Titchmarsh theorem associated with QFT
aa r X i v : . [ m a t h . C A ] J a n Titchmarsh Theorems
Associated with Two-sided QFT
Hakim MONAIM & Said FAHLAOUI
Abstract:
In this paper, we prove both of the Titchmarsh theorems associated withTwo-Sided Quaternionic Fourier Transform and we conclude the first one associatedwith Short-Time Two-Sided Quaternionic Fourier Transform.
Keywords:
Harmonic Analysis; Two-Sided Quaternionic Fourier Transform; Short-Time Two-Sided Quaternionic Fourier Transform; Titchmarsh Theorems.
Mathematics Subject Classification (2020):
Primary 42B35; Secondary 30G35.
1. Introduction
The Quaternionic Fourier Transform (QFT) plays a vital role in the representationof signals. It transforms a real (or quaternionic) R -signal into a quaternion-valued fre-quency domain signal. The four components of the QFT separate four cases of symmetryinto real signals instead of only two as in the classical Fourier Transform. Due to thenoncommutative property of multiplication of quaternions, there are three different typesof QFT: left-sided, right-sided and two-sided QFT. Hitzer [3] introduced these differenttypes of QFT and investigated their important properties. In [8]
Sangwine & A.Ell used the QFT to proceed with color image analysis. In [7] we see the implementation ofthe QFT to design a color image digital watermarking scheme. QFT is applied in [1] and[9] to image pre-processing and neural computing techniques for speech recognition.In [2] (Theorems 84 & 85),
Titchmarsh proved two results, for the classical FourierTransform given by: F ( f )( ξ ) = b f ( ξ ) = 1 √ π Z + ∞−∞ f ( x ) e − ixξ dx (cid:0) f ∈ L ( R ) (cid:1) Theorem 1.1.
Let f ∈ L p ( R ) (1 < p ≤ Z R | f ( x + h ) − f ( x − h ) | p dx = O ( h λp ) when h −→ with (0 < λ ≤ b f ∈ L θ ( R ) such that. p ( λ + 1) p − < θ ≤ pp − akim MONAIM & Said FAHLAOUI Theorem 1.2.
Let 0 < λ <
1. The following conditions are equivalents: i ) − f ∈ ( ϕ ∈ L ( R ) , (cid:18)Z R | ϕ ( x + h ) − ϕ ( x − h ) | dx (cid:19) = O ( | h | λ ) , h −→ ) ii ) − Z | ξ | >r | b f ( ξ ) | dξ = O ( r − λ ) , r −→ ∞ Notations 1.3.
We denote, for h, x ∈ R and p > h p = h p h p , x = ( x , x ) and dx = dx dx .
2. Algebra of Quaternions
Definition 2.1.
The algebra of quaternions is defined by: H = { q = q + iq + jq + kq ; q , q , q , q ∈ R } Where: i = j = k = ijk = − ij = − ji = k ; jk = − kj = i ; ki = − ik = j It’s a four-dimentional associative non-commutative algebra.
Definition 2.2.
Let q = q + iq + jq + kq ∈ H . The quaternionic conjugate& modulus of q are defined as follows: q = q − iq − jq − kq | q | = p qq = q q + q + q + q Properties 2.3.
We have the following results: pq = q p , p + q = p + q , p = p | pq | = | p || q |
3. Two-Sided Quaternionic Fourier Transform
Definition 3.1.
The two-sided QFT, for each f ∈ L ( R , H ), is defined on R as follows: F ( f )( ξ ) = b f ( ξ ) = 12 π Z R e − iξ x f ( x ) e − jξ x dx (1) Lemma 3.2. ( Inversion formula ) ([4] & [9]) If f & b f are both in L ( R , H ), theinversion formula is given by : f ( x ) = 12 π Z R e ix ξ b f ( ξ ) e jx ξ dξ Every quaternion-valued signal f : R −→ H , can be expressed as follows [4]: f = f + if + jf + kf = f + if + f j + if j For 1 ≤ p < ∞ . The L p − Norme of f ∈ L p ( R , H ) is defined by: || f || p = (cid:18)Z R | f ( x ) | p dx (cid:19) p W here | f ( x ) | = vuut X l =0 | f l ( x ) | ichmarsh Theorems associated with two-sided QFT Definition 3.3. If f = f + if + jf + kf ∈ L ( R , H ) .We define a new L -modulus of b f as follows [10]: | b f | Q = vuut X l =0 | b f l | (2)And a new L − norm of b f as follows [10]: || b f || ,Q = (cid:18)Z R | b f ( t ) | Q dt (cid:19) Where b f l = 12 π Z R e − iξ x f l ( x ) e − jξ x dx ( l ∈ { , , , } )Furthermore, we define a new L p − norm of b f as follows [4]: || b f || p,Q = (cid:18)Z R | b f ( t ) | pQ dt (cid:19) p = (cid:18)Z R (cid:16) | b f ( t ) | + | b f ( t ) | + | b f ( t ) | + | b f ( t ) | (cid:17) p dt (cid:19) p (3) Lemma 3.4. ( Plancherel ) ([4], [5] and [10]) If f ∈ L ( R , H ), then : || b f || ,Q = || f || Lemma 3.5. ( Hausdorff-Young ) ([4]) Let 1 ≤ p ≤ f ∈ L p ( R , H ) .Then b f ∈ L pp − ( R , H ) and : || b f || pp − ,Q ≤ || f || p Definition 3.6.
Let f ∈ L ( R , H ) and h ∈ R . We define the operator M h as follows: M h f ( x ) = f ( x + h ) + f ( x − h ) − f ( x + h , x − h ) − f ( x − h , x + h ) Definition 3.7.
Let 0 < λ <
The Cauchy-Lipschitz class is defined by: L ip ( λ,
2) = { f ∈ L ( R , H ) , ||M h f || = O ( h λ ) , h −→ }
4. Short-Time Two-Sided Quaternionic Fourier Transform
Definition 4.1.
Let g a function on R and t ∈ R . We define the following operators: T t g ( x ) = g ( x − t ) and g ∗ ( x ) = g ( − x ) = g ( − x ) Definition 4.2.
The convolution of two functions f and g (such that the function: t Z R f ( x ) g ( t − x ) dx is integrable) is the function f ∗ g defined as follows: f ∗ g ( t ) = Z R f ( x ) g ( t − x ) dx Lemma 4.3. [6] (Pages 80 and 81) Let p ≥ , f ∈ L p ( R , R ) and g ∈ L ( R , R ).Then f ∗ g ∈ L p ( R , R ) and: || f ∗ g || p ≤ || f || p . || g || akim MONAIM & Said FAHLAOUI Definition 4.4.
Fix a non zero window g = 0. For each t, w ∈ R , we define the Short-Time Two-Sided Quaternionic Fourier Transform of a function f , with respect to g, asfollowing: V g f ( t, w ) = Z R e − πiw x f ( x ) g ( x − t ) e − πjw x dt = Z R e − πiw x f T t g ( x ) e − πjw x dt Lemma 4.5.
Let g ∈ L p ( R , H ) and f ∈ L pp − ( R , H ) (0 < p ). Then, for all t, w ∈ R : V g f ( t, w ) = 2 π F ( f T t g )( w ) = 2 π \ ( f T t g )( w ) Proof.
While g ∈ L p ( R , H ) and f ∈ L pp − ( R , H ) , then h¨older’s inequality affirms: f T t g ∈ L ( R , H ) . The rest is given, abviously, by definitions 3.1 & 4.4. (cid:3)
5. Titchmarsh theorems associated with Two-Sided QFT
Theorem 5.1.
Let p ∈ ]1 , , λ ∈ ]0 ,
1] and f ∈ L p ( R , H ) such that: Z R |M h f ( x ) | p dx = O ( h λp ) when h −→ b f ∈ L θ ( R , H ) with: p ( λ + 1) p − < θ ≤ pp − Proof.
Let f = f + if + jf + kf with f l : R −→ R , ( l ∈ { , , , } ) .We have, for all h ∈ R + × R + : \ M h f l ( ξ ) = 2 π sin ( h ξ ) sin ( h ξ ) i b f l ( ξ ) j Then 2 π | sin ( h ξ ) sin ( h ξ ) || b f l ( ξ ) | = | \ M h f l ( ξ ) | And by (2), we get: 2 π | sin ( h ξ ) sin ( h ξ ) || b f ( ξ ) | Q = | [ M h f ( ξ ) | Q The lemma 3.5 & the assertion (4) yield: || [ M h f || pp − ,Q ≤ ||M h f || p Z R | π sin ( h ξ ) sin ( h ξ ) | pp − | b f ( ξ ) | pp − Q dξ ≤ (cid:18)Z R |M h f ( x ) | p dx (cid:19) p − ≤ O (cid:16) h λ pp − (cid:17) (5)While | sin ( t ) | ≥ t on [0 ,
1] . Then | sin ( ξ n h n ) | ≥ ξ n h n for all ξ n ∈ [0 , h n ] , ( n ∈ { , } ).Thus Z h Z h | π ξ h ξ h | pp − | b f ( ξ ) | pp − Q dξ ≤ Z h Z h | π sin ( h ξ ) sin ( h ξ ) | pp − | b f ( ξ ) | pp − Q dξ ichmarsh Theorems associated with two-sided QFT Using (5), we get Z h Z h | ξ ξ | pp − | b f ( ξ ) | pp − Q dξ = O (cid:16) h ( λ − pp − (cid:17) (6)Let’s define, (for θ ≤ pp − and t , t ≥
1) , the function: ψ ( t ) = Z t Z t | ξ ξ | θ | b f ( ξ ) | θQ dξ The H¨older’s inequality gives: ψ ( t ) ≤ (cid:18)Z t Z t | ξ ξ | pp − | b f ( ξ ) | pp − Q dξ (cid:19) θ ( p − p (cid:18)Z t Z t dξ (cid:19) θ (1 − p ) p This last inequality, combined with (6), gives: ψ ( t ) = O (cid:16) t − λθ + θp (cid:17) (7)We notice that (on ( R ∗ + ) ) : | b f ( ξ ) | θQ = ξ − θ ∂ ∂ξ ∂ξ ψ ( ξ )By Fubini’s theorem and integration by parts, with respect to ξ , we obtain: Z t Z t | b f ( ξ ) | θQ dξ = Z t Z t ξ − θ ∂ ∂ξ ∂ξ ψ ( ξ ) dξ = Z t ξ − θ (cid:18)Z t ξ − θ ∂∂ξ ∂∂ξ ψ ( ξ , ξ ) dξ (cid:19) dξ = Z t ξ − θ (cid:18) t − θ ∂∂ξ ψ ( t , ξ ) + θ Z t ξ − θ − ∂∂ξ ψ ( ξ , ξ ) dξ (cid:19) dξ = Z t ξ − θ t − θ ∂∂ξ ψ ( t , ξ ) dξ + θ Z t Z t ξ − θ − ∂∂ξ ψ ( ξ , ξ ) dξ dξ The first integral (by parts with respect to ξ ) and (7) give: Z t t − θ ξ − θ ∂∂ξ ψ ( t , ξ ) dξ = t − θ t − θ ψ ( t , t ) + θ Z t t − θ ξ − θ − ψ ( t , ξ ) dξ = O (cid:16) t − θ − λθ + θp (cid:17) + θt − θ − λθ + θp O (cid:18)Z t ξ − θ − λθ + θp dξ (cid:19) = O (cid:16) t − θ − λθ + θp (cid:17) (8)The second (by parts with respect to ξ ) and (7) give also: Z t ξ − θ θ Z t ξ − θ − ∂∂ξ ψ ( ξ , ξ ) dξ dξ = θ Z t ξ − θ − Z t ξ − θ ∂∂ξ ψ ( ξ , ξ ) dξ dξ = θ Z t ξ − θ − (cid:18)Z t ξ − θ ∂∂ξ ψ ( ξ , ξ ) dξ (cid:19) dξ = θ Z t ξ − θ − (cid:18) t − θ ψ ( ξ , t ) + θ Z t ξ − θ − ψ ( ξ , ξ ) dξ (cid:19) dξ = O (cid:18) t − θ t − λθ + θp Z t ξ − θ − ξ − λθ + θp dξ (cid:19) + O (cid:18)Z t Z t ξ − θ − ξ − λθ + θp dξ (cid:19) = O (cid:16) t − θ − λθ + θp (cid:17) (9) akim MONAIM & Said FAHLAOUI (8) and (9) give Z t Z t | b f ( ξ ) | θQ dξ = O (cid:16) t − θ − λθ + θp (cid:17) The monomial t − θ − λθ − θ can be bounded, on [1 , + ∞ ) × [1 , + ∞ ) , if and only if:1 − θ − λθ + θp < p ( λ + 1) p − < θ On the domain [1 , + ∞ ) × [0 ,
1] (Similarly on [0 , × [1 , + ∞ ) . )Let’s define the function: ψ ( t , ξ ) = Z t | ξ | θ | b f ( ξ ) | θ dξ If we take ( h = 1) in (6), we obtain: Z Z t | ξ | pp − | b f ( ξ ) | pp − Q dξ = O (cid:16) t (1 − λ ) pp − (cid:17) (10)By h¨older inequality, we get: Z ψ ( t , ξ ) dξ ≤ (cid:18)Z Z t | ξ | pp − | b f ( ξ ) | pp − Q dξ (cid:19) θ ( p − p (cid:18)Z t dξ (cid:19) θ (1 − p ) p From (10) we conclude: Z ψ ( t , ξ ) dξ = O (cid:18) t − λθ + θp (cid:19) (11)We notice that (on ( R ∗ + ) ) : | b f ( ξ ) | θQ = ξ − θ ∂∂ξ ψ ( ξ , ξ )Fubini’s theorem, an integration by parts (with respect to ξ ) and (11) give: Z t Z | b f ( ξ ) | θQ dξ = Z t Z ξ − θ ∂∂ξ ψ ( ξ ) dξ = Z (cid:18)Z t ξ − θ ∂∂ξ ψ ( ξ , ξ ) dξ (cid:19) dξ = Z (cid:18) t − θ ψ ( t , ξ ) + θ Z t ξ − θ − ψ ( ξ , ξ ) dξ (cid:19) dξ = t − θ Z ψ ( t , ξ ) dξ + θ Z t ξ − θ − Z ψ ( ξ , ξ ) dξ dξ = O (cid:18) t − θ − λθ + θp (cid:19) Again, this can be bounded if only if:1 − θ − λθ + θp < ichmarsh Theorems associated with two-sided QFT Yields: p ( λ + 1) p − < θ By the same way, the integrals of b f on the 2 nd , 3 th and 4 th quadrants are bounded.Finally: b f ∈ L θ ( R ) with p ( λ + 1) p − < θ ≤ pp − (cid:3) Corollary 5.2.
Fix a non-zero window g ∈ L ( R , H ) ∩ L ∞ ( R , H ). Let 1 < p ≤ < λ ≤
1. For all f ∈ L p ( R p , H ) , such that: Z R |M h f T t g ( x ) | p dx = O ( h λp ) when h −→ i- ∀ t ∈ R , V g f ( t, . ) ∈ L θ ( R , H ) with p ( λ +1) p − < θ ≤ pp − . ii- V g f ∈ L ( R , H ), and: ||V g f || ≤ π || g || . || f || Proof. i-
We have g ∈ L ∞ ( R , H ) and f ∈ L p ( R , H ) , Then we get: ∀ t ∈ R , f T t g ∈ L p ( R , H )The lemma 4.5 yields: F ( f T t g )( w ) = [ f T t g ( w ) = 12 π V g f ( t, w )And by theorem 5.1 above, we obtain: V g f ( t, . ) ∈ L θ ( R , H ) with p ( λ + 1) p − < θ ≤ pp − ii- The lemma 3.4 gives: 1(2 π ) Z R |V g f ( t, w ) | Q dw = Z R | f T t g ( x ) | dx The definitions 4.1 & 4.2 give us: Z R | f T t g ( x ) | dx = | f | ∗ | g ∗ | ( t )And the lemma 4.3 affirms: Z R | f | ∗ | g ∗ | ( t ) dt ≤ || ( | g ∗ | ) || . || ( | f | ) || Then: 1(2 π ) Z R Z R |V g f ( t, w ) | Q dwdt ≤ || g ∗ || . || f || = || g || . || f || (12)Finally: V g f ∈ L ( R , H ) and ||V g f || ≤ π || g || . || f || akim MONAIM & Said FAHLAOUI (cid:3) Remark 5.3.
The inequality (12) is satisfied independently of the theorem 5.1.If f, g ∈ L ( R , H ) , then V g f ∈ L ( R , H ) and: ||V g f || ,Q ≤ π || g || . || f || (cid:3) Theorem 5.4.
Let f ∈ L ( R , H ) . Then the following conditions are equivalents. i- f ∈ L ip ( λ, ii- Z R \B (0 , || r || ) | b f ( ξ ) | Q dξ = O ( r − λ ) , r −→ ∞ Proof.
Let f ∈ L ip ( λ,
2) , then: ||M h f || = O ( h λ ) , h −→ Plancherel formula || [ M h f || ,Q = ||M h f || = O ( h λ )We know that ∃ C > C Z h h Z h h | b f ( ξ ) | Q dξ ≤ Z R | π sin ( ξ h ) sin ( ξ h ) b f ( ξ ) | Q dξ Then: Z h h Z h h | b f ( ξ ) | Q dξ = O ( h λ ) (13)Which gives for r , r > Z ∞ r Z ∞ r | b f ( ξ ) | Q dξ = (cid:18)Z r r Z r r + Z r r Z r r + Z r r Z r r ... (cid:19) | b f ( ξ ) | Q dξ = O (cid:0) r − λ r − λ + 2 r − λ r − λ + 2 r − λ r − λ + 4 r − λ r − λ + ... (cid:1) = O (cid:0) r − λ (cid:1) (14)On the domains [ r , + ∞ [ × [0 , r ] , (The same way on [0 , r ] × [ r , + ∞ [).For some positive constant C > C Z h h Z | ξ h b f ( ξ ) | Q dξ ≤ Z R | π sin ( ξ h ) sin ( ξ h ) b f ( ξ ) | Q dξ Then: Z h h Z | ξ b f ( ξ ) | Q dξ = O ( h λ )The mean value theorem gives: Z h h Z | b f ( ξ ) | Q dξ = O ( h λ ) (15) ichmarsh Theorems associated with two-sided QFT Which means, for r > Z ∞ r Z | b f ( ξ ) | Q dξ = (cid:18)Z r r Z + Z r r Z ... (cid:19) | b f ( ξ ) | Q dξ = O (cid:0) r − λ + 2 r − λ + ... (cid:1) = O (cid:0) r − λ (cid:1) And similarly on the other quadrants. Which means: Z R \B (0 , || r || ) | b f ( ξ ) | Q dξ = O (cid:0) r − λ (cid:1) Then ii follows from i .On the other hand, if: Z R \B (0 , || r || ) | b f ( ξ ) | Q dξ = O ( r − λ ) , r −→ ∞ (16)Let’s define the function, for t , t > ψ ( t ) = Z ∞ t Z ∞ t | b f ( ξ ) | Q dξ By Fubini’s theorem, we get: Z r Z r ξ | b f ( ξ ) | Q dξ = Z r ξ (cid:18)Z r ξ ∂∂ξ ∂∂ξ ψ ( ξ , ξ ) dξ (cid:19) dξ Using an integration by parts, we obtain: Z r ξ ∂∂ξ ∂∂ξ ψ ( ξ , ξ ) dξ = r ∂∂ξ ψ ( r , ξ ) − Z r ξ ∂∂ξ ψ ( ξ , ξ ) dξ And Z r r ξ ∂∂ξ ψ ( r , ξ ) dξ = r r ψ ( r , r ) − Z r r ξ ψ ( r , ξ ) dξ (17)Also2 Z r ξ Z r ξ ∂∂ξ ψ ( ξ , ξ ) dξ dξ = 2 Z r ξ Z r ξ ∂∂ξ ψ ( ξ , ξ ) dξ dξ = 2 Z r ξ (cid:18)Z r ξ ∂∂ξ ψ ( ξ , ξ ) dξ (cid:19) dξ = 2 Z r ξ (cid:18) r ψ ( ξ , r ) − Z r ξ ψ ( ξ , ξ ) dξ (cid:19) dξ = 2 r Z r ξ ψ ( ξ , r ) dξ − Z r Z r ξ ξ ψ ( ξ , ξ ) dξ dξ (18)Combining (17) & (18), we obtain: Z r Z r ξ | b f ( ξ ) | Q dξ = r r ψ ( r , r ) − r Z r ξ ψ ( r , ξ ) dξ − r Z r ξ ψ ( ξ , r ) dξ + 4 Z r Z r ξ ξ ψ ( ξ , ξ ) dξ dξ akim MONAIM & Said FAHLAOUI Hence Z r Z r ξ | b f ( ξ ) | Q dξ ≤ r r ψ ( r , r ) + 4 Z r Z r ξ ξ ψ ( ξ , ξ ) dξ dξ = O ( r − λ )Then Z B (0 , || r || ) ξ | b f ( ξ ) | Q dξ = O ( r − λ ) (19)(16) & (19) give: Z R π sin ( ξ h ) sin ( ξ h ) | b f ( ξ ) | Q dξ = O h Z B (0 , || ( h , h ) || ) | b f ( ξ ) | Q dξ ! + O Z R \B (0 , || ( h , h ) || ) | b f ( ξ ) | Q dξ ! = O ( h λ ) (20)By lemma 3.4 & (20) we get: ||M h f || = || [ M h f || Q = O ( h λ )In other words: f ∈ L ip ( λ,
2) . And i now follows from ii . (cid:3) Remark 5.5.
Let f ∈ L ip ( λ,
2) . Then b f ∈ L θ ( R , H ) such that.22 λ + 1 < θ ≤ Proof.
Let θ ≤ r , r > Z r r Z r r | b f ( ξ ) | θQ dξ ≤ (cid:18)Z r r Z r r | b f ( ξ ) | Q dξ (cid:19) θ (cid:18)Z r r Z r r dξ (cid:19) − θ (21)From (13) and (21), we get: Z ∞ r Z ∞ r | b f ( ξ ) | θQ dξ = O ( r − λθ ) O ( r − θ ) = O ( r − λθ − θ )The monomial r − λθ − θ can be bounded if and only if 1 − λθ − θ < λ +1 < θ .On [ r , + ∞ [ × [0 , , ( By the same way on [0 , × [ r , + ∞ [ ).By H¨older’s inequality, we obtain: Z r r Z | b f ( ξ ) | θQ dξ ≤ (cid:18)Z r r Z | b f ( ξ ) | Q dξ (cid:19) θ (cid:18)Z r r dξ (cid:19) − θ This last inequality and (15) affirm: Z ∞ r Z | b f ( ξ ) | θQ dξ = O ( r − λθ ) O ( r − θ ) = O ( r − λθ − θ ) ichmarsh Theorems associated with two-sided QFT REFERENCES
Again, this can be bounded if and only if 1 − λθ − θ < λ +1 < θ .And similarly for the other quadrants. It follows b f ∈ L θ ( R , H ) such that.22 λ + 1 < θ ≤ (cid:3) Conclusion
This Paper presented a proof of the Titchmarsh theorems in the two-sided Quater-nionic Fourier Transform. The results obtained can extend the qualitative property pre-sented by Titchmarsh to the algebra of quaternions. We have concluded about short-timetwo-sided Quaternionic Fourier Transform. This research can be extended to new excitingresearch areas of harmonic analysis.
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EFERENCES
Hakim MONAIM & Said FAHLAOUI
Hakim MONAIM
Harmonic Analysis & Probabilities team.Department of Mathematics.Faculty of Sciences.Moulay Ismail University.BP 11201 Zitoune, Meknes, Morocco.e-mail: [email protected]
Said FAHLAOUI