx, for f ∈ L ( R ) . The Benedicks-Amrein-Berthier’s theorem is the following:Let f ∈ L ( R ) , and F { f } be its Fourier transform.Suppose that the sets { x ∈ R ∶ f ( x ) ≠ } and { ξ ∈ R ∶ F { f } ( ξ ) ≠ } both have finite Lebesgue measure,then f = . In this paper, we extend the validity of the Benedicks-Amrein-Berthiers theorem to the two-sided QFT.Other theorems of the UPs: Hardy, Beurling, Cowling-Price, Gelfand-Shilov, Miyachi have been extendedfor this transform (e.g. [6, 7, 8]) .We further emphasize that our generalization is non-trivial because of the non-commutative property ofquaternion multiplication.Our paper is organized as follows: In the next section, we collect some basic concepts in quaternionalgebra. In section 3, we recall the definition and some results for the two-sided QFT useful in the sequel.In section 4, we prove Benedicks-Amrein-Berthiers theorem for the two-sided QFT.For the sake of further simplicity, we will use the real vector notations x = ( x , x ) ∈ R , dx = dx dx the Lebesgue-measure, ∣ A ∣ is the normed Lebesgue measure of a mea-surable set A ⊆ R n . * Corresponding author. The algebra of quaternions
The quaternion algebra H was formally invented by by Irish mathematician Sir William Rowan Hamil-ton in 1843, it is an associative non-commutative four-dimensional algebra which is a generalization ofcomplex numbers. H = { q = q + iq + jq + kq ; q , q , q , q ∈ R } where the elements i, j, k satisfy Hamilton’s rules ij = − ji = k ; jk = − kj = i ; ki = − ik = j ; i = j = k = − . The algebra of quaternions is isomorphic to the Clifford algebra Cl ( , ) of R ( , ) : H ≅ Cl ( , ) . (2.1)The scalar part of q = q + iq + jq + kq ∈ H denoted by Sc ( q ) is q , and iq + jq + kq is called the vectorpart(or pure quaternion) of q denoted by V ec ( q ) .The conjugate q of q is defined by : q = q − iq − jq − kq . and we get the norm od modulus of q ∈ H as ∣ q ∣ Q = √ qq = √ q + q + q + q . Then we get ∣ pq ∣ Q = ∣ p ∣ Q ∣ q ∣ Q . Applying the conjugate and the modulus of q , we can define the inverse of q ∈ H /{ } as q − = q ∣ q ∣ Q . Thus, H is a normed division algebra.Let q ∈ H , we have a natural generalization of Euler’s formula for quaternions q = ∣ q ∣ Q e µθ , (2.2)where e µθ = cos ( θ ) + µ sin ( θ ) , with θ = artan ∣ V ec ( q )∣ Q Sc ( q ) , ≤ θ ≤ π, and µ ∶= V ec ( q )∣ V ec ( q )∣ Q verifying µ = − . We introduce the quaternion space L p ( R , H ) , ≤ p ≤ ∞ , as the Banach space of all quaternion-valued functions, given by L p ( R , H ) = { f ∶ R → H , f = f + if + jf + kf ∣ f m ∈ L p ( R , R ) , m = , , , } The L p ( R , H ) − inner product and norrm are defined as ∀ f, g ∈ L p ( R , H ) , ( f, g ) = ∫ R f ( x ) g ( x ) dx ∣ f ∣ p,Q = ( ∫ R ∣ f ( x )∣ pQ dx ) p < ∞ f or ≤ p < ∞ , ∣ f ∣ ∞ ,Q = ess sup x ∈ R ∣ f ( x )∣ Q < ∞ , p = +∞ Quaternion Fourier transforms
Ell [9] introduced the quaternion Fourier transform (QFT) as an hyper-complex transform which gen-eralizes the classical Fourier transform (CFT) using the framework of quaternion algebra. Several knownand useful properties, and theorems of this extended transform are generalizations of the correspondingproperties, and theorems of the CFT with some modifications (e.g., [3, 4, 9, 11]).The QFT plays a significant role in sgnal analysis due to transforming a real 2D signal into a quaternion-valued frequency domain signal, and it belongs to the family of Clifford Fourier transformations becauseof (2.1).There are three different types of QFT, the left-sided QFT , the right-sided QFT , and two-sided QFT.For more details, we refer the reader to [12].In this paper we only treat the two-sided QFT.In the following, we will recall the definition and some properties of the two-sided QFT needed to provethe main theorem.
Definition 3.1.
Let f ∈ L ( R , H ) .Then two-sided quarternion Fourier transform of the function f is given by F { f ( x )}( ξ ) = ∫ R e − i πξ x f ( x ) e − j πξ x dx, (3.1)Where ξ, x ∈ R . Lemma 3.2. (Continuity)If f ∈ L ( R , H ) , then F { f } is a continuous function on R . Proof. See [[4], Proposition 3.1(ix)].
Lemma 3.3.
Inverse QFT ([3], Thm. 2.5)If f ∈ L ( R , H ) , and F { f } ∈ L ( R , H ) ,then the two-sided QFT is an invertible transform and its inverse is given by f ( x ) = ∫ R e i πξ x F { f ( x )}( ξ ) e j πξ x dξ, dξ = dξ dξ . Lemma 3.4. (Dilation property), see example 2 on page 50 [3]Let k , k be a positive scalar constants, we have F { f ( t , t )} ( u k , u k ) = k k F { f ( k t , k t )} ( u , u ) . The uncertainty principle is a significant mathematical result that gives limitations on the exact mea-surement of a signal in both the time domain and frequency domain simultaneously. It has implications indifferent disciplines such as quantum physics, information processing, signal analysis, etc.Because of their great importance for signal and image analysis, researchers interested to extend thevalidity of the of the uncertainty principles in the framework of the algebra of Quaternions (see, e.g.,[11, 4, 6].In this context, this section deal with proving BenedicksAmreinBerthier’s uncertainty principle the two-sided QFT by virtue of the quaternion Fourier series expansion .
Definition 4.1.
The quaternion-valued signal f ∶ R → H is said to be periodic of period K = ( L, L ) , L > , if it verifies f ( x + K ) = f ( x ) ∀ x ∈ R . f ( x ) ≃ a + ∑ ( r,s )∈ Z cos ( rπx L ) cos ( sπx L ) A rs + ∑ ( r,s )∈ Z sin ( rπx L ) cos ( sπx L ) B rs + ∑ ( r,s )∈ Z cos ( rπx L ) sin ( sπx L ) C rs + ∑ ( r,s )∈ Z sin ( rπx L ) sin ( sπx L ) D rs , with a , A rs , B rs , C rs , D rs ∈ H . We have according to the quaternion Euler formula (2.2) cos ( nπx r L ) = e i nπxrL + e − i nπx L and sin ( nπx r L ) = e j nπxrL − e − j nπxrL j , r = , . Therfore , we can rewrite the Fourier series in the exponential notations: f ( x ) ≃ ∑ ( r,s )∈ Z e i rπx L C rs e j sπx L , with C rs ∈ H . (4.1) Lemma 4.2.
The quaternion Fourier coefficients are given by: C rs ≃ ( L ) ∫ L − L ∫ L − L e − i rπx L f ( x ) e − j sπx L dx Proof. we have by (4.1) : ( L ) ∫ L − L ∫ L − L e − i rπx L f ( x ) e − j sπx L dx ≃ ( L ) ∫ L − L ∫ L − L e − i rπx L ∑ ( m ,m )∈ Z e i m πx L C m m e j m πx L e − j sπx L dx dx = ( L ) ∑ ( m ,m )∈ Z ∫ L − L e − i rπx L e i m πx L dx C m m ∫ L − L e j m πx L e − j sπx L dx = ( L ) ∑ ( m ,m )∈ Z C m m Lδ rm Lδ sm = C rs . ◻ The next two lemmas will be needed to prove the main theorem.
Lemma 4.3.
Let T = [ , ] . If f ∈ L ( R , H ) ,Then ϕ ( t ) = ∑ k ∈ Z f ( t + k ) converges in L ( T , H ) , andthe quaternion Fourier series of ϕ is ∑ k =( k ,k )∈ Z e πit k F { f }( k ) e πjt k Proof. As f ∈ L ( R , H ) , we have 4 [ , ] ∣ ∑ ∣ k ∣ ≤ N f ( t + k )∣ Q ≤ ∫ [ , ] ∑ ∣ k ∣ ≤ N ∣ f ( t + k ) ∣ Q dt = ∑ ∣ k ∣ ≤ N ∫ [ , ] ∣ f ( x + k ) ∣ Q dt = ∑ ∣ k ∣ ≤ N ∫ k + k ∫ k + k ∣ f ( t ) ∣ Q dt = ∑ ∣ k ∣ ≤ N ∫ k + k ∫ k + k ∣ f ( t ) ∣ Q dt ≤ ∫ R ∣ f ( t ) ∣ Q dt < ∞ . Where ∣ k ∣ = ∣ k ∣ + ∣ k ∣ and N ∈ N is big enough.Thus we have shown that ϕ ∈ L ( T , H ) . Since ϕ is periodic with period = 1, according to lemma 4.2 the qutaernion Fourier coefficients are ∫ − ∫ − e − πit m ϕ ( t ) e − πjt m dt = ∫ − ∫ − e − πit m ∑ k ∈ Z f ( t + k ) e − πjt m dt = ∫ − ∫ − e − πi ( t + k ) m ∑ k = ( k ,k ) ∈ Z f ( t + k ) e − πj ( t + k ) m dt = ∑ k ∈ Z ∫ k + k − ∫ k + k − e − πim y f ( y ) e − πjm y dy dy = F { f ( y )}( m , m ) . ◻ Lemma 4.4.
Let f ∶ [ , ] → H be a measurable function.If ∫ [ , ] ∣ f ( x )∣ Q dx < ∞ , then there is a set E ⊆ [ , ] with ∣ E ∣ = and ∣ f ( a )∣ Q < ∞ ∀ a ∈ E. Proof. Let E = { x ∈ [ , ] ∶ ∣ f ( x )∣ Q < ∞ } and F = [ , ] / E. By absurd, if ∣ F ∣ > then ∫ F ∣ f ( x )∣ Q dx = ∞ , contradiction with ∫ [ , ] ∣ f ( x )∣ Q dx < ∞ . Then necessarily ∣ F ∣ = , therefore ∣ E ∣ = . ◻ will also need the following result of the zeros of an analytic function. Lemma 4.5.
Let f be an analytic function on C , and Z ( f ) ∶= { z ∈ C ∶ f ( z ) = } . One of these properties can occur(i) Z ( f ) is of zero measure on C .(ii) Z ( f ) = C . Proof. The lemma holds for a function of one variable, see [[13], Thm. 10.18].Let µ r be the Lebesgue-measure on C r , r = , and f be a function of two complex variables z , z .We define f z ( z ) = f ( z , z ) , and f z ( z ) = f ( z , z ) .We have Z ( f ) = Z ( f z ) ⊗ Z ( f z ) and µ ( Z ( f )) = µ ( Z ( f z )) µ ( Z ( f z )) . µ ( Z ( f z )) = or µ ( Z ( f z )) = then µ ( Z ( f )) = . If not, then Z ( f z ) = C and Z ( f z ) = C such that Z ( f ) = C . ◻ Theorem 4.6.
Let f ∈ L ( R , H ) ,We define Σ ( f ) ∶= { x ∈ R ∶ f ( x ) ≠ } and Σ ( F { f }) ∶= { ξ ∈ R ∶ F { f }( ξ ) ≠ } . If ∣ Σ ( f )∣ ∣ Σ ( F { f })∣ < ∞ , then f = . Proof. Since ∣ Σ ( f )∣ is finite, by composing f with a dilation, lemma 3.4, we can assume that ∣ Σ ( f )∣ < without loss of generality.Hence we have ∫ [ , ] ∑ k ∈ Z Σ ( F { f }) ( y + k ) dy = ∑ k ∈ Z ∫ [ , ] Σ (F{ f }) ( y + k ) dy = ∑ k = ( k ,k ) ∈ Z ∫ − k − k ∫ − k − k Σ (F{ f }) ( z ) dz = ∫ R Σ (F{ f }) ( z ) dz = ∣ Σ ( F { f })∣ < ∞ (4.2)(by assumption)Also ∫ [ , ] ∑ k ∈ Z Σ ( f ) ( x + k ) dx = ∫ R Σ ( f ) ( x ) dx = ∣ Σ ( f )∣ < (4.3)According to (4.2) and lemma 4.4, there exists an E ⊆ [ , ] , with ∣ E ∣ = and ∑ k ∈ Z Σ (F{ f }) ( a + k ) < ∞ for all a ∈ E .Let F = { x ∈ [ , ] ∶ ∑ k ∈ Z Σ ( f ) ( x + k ) = } We have ∣ F ∣ > ( otherwise we would have ∫ [ , ] ∑ k ∈ Z Σ ( f ) ( x + k ) dx ≥ , contradiction with (4.3)).Given a = ( a , a ) ∈ E, We define ϕ a ( t ) = ∑ k = ( k ,k ) ∈ Z e − πia ( t + k ) f ( t + k ) e − πja ( t + k ) , for t = ( t , t ) ∈ R .As f ∈ L ( R , H ) by assumption, applying lemma 4.3 gives ϕ a ∈ L ( T , H ) , Now, Let g ( t ) = e − πia t f ( t ) e − πja t . Straightforward computations show that F { g }( y ) = F { f }( y + a ) . (4.4)6herefore, the quaternion Fourier series of ϕ a is ∑ k = ( k ,k ) ∈ Z e πit k F { f }( a + k ) e πjt k . Indeed, ϕ a ( t ) = ∑ k = ( k ,k ) ∈ Z g ( t + k ) = ∑ k = ( k ,k ) ∈ Z e πit k F { g }( k ) e πjt k = ∑ k = ( k ,k ) ∈ Z e πit k F { f }( a + k ) e πjt k . The second equality follows from lemma 4.3, the last by (4.4).Complexifying the variable t = a + i C b ; a =( a , a ), b =( b , b ) ∈ R (we note by i C the complex numberchecking i C = -1).Since a ∈ E , ϕ a ( t ) is well defined for all t ∈ C , and it is a trigonometric polynomial (with quaternioncoefficients).Hence ϕ a is analytic.So, by lemma 4.5 either ϕ a = everywhere, or ϕ a ≠ a.e.On the other hand, ∣ ϕ a ( t )∣ Q ≤ ∑ k = ( k ,k ) ∈ Z ∣ f ( t + k )∣ Q = f or t ∈ F, and F has measure greater than zero, thus ϕ a = everywhere for all a ∈ E .Whence F { f }( a + k ) = for all a ∈ E and k ∈ Z .Then F { f } = a.e.Since F { f } is continuous (lemma 3.2), we obtain F { f } = . Then by lemma 3.3, we conclude that f = .This completes the proof of BenedicksAmreinBerthier type theorem. ◻ Corollary 4.7.
Let f ∈ L p ( R , H ) f or p > ,If ∣ Σ ( f )∣ ∣ Σ ( F { f })∣ < ∞ , then f = . Proof. Suppose f ∈ L p ( R , H ) with p > ,since ∣ Σ ( f )∣ < ∞ , then by H ˆolder inequality we obtain f ∈ L ( R , H ) .So by the theorem 4.6 we get f = . which was to be proved. ◻ In this paper, based on quaternionic Fourier series representation, a generalization of the Benedicks-Amrein-Berthier’s UP associated with the QFT is proposed. The extension of this qualitative UP to thequaternionic algebra framework shows that a quaternionic 2D signal and its QFT cannot both simultane-ously have support of finite measure .In the future work, we will consider Beurling’s UP, Hardys UP, Cowling-Prices UP and Gelfand-Shilovs UPfor the two-dimensional quaternionic windowed Fourier transform.
Acknowledgements
The authors would like to thank Eckhard Hitzer for several helpful comments and Rajakumar Roopku-mar for proofreading and discussions related to this work.7 eferenceseferences