Miyachi's Theorem for the Quaternion Fourier Transform
aa r X i v : . [ m a t h . C A ] S e p Miyachi’s Theorem For the Quaternion FourierTransform
Youssef El Haoui , * , Said Fahlaoui Department of Mathematics and Computer Sciences, Faculty of Sciences, Equipe dAnalyseHarmonique et Probabilits, University Moulay Ismail, BP 1120, Zitoune, Meknes, MoroccoE-MAIL: [email protected], [email protected]
Abstract
The quaternion Fourier transform (QFT) satisfies some uncertainty principles similar to the Eu-clidean Fourier transform. In this paper, we establish Miyachi’s theorem for this transform and conse-quently generalize and prove the analogue of Hardy’s theorem and Cowling-Price uncertainty principlein the QFT domain.
Key words:
Quaternion Fourier transform, Miyachi’s theorem.
Uncertainty principle (UP) is an important tool in harmonic anlysis; it states that a nonzero functionand its Fourier transform cannot both be very rapidly decreased. UP has implications in different areaslike quantum physics, information processing, signal analysis, etc.In signal analysis, it gives in general a lower bound for the simultaneous localization of signals in phaseand frequency spaces. There are many advantageous ways to get the statement about localization pre-cise; examples include theorems of Hardy theorem [11], Beurling [13], and Miyachi [17] which interpreteddifferently the localisation, as sharp pointwise estimates of a signal and its Fourier transform. More pre-cisely Miyachi’s theorem asserts that if f is a measurable function on R satisfying : e αx f ∈ L ( R ) + L ∞ ( R ) , and ∫ R log + ⎛⎜⎜⎜⎝ ∣ ˆ f ( y ) e π α y ∣ ρ ⎞⎟⎟⎟⎠ d y < ∞ , for some positive constants α and ρ , where log + ( x ) = { log ( x ) , if x > . otherwise . and ˆ f stands for the classical Fourier transform of f ,then f is a constant multiple of the Gaussian e − αx .Miyachi’s theorem has been extended in several different directions in recent years, including exten-sions to Dunkl transform [5], Clifford–Fourier transform[9], and much more generally, to nilpotent lie groups[1] and Heisenberg motion groups [2].The quaternion Fourier transform (QFT) is a non-trivial extension of the real and complex classi-cal Fourier transform to the algebra of the quaternions. Since the quaternion multiplication is non-commutative, there are three types of the QFT depending on which side multiplication of the kernel is * Corresponding author.
In order to extend complex numbers to a four-dimensional algebra, the Irish W. R. Hamilton inventedin 1843 the quaternion algebra H .Any quaternion q ∈ H can be expressed by q = q + i q + j q + k q ; q , q , q , q ∈ R , where i , j , k satisfy Hamilton’s rules i = j = k = − , ij = − ji = k , jk = − kj = i ; ki = − ik = j . Quaternions are isomorphic to the Clifford algebra Cl ( , ) of R ( , ) : H ≅ Cl ( , ) . (2.1)We define the conjugation of q ∈ H by: q def = q − i q − j q − k q and its modulus ∣ q ∣ Q is defined by ∣ q ∣ Q def = √ qq = √ q + q + q + q . Particularly, when q = q is a real number, the module ∣ q ∣ Q reduces to the ordinary Euclidean module ∣ q ∣ = √ q . Also, we observe that for x ∈ R , ∣ x ∣ Q = ∣ x ∣ , where ∣ . ∣ is the Euclidean norm ∣( x , x )∣ = x + x Moreover, for arbitrary p, q ∈ H the following identity holds2 pq ∣ Q = ∣ p ∣ Q ∣ q ∣ Q . Clearly, the inverse of ≠ q ∈ H is defined by : q − = q ∣ q ∣ Q . which shows that H is a normed division algebra.Due to (2.1), we recall the following properties:if x is a vector in Cl ( , ) , then ∣ x ∣ = − x . (2.2)Let (,) be the inner product on R ( , ) ; for 2.3rs x and y we have ( x , y ) def = ∑ l = x l y l = − ( xy + yx ) . (2.3)In this paper, we will study the quaternion-valued signal f ∶ R → H , which can be written in this form f = f + i f + j f + k f , with f m ∶ R → R f or m = , , , . We introduce the Banach spaces L p ( R , H ) , ≤ p ≤ ∞ , L p ( R , H ) def = { f ∣ f ∶ R → H , ∣ f ∣ p,Q def = ( ∫ R ∣ f ( x )∣ pQ d x ) p < ∞ } , ≤ p < ∞ ,L ∞ ( R , H ) def = { f ∣ f ∶ R → H , ∣ f ∣ ∞ ,Q def = ess sup x ∈ R ∣ f ( x )∣ Q < ∞ } . where d x = dx dx , refers to the usual Lebesgue measure in R . If f ∈ L ∞ ( R , H ) is continuous, then ∣ f ∣ ∞ ,Q = sup x ∈ R ∣ f ( x )∣ Q Furthermore, we define naturally the two following Banach spaces L ( R , H ) ∩ L ∞ ( R , H ) = { f ∣ f ∈ L ( R , H ) and f ∈ L ∞ ( R , H )} ,L ( R , H ) + L ∞ ( R , H ) = { f = f + f , f ∈ L ( R , H ) , f ∈ L ∞ ( R , H )} . We denote by S ( R , H ) the quaternion Schwartz test function space, i.e., the set C ∞ of smoothfunctions f , from R to H , given by S ( R , H ) def = { f ∈ C ∞ ( R , H ) ∶ sup x ∈ R , ∣ α ∣≤ n (( + ∣ x ∣ m ) ∂ α ∣ f ( x )∣ Q ) < ∞ , m, n ∈ N } , where ∂ α def = ∂ ∣ α ∣ ∂ α x ∂ α x , ∣ α ∣ = α + α for a multi-index α = ( α , α ) ∈ N . In this section, we review the definition and some properties of the two-sided QFT.
Definition 3.1.
Let f in L ( R , H ) . Then, the two-sided quaternion Fourier transform of the function f isgiven by F { f }( ξ ) def = ∫ R e − i πξ x f ( x ) e − j πξ x d x , (3.1)where ξ , x ∈ R . emma 3.2. 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 itsinverse is given by f ( x ) = ∫ R e i πξ x F { f }( ξ ) e j πξ x d ξ , d ξ = dξ dξ . Lemma 3.3.
Scaling propertyLet α be a positive scalar constant; then, the two-sided QFT of f α ( x ) = f ( α x ) becomes F { f α } ( ξ ) = ( α ) F { f } ( α ξ ) . (3.2)Proof. Equation (3.1) gives F { f α }( ξ ) = ∫ R e − i πξ x f ( α x ) e − j πξ x d x . We substitute y for α x and get F { f α }( ξ ) = ( α ) ∫ R e − i π ( α ξ ) y f ( y ) e − j π ( α ξ ) y d y = ( α ) F { f } ( α ξ ) . The next lemma states that the QFT of a Gaussian quaternion function is another quaternion Gaussianquaternion function.
Lemma 3.4.
QFT of a Gaussian quaternion function.Consider a two-dimensional Gaussian quaternion function f given by f ( x ) = qe −( α x + α x ) , where q = q + iq + jq + kq is a constant quaternion, and α , α are positive real constants.Then F { f }( ξ ) = q π √ α α e − π α ξ − π α ξ . (3.3) where x , ξ ∈ R . Proof. Let g be defined by g ( x ) def = e −( α x + α x ) , we have F { f }( ξ ) = q F { g }( ξ ) + i q F { g }( ξ ) + q F { g }( ξ ) j + q i F { g }( ξ ) j , where we used the R -linearity of the QFT and the properties F { i h } = i F { h } , F { j h } = F { h } j , and F { k h } = i F { h } j for h real-valued function.On the other hand, we have F { g }( ξ ) = ∫ R e − i πξ x g ( x ) e − j πξ x d x = ∫ R e − α x e − i πξ x dx ∫ R e − α x e − j πξ x dx = √ πα e − π α ξ √ πα e − π α ξ = π √ α α e − π α ξ − π α ξ . This completes the proof of Lemma 3.4. 4 emma 3.5. [8, Lemma 3.11]Let f ∶ R → R be of the form f ( x ) = P ( x ) e − πα ∣ x ∣ , where P is a polynomial and α > , Then F { f }( ξ ) = Q ( ξ ) e − πα ∣ ξ ∣ , where Q is a quaternion polynomial with degP = degQ. In this section, we prove Miyachi’s theorem for the quaternion Fourier transform. For this, we need thefollowing technical lemma of the complex analysis.
Lemma 4.1. [5, Lemma 1]Let h be an entire function on C such that ∣ h ( z )∣ ≤ Ae B ∣ Re ( z )∣ f or all z ∈ C and ∫ R log + (∣ h ( y )∣) d y < ∞ , for some positive constants A and B .Then h is a constant function. Theorem 4.2. (Miyachis Theorem).Let α , β > . Suppose that f is a measurable function such that e α ∣ x ∣ f ∈ L ( R , H ) + L ∞ ( R , H ) , (4.1) and ∫ R log + ( ∣ F { f } ( y ) e β ∣ y ∣ ∣ Q ρ ) d y < ∞ , (4.2) for some ρ, < ρ < +∞ Then, three cases can occur:(i) If αβ > π , then f = almost everywhere.(ii) If αβ = π , then f ( x ) = Ce − α ∣ x ∣ , where C is a constant quaternion.(iii) If αβ < π , then there exist infinitely many functions satisfying (4.1) and (4.2) . Proof.(i) We first prove the result for the case αβ = π . By scaling, we can assume that α = β = π. Indeed, let g ( x ) = f (√ πα x ) ; then, by Lemma 3.3 we obtain F { g } ( t ) = απ F { f } (√ απ t ) , t ∈ R . (4.3)In addition, we get by (4.1) e π ∣ x ∣ g ( x ) = e π ∣ x ∣ f (√ πα x ) ∈ L ( R , H ) + L ∞ ( R , H ) . ∫ R log + ⎛⎝ ∣ F { f } ( y ) e β ∣ y ∣ ∣ Q ρ ⎞⎠ d y = απ ∫ R log + ⎛⎜⎜⎝ ∣ F { f } (√ απ t ) e αβπ ∣ t ∣ ∣ Q ρ ⎞⎟⎟⎠ d t (4.3) = απ ∫ R log + ⎛⎝ ∣ F { g } ( t ) e π ∣ t ∣ ∣ Q ρ ′ ⎞⎠ d t < ∞ , where ρ ′ = απ ρ. If the result is shown for α = β = π, then g ( x ) = Ce − π ∣ x ∣ , and thus, f ( x ) = g (√ απ x ) = C e − α ∣ x ∣ . Now, we assume that α = β = π ∶ Applying the same method as in [7, Thm. 5.3], by complexifying the variable z = a + i C b , where a = ( a , a ) , b = ( b , b ) ∈ R , and we note by i C the complex number which satisfies i C = − . We have ∣ z ∣ Q = ∣ a ∣ + ∣ b ∣ = ∣ z ∣ , where ∣ . ∣ is the Euclidean norm in C . Let w ( x ) = e π (∣ a ∣ +∣ b ∣ ) e π (∣ a ∣ +∣ b ∣ ) e − π (∣ x ∣−(∣ a ∣+∣ b ∣)) e − π (∣ x ∣−(∣ a ∣+∣ b ∣)) = e π ∣ z ∣ e − π (∣ x ∣−(∣ a ∣+∣ b ∣)) e − π (∣ x ∣−(∣ a ∣+∣ b ∣)) . Clearly, w belongs to L ( R , H ) ∩ L ∞ ( R , H ) . By assumption, e π ∣ x ∣ f belongs to L ( R , H ) + L ∞ ( R , H ) . As F { f } ( z ) = ∫ R e − π i x ( a + i C b ) f ( x ) e − π jx ( a + i C b ) d x , we have ∣ F { f } ( z )∣ Q ≤ ∫ R ∣ f ( x )∣ Q e π (∣ x a ∣+∣ x b ∣+∣ x a ∣+∣ x b ∣) d x = ∫ R ∣ e π ∣ x ∣ f ( x )∣ Q w ( x ) d x . Hence, F { f } ( z ) is well defined, and is an entire function on C . Furthermore, by (4.1) there exists u ∈ L ( R , H ) and v ∈ L ∞ ( R , H ) such that e π ∣ x ∣ f ( x ) = u ( x ) + v ( x ) , Using the triangle inequality and the linearity of the integral we get ∣ F { f } ( z )∣ Q ≤ ∫ R ∣ u ( x )∣ Q w ( x ) d x + ∫ R ∣ v ( x )∣ Q w ( x ) d x . Then according to the H ˆolder’s inequality, we have ∣ F { f } ( z )∣ Q ≤ ∣ u ∣ ,Q ∣ w ∣ ∞ ,Q + ∣ v ∣ ∞ ,Q ∣ w ∣ ,Q , ∫ R e − π (∣ t ∣+ m ) dt ≤ , where m ∈ R , we obtain ∣ w ∣ ,Q ≤ e π ∣ z ∣ , and ∣ w ∣ ∞ ,Q ≤ e π ∣ z ∣ . Then ∣ F { f } ( z )∣ Q ≤ e π ∣ z ∣ ∣ u ∣ ,Q + e π ∣ z ∣ ∣ v ∣ ∞ ,Q ≤ K e π ∣ z ∣ , where K is a positive constant independent of z. Now, let h ( z ) = e − πz ∣ F { f } ( z ) ∣ Q , for z ∈ C , then, h is an entire function.By (2.2) and (2.3), we have z = ( a + i C b ) ( a + i C b ) = − ∣ a ∣ + ∣ b ∣ − i C ( a , b ) , then ∣ e − π z ∣ Q ≤ e π ∣ a ∣ e − π ∣ b ∣ , where we used ∣ e πi C ( a , b ) ∣ Q = .As a result ∣ h ( z )∣ ≤ Ke π ∣ a ∣ e − π ∣ b ∣ e π ∣ a ∣ e π ∣ b ∣ , = Ke π ∣ a ∣ . (4.4)On the other hand, by (2.2) and by assumption ∫ R log + ( ∣ h ( y )∣ ρ ) d y = ∫ R log + ⎛⎝ ∣ e π ∣ y ∣ F { f } ( y )∣ Q ρ ⎞⎠ d y < ∞ . (4.5)Then by (4.4),(4.5) and by applying Lemma 4.1 to the function h ( y ) / ρ , we deduce that h ( y ) = const. i.e ∣ F { f } ( y ) e − π ∣ y ∣ ∣ Q = const. Therefore F { f } ( y ) = Ce − π ∣ y ∣ , where C is a constant quaternion.Then by Lemmas 3.2 and 3.4, we have f ( x ) = C e − π ∣ x ∣ . (ii) If αβ > π . Let g ( x ) = f (√ πα x ) , a simple calculation shows that ∫ R log + ⎛⎝ ∣ F { g } ( t ) e π ∣ t ∣ ∣ Q ρ ′ ⎞⎠ dt < πα ∫ R log + ⎛⎝ ∣ F { f } ( y ) e β ∣ y ∣ ∣ Q ρ ⎞⎠ d y < ∞ , ( by (4.2) ) ρ ′ = απ ρ. Then, according to the first case g ( x ) = C e − π ∣ x ∣ , where C is a constant quaternion.Consequently f ( x ) = C e − α ∣ x ∣ . Hence, by Lemma 3.4 we get F { f } ( y ) = Ce − π α ∣ y ∣ . Refering to (4.2), C must be zero.(iii) For the final case, αβ < π Let f ( x ) = ϕ k,l ( x ) e − πγ ∣ x ∣ with απ < γ < πβ , where { ϕ k,l } k,l ∈ N is a basis of S ( R , H ) , which is defined by ϕ k,l ( x , x ) def = ϕ k ( x ) ϕ l ( x ) , for ( x , x ) ∈ R , and ϕ k ( x ) = ( − ) k k ! e πx d k dx k ( e − π x ) , x ∈ R . It is important to see that { ϕ k,l } k,l ∈ N is a basis of S ( R , H ) (see [6, 7]).We have e α ∣ x ∣ f = e α ∣ x ∣ ϕ k,l ( x ) e − πγ ∣ x ∣ = ϕ k,l ( x ) e ( α − πγ )∣ x ∣ ∈ L ( R , H ) + L ∞ ( R , H ) . Lemma 3.5 implies F { f } ( y ) = Q ( y ) e − πγ ∣ y ∣ , where Q is a quaternion polynomial.Then, since β < πα , ∫ R log + ⎛⎝ ∣ F { f } ( y ) e β ∣ y ∣ ∣ Q ρ ⎞⎠ d y = ∫ R log + ⎛⎜⎜⎝ ∣ Q ( y ) e ( β − πγ )∣ y ∣ ∣ Q ρ ⎞⎟⎟⎠ d y < ∞ . This completes the proof of Theorem 4.2.In the following, we illustrate the effectiveness of Theorem 4.2, by giving an example, and derive twogeneralizations of uncertainty principle associated with the QFT.
Example 4.3.
Consider α, β two positive numbers with αβ = π , and the quaternion Gaussian function f ( x ) = qe − α ∣ x ∣ , where q = q + i q + j q + k q is a constant quaternion.Obviously, we have e α ∣ x ∣ f = q ∈ L ∞ ( R , H ) ⊂ L ( R , H ) + L ∞ ( R , H ) . f satisfies condition (4.1).By lemma 3.4, we get F { f }( y ) = q πα e − β ∣ y ∣ . Then ∫ R log + ( ∣ F { f }( y )∣ Q e β ∣ y ∣ ρ ) d y = ∫ R log + ( ∣ q ∣ Q πα ρ ) d y < ∞ , whenever ρ > ∣ q ∣ Q πα . Corollary 4.4.
Hardy’s uncertainty principle for the QFT.Let α and β be both positive constants. Suppose f be in L ( R , H ) with(i) ∣ f ( x )∣ < Ce − α ∣ x ∣ . (ii) ∣ F { f }( y )∣ < C ′ e − β ∣ y ∣ . for some constants C > and C ′ > . Then, three cases can occur :1. If αβ > π , f = almost everywhere on R .
2. If αβ = π , then f is a constant quaternion multiple of e − α ∣ x ∣ .3. If αβ < π , there are infinitely many linearly independent functions satisfying both conditions ( i ) and ( ii ) . Proof. Immediately using the decay condition ( i ) one has f e α ∣ x ∣ ∈ L ∞ ( R , H ) . Hence f verifiescondition (4.1) of Theorem 4.2.Moreover, for ρ > we have ∣ F { f }( y )∣ Q e β ∣ y ∣ ρ ≤ C ′ ρ . Thus log + ( ∣ F { f }( y )∣ Q e β ∣ y ∣ ρ ) ≤ log + ( C ′ ρ )´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ = , whenver ρ > C ′ . So condition (4.2) of Theorem 4.2 is verified. Then, direct application of Theorem 4.2 enables us toachieve the proof.
Corollary 4.5.
Cowling-Price’s uncertainty principle for the QFT.Let α and β be positive real numbers, ≤ p, q ≤ ∞ such that min ( p, q ) is finite, and let f are a squareintegrable quaternion-valued function satisfying the following decay conditions Suppose f be in L ( R , H ) (i) ∫ R (∣ f ( x )∣ Q e α ∣ x ∣ ) p d x < ∞ . (ii) ∫ R (∣ F { f }( y )∣ Q e β ∣ y ∣ ) q d y < ∞ . Then the three following conclusions hold:1. f = almost everywhere whenever αβ > π .
2. If αβ = π , then f is a constant quaternion multiple of e − α ∣ x ∣ .3. If αβ < π , there are infinitely many linearly independent functions satisfying both conditions ( i ) and ( ii ) . Proof. According to (i), we get f e α ∣ x ∣ ∈ L p ( R , H ) , and using the fact that L p ⊂ L + L ∞ we obtain that f fulfills condition (4.1) of Theorem 4.2.Furthermore, based on the inequality log + ( x ) ≤ x for x ∈ R + , we can easily see that f satisfies the second condition (4.2).Therefore, by applying Theorem 4.2 we conclude the proof.9 Conclusions