Properties of Quaternion Fourier Transforms

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1 Properties of Quaternion Fourier Transforms Dong Cheng and Kit Ian Kou Department of Mathematics, Faculty of Science and Technology, University of Macau, Macao, China arxiv: v [math.ca] 8 Jun 06 Abstract With the recent popularity of quaternion and quaternion Fourier transforms (QFTs technique in physics and engineering applications, there tends to be an inordinate degree of interest placed on the properties of QFTs. Due to the noncommutativity of multiplication of quaternions, there are different types of QFTs and we focus on the right-sided QFT (QFT and two-sided QFT (SQFT. In this paper, we begin by considering the behavior of the QFT on the space L (,H. Theenergy-preservedpropertydeducestotheextension ofqftonl (,H. We are largely exploiting the relation between QFT and SQFT and some elementary properties of QFT can be transformed to SQFT. As a generalization of QFTs, the properties of quaternion linear canonical transforms (QLCTs are also studied on the basis of existing properties of QFTs. Keywords: quaternion Fourier transforms; quaternion linear canonical transforms; inversion theorem; Plancherel theorem 000 Mathematics Subject Classification: 4A38, 4B0, 43A3, 43A50 Introduction The quaternion Fourier transforms (QFTs play a vital role in the representation of (hypercomplex signals. They transform real (or quaternion-valued D signals into a quaternion-valued frequency domain signals. The four components of the QFT separate four cases of symmetry into real signals instead of only two as in the complex Fourier Transform (FT. In [6, 8] authors used the QFT to proceed color image analysis. The study [, 7] implemented the QFT to design a color image digital watermarking scheme. chengdong70@63.com kikou@umac.mo

2 esearchers in [3] applied the QFT to image pre-processing and neural computing techniques for speech recognition. ecently, certain asymptotic properties of the QFT were analyzed and a straightforward generalization of the classical Bochner-Minlos theorem to the framework of quaternion analysis was derived []. In classical Fourier theory, if f is integrable (L (,C, then the Fourier transform f(ξ is well-deمحned by f(ξ = f(te iξt dt. (. Moreover, if f is also integrable, then f(t = f(ξe iξt dξ for almost every t. The integral (. deمحning the Fourier transform is not suitable for square integrable function (L (,C. But There is a natural and elegant theory of Fourier transform for square integrable functions. In fact, the Fourier transform F on L (,C is not only isometric but also onto. This desirable property is very important in signal processing. One may naturally asks what are the analogous results for QFTs of quaternion-valued signals? The previous contributions on inversion theorem and energy-preserved property of QFTs were developed in [6, ]. On one hand, however, the existing results are not well established systematically. On the other hand, the prerequisites for setting up of the established theorem did not be studied completely. Therefore it is of great interest to progress with a function theory of QFT. In light of this, the inversion theorem on L (,H and Plancherel theorem of QFTs are investigated thoroughly in this paper. Moreover, as a generalization of QFTs, the properties of quaternion linear canonical transforms (QLCTs are also developed combining with the existing properties of QFTs. The rest of the paper is organized as follows. In the next section, we recall some basic knowledge of quaternion algebra. Section 3 investigates the inversion theorem on L (,H and Plancherel theorem of right-sided QFT. In section 4, we establish the relation between QFT and SQFT and study the elementary properties of SQFT. In section 5, the properties of quaternion linear canonical transforms (QLCTs are studied on the basis of existing properties of QFTs. Preliminaries. Quaternion algebra Throughout the paper, let H := {q = q 0 +iq +jq +kq 3 q 0,q,q,q 3 }, be the Hamiltonian skew field of quaternions, where the elements i, j and k obey the Hamiltonلاs multiplication rules: ij = ji = k, jk = kj = i, ki = ik = j, i = j = ijk =.

3 For every quaternion q = q 0 + q, q = iq + jq + kq 3, the scalar and vector parts of q, are deمحned as Sc(q = q 0 and Vec(q = q, respectively. If q = Vec(q, then q is called pure imaginary quaternion. The quaternion conjugate is deمحned by q = q 0 q = q 0 iq jq kq 3, and the norm q of q deمحned as q = qq = qq = m=3 m=0 q m. Then we have q = q, p+q = p+q, pq = q p, pq = p q, p,q H. Using the conjugate and norm of q, one can deمحne the inverse of q H\{0} by q = q/ q. The multiplication of two quaternions is noncommutative, but Sc(pq = Sc(qp p,q H. (. The quaternion exponential function e q is deمحned by means of an inمحnite series as e q := Analogous to the complex case one may derive a closed-form representation: n=0 q n n!. e q = e q 0 (cos q + q q sin q. The quaternion has subsets C µ := {a+bµ : a,b, µ =,µ = Vec(µ}. For each complex. xedمح unit pure imaginary quaternion µ, C µ is isomorphic to the. Quaternion module L p (,H The left quaternion module L p (,H(p =, consists of all H-valued functions whose pth power is Lebesgue integrable on : } L p (,H := {f f : H, f plp(,h := f(x,x p dx dx <. X Like the complex case, the left quaternion module L (,H consists of all essentially bounded measurable H-valued functions on. The left quaternionic inner product of f,g L (,H is deمحned as < f,g > L (,H= f(x,x g(x,x dx dx. This inner product leads to a scalar norm f =< f,f > L (X,H which coincides with the usual L -norm for f, considered as a vector-valued function. In fact, L (,H is a left quaternionic Hilbert space with inner product <, > L (,H (see [5]. For simplicity of notation, we use <, > in place of <, > L (,H in the following. It is not difficult to verify that < pf,qg >= p < f,g > q (. 3

4 In fact, f = f 0 + if + jf + kf 3 L p (,H(p =,, if and only if f m (m = 0,,,3 L p (,. Then we can consider every f L p (,H(p =,, as a linear combination of real (complex L p functions. Therefore, some properties of real (complex L p functions can be natural extended to quaternionic L p functions. The corresponding complex versions of following results can be found in [5, 7]. Proposition. If {f n } is a cauchy sequence in L p (,H(p =,,, with limit f, then {f n } has a subsequence which converges pointwise almost every (x,x to f. Proposition. Let p ǫ (x,x = ǫ be the Poisson kernel, then π (ǫ +x (ǫ +x. lim ǫ 0 f p ǫ f p = 0 for every f L p (,H(p =,;. lim ǫ 0 (f p ǫ (x,x = f(x,x, if f(x,x L (,H is continuous at a point (x,x. 3 The right-sided quaternion Fourier transform The quaternion Fourier transform (QFT was rstمح deمحned by Ell to analyze linear timeinvariant systems of partial differential equations [9]. Later, some constructive works related to the QFT and its application to color image processing were studied (see [6, 4, 6,, 8, ]. The non-commutativity of the quaternion multiplication leads to different types of quaternion Fourier transformations (see [0]. In this section, we consider the right-sided quaternion Fourier transform. 3. Theright-sided quaternion Fourier transform pairs in L (,H Definition 3. (QFT For every f L (,H, the factored, right-sided quaternion Fourier transform of f is defined by (F r f(ω,ω := f(x,x e iω x dx dx. π The following results are related to our study of the inversion theorem and Plancherel theorem of QFT. We use the integral representations to express the convolutions. Proposition 3. If f L (,H, put f(x,x := f( x, x and define g(x,x := ( f f(x,x = f(y,y f(x +y,x +y dy dy, then (f p ǫ (x,x = P(ǫω,ǫω (F r f(ω,ω e jω x e iω x dω dω π and Sc((g p ǫ (0,0 = P(ǫω,ǫω (F r f(ω,ω dω dω, where P(ω,ω = e ω ω. 4

5 Proof. Since p ǫ (x y,x y = π ǫ e ǫω e iω y e iω x dω ǫ +(x y = e ǫω e iω ǫ y x dω π ǫ +(x y eiω = e ǫω e iω y e ǫω e jω (x y dω 4π e iω x dω = P(ǫω 4π,ǫω e iω y e jω y e jω x e iω x dω dω, then Then (f p ǫ (x,x = dy dy f(y,y P(ǫω 4π,ǫω e iω y e jω y e jω x e iω x dω dω = P(ǫω 4π,ǫω dω dω f(y,y e iω y e jω y e jω x e iω x dy dy = P(ǫω,ǫω (F r f(ω,ω e jω x e iω x dω dω. π Note that g(x,x = ( f f(x,x = f( y, y f(x y,x y dy dy = f(y,y f(x +y,x +y dy dy. Sc((g p ǫ (0,0 ( = Sc g(y,y p ǫ ( y, y dy dy ( [ = Sc f(s,s f(s +y,s +y ds ds ]p ǫ ( y, y dy dy ( = Sc ds ds f(s,s f(s +y,s +y p ǫ ( y, y dy dy ( = Sc ds ds f(s,s f(z,z p ǫ (s z,s z dz dz ( = Sc ds 4π ds f(s,s f(z,z dz dz P(ǫω,ǫω e iω z e jω z e jω s e iω s dω dω ( 6 = Sc dω 4π dω P(ǫω,ǫω ds ds e jω s e iω s f(s,s f(z,z e iω z e jω z dz dz 6 = P(ǫω,ǫω (F r f(ω,ω dω dω 5

6 which completes the proof. To لاauthors knowledge, the inversions of QFT have not been investigated for absolutely integrable quaternion-valued function thoroughly. By Proposition.,. and 3., we give an inversion theorem about QFT as follows. Theorem 3.3 (Inversion of QFT If f,f r f L (,H and g(x,x = (F r f(ω,ω e jω x e iω x dω dω, π then f(x,x = g(x,x for almost every (x,x. Proof. Form Proposition 3., (f p ǫ (x,x = π P(ǫω,ǫω (F r f(ω,ω e jω x e iω x dω dω. (3. The integrands on the right side of (3. are bounded by (F r f(ω,ω, and since P(ǫω,ǫω as ǫ 0, then π P(ǫω,ǫω (F r f(ω,ω e jω x e iω x π (F rf(ω,ω e jω x e iω x as ǫ 0. Hence, the right side of (3. converges to g(x,x, for every (x,x, by the dominated convergence theorem. From Proposition. and., we see that that there is a sequence {ǫ n } such that ǫ n 0 and lim n (f p ǫn (x,x = f(x,x, for almost every (x,x. Hence f(x,x = g(x,x for almost every (x,x. If f L (,H, Theorem 3.3 indicates that F r f = 0 for almost every (ω,ω implies f = 0 for almost every (x,x. Corollary 3.4 (Uniqueness of QFT If f,g L (,H and(f r f(ω,ω = (F r g(ω,ω for almost every (ω,ω, then f(x,x = g(x,x for almost every (x,x. Now we see that, under suitable conditions, the original signal f can be reconstructed from F r f by the inverse right-sided quaternion Fourier transform (IQFT. Definition 3.5 (IQFT For every F r L (,H, the inverse right-sided quaternion Fourier transform of F r is defined by (Fr F(x,x := F r (ω,ω e jω x e iω x dω dω. π emark 3.6 Both F r and Fr are bounded linear transformation from L (,H into L (,H. We will see later, both F r L L and F r L L can be extended to L (,H. As an operator on L (,H, Fr is the inversion of F r. 6

7 Next we present some properties of inverse QFT. Proposition 3.7 Suppose that F r L (,H.. If G r (ω,ω = F r (ω,ω iω L (,H, then the partial derivative of F with respect to x exists and (F r F r (x, x = (Fr G r (x,x. x r F r. If G r (ω,ω = F r (ω,ω jω L (,H, then the partial derivative of F r F r with respect to x exists and (F F r x (x,x = (F r G r (x,x. 3. If G 3r (ω,ω = F r (ω,ω kω ω L (,H, then (F r F r x x (x, x = (F r G 3r (x,x. An important result called the multiplication formula in classical Fourier analysis can be generalized to QFT. Before to give the formula, we introduce an auxiliary transform for f(x,x = f 0 (x,x + if (x,x + jf (x,x + kf 3 (x,x which is deمحned by αf(x,x := f 0 (x,x +if (x, x +jf ( x,x +kf 3 ( x, x. Then we obtain the following result. Proposition 3.8 Suppose that f,g L (,H, h := αg, H r := F r h and F r := F r f, then F r (x,x g(x,x dx dx = f(x,x H r (x,x dx dx. (3. Moreover, if g is contained in L (,H, we have g = h. Proof. Write g = g 0 +ig +jg +kg 3, then e iω x g(ω,ω dω dω = g 0 (ω,ω e iω x dω dω + ig (ω,ω e iω x e jω x dω dω + jg (ω,ω e iω x dω dω + kg 3 (ω,ω e iω x e jω x dω dω = g 0 (ω,ω e iω x dω dω + ig (ω, ω e iω x dω dω + jg ( ω,ω e iω x dω dω + kg 3 ( ω, ω e iω x dω dω = h(ω,ω e iω x dω dω = H r (x,x. 7

8 Applying Fubiniلاs theorem, we have F r (ω,ω g(ω,ω dω dω ( = f(x,x e iω x dx dx g(ω,ω dω dω ( = f(x,x e iω x g(ω,ω dω dω dx dx = f(x,x H r (x,x dx dx. If g L (,H, it is easy to verify that g = h by deمحnition of α. 3. The Plancherel theorem of QFT In the complex case, the Plancherel theorem indicates that if f L L, and it turns out that f L and f = f, where f is the classical Fourier transform of f. Moreover, this isometry of L L into L extends to an isometry of L onto L, and this extension deمحnes the Fourier transform of every f L. The convolution theorem plays a vital role in proving the Plancherel theorem (see [5, 7]. However, the classical convolution theorem no longer holds for the QFT. The Plancherel theorem of QFT was discussed in recent research papers (see [6],[],[]. We give a restatement of Plancherel theorem here since the prerequisites for setting up of the theorem may not be put forward so clearly in recent research papers. It is probably worth pointing out that Proposition 3. plays a key role in our proof. Theorem 3.9 If f L (,H L (,H, then F r f L (,H and Parseval s identity F r f = f holds. Proof. We xمح f L (,H L (,H, put f(x,x := f( x, x and deمحne g(x,x := ( f f(x,x =< f,f x, x >= f(y,y f(x +y,x +y dy dy, where f x, x (y,y denotes a translate of f. Since (x,x f x, x is a continuous mapping of into L (,H and noticing that the continuity of the inner product, we see that g(x,x is a continuous function. By Cauchy-Schwarz inequality, g(x,x f x, x f = f. Thus g is bounded. Furthermore, g L (,H since f L (,H and f L (,H. Since g is continuous and bounded, Proposition. shows that limsc((g p ǫ (0,0 = Sc(g(0,0 = f ǫ 0. 8

9 On the other hand, since g L (,H, by Proposition 3.: Sc((g p ǫ (0,0 = P(ǫω,ǫω (F r f(ω,ω dω dω. Since0 P(ǫω,ǫω (F r f(ω,ω increases to (F r f(ω,ω asǫ 0, themonotone convergence theorem gives limsc((g p ǫ (0,0 = lim P(ǫω,ǫω (F r f(ω,ω dω dω ǫ 0 = limp(ǫω,ǫω (F r f(ω,ω dω dω ǫ 0 = (F r f(ω,ω dω dω = F r f. ǫ 0 Therefore, F r f L (,H and F r f = f. From Theorem 3.9, F r L L is an isometry of L (,H L (,H into L (,H. Since L (,H L (,H is a dense subset of L (,H. Therefore, there exists a unique bounded (continuous extension, Ψ r, of F r L L to all of L (,H. If f L (,H, F r = Ψ r f is deمحned by the L limit of the sequence {F r f k }, where {f k } is any sequence in L (,H L (,H converging to f in the L norm. If we choose f k (x,x = f(x,x χ [ k,k] (x,x, we have F r (ω,ω = l.i.m. k f(x,x e ix ω e jx ω dx dx. π [ k,k] where f = l.i.m. k f k means f f k 0 as k. We call F r = Ψ r f is the QFT of f on L (,H. The multiplication formula (3. easily extends to L (,H. The left H-linear operator Ψ r on L (,H is an isometry. So Ψ r is a one-to-one mapping. Moreover, we can show that Ψ r is onto. Theorem 3.0 The QFT Ψ r is a unitary operator on L (,H. Proof. Firstly, we show that the range of Ψ r denoted by (Ψ r is a closed subspace of L (,H. Let {F k } be a sequence in (Ψ r converging to F r in L norm sense. we now prove that F r (Ψ r. Suppose that Ψ r f k = F k. Then the isometric property shows that Ψ r is continuous and {f k } is also a Cauchy sequence. The completeness of L (,H implies that {f k } converges to some f L (,H, and the continuity of Ψ r shows that Ψ r f = l.i.m. k Ψ rf k = F r. If (Ψ r were not all of L (,H, as every closed subspace of Hilbert space L (,H has an orthogonal complement, we could ndمح a function u such that F r (x,x u(x,x dx dx = 0 9

10 for all f L (,H and u 0. Let g = u,h = αg, by multiplication formula, f(x,x H r (x,x dx dx = F r (x,x g(x,x dx dx = 0 for all f L (,H. Pick f = H r, this implies that H r (x,x = 0 for almost every (x,x, contradicting the fact that H r = h = g = u 0. Next result shows that the mapping Ψ r is a Hilbert space isomorphism of L (,H, that is, preserving inner product. Theorem 3. Let f,g L (,H and F r = Ψ r f,g r = Ψ r g. Then f(x,x g(x,x dx dx = F r (ω,ω G r (ω,ω dω dω. Proof. Let p 0 +ip +jp +kp 3 = f(x,x g(x,x dx dx and q 0 +iq +jq +kq 3 = F r (ω,ω G r (ω,ω dω dω. From the Parsevalلاs identity, we have f +g = f + g +p 0 = F r +G r = F r + G r +q 0. Thus p 0 = q 0. By using (. and applying Parsevalلاs identity to f+ig = F r+ig r, f +jg = F r +jg r, f +kg = F r +kg r respectively, we can get p m = q m, (m =,,3 which completes the proof. Theorem 3. The inverse f = Ψ r F r is the L limit of the sequence {Fr F rk } k, where {F rk } k is any sequence in L (,H L (,H converging to F r in the L norm. If we choose F rk = F r χ [ k,k], we have f(x,x = l.i.m. k F r (ω,ω e jx ω e ix ω dω dω. π [ k,k] In particular, if F r L (,H L (,H, then f(x,x = F r (ω,ω e jx ω e ix ω dω dω. π 0

11 Proof. For Ψ r B(L (,H, the quaternionic iesz representation theorem (see [5] guarantees that there exists a unique operator Ψ r B(L (,H, which is called the adjoint of Ψ r, such that for all f,g L (,H, (Ψ r f,g = (f,ψ r g. Since Ψ r is unitary, then Ψ r = Ψ r. For any xedمح F r L (,H, let {F rk } k be an arbitrary sequence in L (,H L (,H converging to F r in the L norm, we have < g,ψ r F r > =< G r,f r > = lim < G r,f rk > k ( = lim k ( = lim g(x,x k = lim < g,fr F rk >=< g,l.i.m. k k F r F rk > g(x,x e ix ω e jx ω dx dx F rk (ω,ω e jx ω e ix ω dω dω F rk (ω,ω dω dω dx dx for all g L (,H L (,H. Thus f = Ψ r F = Ψ rf r = l.i.m. k F r F rk. In particular, if F r L (,H L (,H, then which completes the proof. f(x,x = π F r (ω,ω e jx ω e ix ω dω dω emark 3.3 Since Ψ r (Ψ r coincides with F r (F r in L (,H L (,H. For simplicity of notations, in the following, by capital letter F r, we mean the QFT of f L (,H L (,H if no otherwise specified. 4 The two-sided quaternion Fourier transform In this section, we study the two-sided (sandwich quaternion Fourier transform (SQFT. 4. Thetwo-sided quaternion Fouriertransform pairs in L (,H Definition 4. (SQFT For every f L (,H, the factored, two-sided quaternion Fourier transform of f is defined as (F s f(ω,ω := e iω x f(x,x dx dx. π Unlike the QFT, SQFT is not a left H-linear operator. But SQFT is left C i -linear and right C j -linear. Moreover, SQFT could establish relationship with QFT through the following transform.

12 Definition 4. Iff(x,x = f 0 (x,x +if (x,x +jf (x,x +kf 3 (x,x L p (X,H(p =,, the transform β is defined by βf(x,x := f 0 (x,x +if (x,x +jf ( x,x +kf 3 ( x,x. Proposition 4.3 Suppose that f,g L p (X,H(p =,. Then the following assertions hold.. β is a left C i -linear bijection mapping on L p (X,H. So the inverse of β can be defined and β is actually equal to β itself.. If f L (,H then F s f = F r (βf. (4. Moreover, if f is C i -valued or f is even with respect to first variable, then F s f = F r f. 3. If f,g L (,H then Sc(< βf,βg > = Sc(< f,g >, Sc(i < βf,βg > = Sc(i < f,g >. In particular f = βf. Proof. The assertion ( is a direct consequence of the deمحnition of β. To prove (, write f in form of f 0 +if +jf +kf 3 and let h := βf. Then F s f(ω,ω = e iω x f(x,x dx dx = f 0 (x,x e iω x dx dx + if (x,x e iω x dx dx + jf (x,x e iω x dx dx + kf 3 (x,x e iω x ddx dx = f 0 (x,x e iω x dx dx + if (x,x e iω x dx dx + jf ( x,x e iω x dx dx + kf 3 ( x,x e iω x ddx dx = h(x,x e iω x dx dx = F r h(ω,ω. If f is C i -valued or f is even with respect to rstمح variable, then βf = f. It follows that F s f = F r (βf.

13 Now we prove (iii. If f,g L (,H, then Sc(< βf,βg > = f 0 (x,x g 0 (x,x dx dx + f (x,x g (x,x dx dx + f ( x,x g ( x,x dx dx + f 3 ( x,x g 3 ( x,x dx dx = f 0 (x,x g 0 (x,x dx dx + f (x,x g (x,x dx dx + f (x,x g (x,x dx dx + f 3 (x,x g 3 (x,x dx dx = Sc(< f,g >. Since β is left C i -linear, then Sc(i < f,g > = Sc(< if,g > = Sc(< β(if,g > = Sc(< iβf,g > = Sc(i < βf,βg >. At last which completes the proof. βf = f 0 + f + f + f 3 = f Theorem 4.4 (Inversion of SQFT If f,f s f L (,H and g(x,x = e iω x (F s f(ω,ω e jω x dω dω, (4. π then f(x,x = g(x,x for almost every (x,x. Proof. By invoking (ii of Proposition 4.3 we have F s f = F r (βf. Let h := Fr (F s f, then (βf(x,x = h(x,x for almost every (x,x by Theorem 3.3. To prove f(x,x = g(x,x for almost every (x,x, it suffices to verify βg = h. Note that h(x,x = (F s f(ω,ω e jω x e iω x dω dω. π It is easy to see that Sc(h(x,x = Sc(g(x,x by invoking equation (.. Similarly we have Sc(h(x,x i = Sc(g(x,x i. Since Then h(x,x j = π (F s f(ω,ω e jω x je iω x dω dω. ( Sc(h(x,x j = Sc (F s f(ω,ω e jω x je iω x dω dω π ( = Sc e iω x (F s f(ω,ω e jω x jdω dω π = Sc(g( x,x j. 3

14 Analogously, we have Sc(h(x,x k = Sc(g( x,x k. Then we conclude that βg = h which completes the proof. So we can deمحne the inverse two-sided quaternion Fourier transform by equation (4. or equivalently by β F r F s. Definition 4.5 (ISQFT For every F s L (,H, the inverse two-sided quaternion Fourier transform of F s is defined by (Fs F s (x,x := e iω x F s (ω,ω e jω x dω dω. π 4. The Plancherel theorem of SQFT In section 3., we extend F r L L to L (,H. The QFT on L (,H has more symmetry than QFT in L (,H. The relation F s f = F r (βf drives us to extend F s L L to L (,H. Definition 4.6 For every f L (,H, the SQFT of Ψ s f is defined by Ψ s f := Ψ r (βf (4.3 In fact, we can deمحne Ψ s starting from the original deمحnition 4. by taking L norm limit. (4.3 gives us a different but actually equivalent form of Ψ s. Theorem 4.7 Suppose that f,g s L (,H, Then the following assertions hold.. Ψ s f defined by (4.3 is equal to the L limit of the sequence {F s f k } k, where {f k } k is any sequence in L (,H L (,H converging to f in the L norm. If f L (,H L (,H, then Ψ s f = F s f.. Ψ s is a bijection on L (,H and Ψ s G s = β Ψ r G s. Furthermore, Ψ s G s is equal to the L limit of the sequence {Fs G sk } k, where {G sk } k is any sequence in L (,H L (,H converging to G s in the L norm. If G s L (,H L (,H, then Ψ s G s = Fs G s. Proof. The assertion (i is a consequence of (4.3 and deمحnition of Ψ r. The assertion (ii is a consequence of (4.3 and Theorem 3.. As an immediate consequence of Theorem 3. and Theorem 3., we present the following result. Theorem 4.8 If f,g L (,H, then < Ψ s f,g >=< βf,ψ r g >. HavingdeمحnedtheSQFTforfunctionsinL (,H, weobtainthefollowingparsevalلاs identity. 4

15 Theorem 4.9 Suppose that f,g L (,H, F s = Ψ s f,g s = Ψ s g. Let p 0 +ip +jp +kp 3 = f(x,x g(x,x dx dx and q 0 +iq +jq +kq 3 = F s (ω,ω G s (ω,ω dω dω. Then F s = f and p m = q m, (m = 0,. Moreover, if both f and g are C i -valued or even with respect to first variable, then p m = q m, (m = 0,,,3. Proof. Firstly, we show that Parsevalلاs identity of SQFT holds. Applying Parsevalلاs identity of QFT and (iii of Proposition 4.3, we have F s =< Ψ sf,ψ s f > =< Ψ r (βf,ψ r (βf > =< βf,βf > = βf = f. Byinvoking Parsevalلاs identity ofsqftto f+g = F s+g s, f+ig = F s+ig s respectively, we get p m = q m, (m = 0,. If both f and g are C i -valued or even with respect to rstمح variable, we have Ψ s f = Ψ r f and Ψ s g = Ψ r g, therefore < Ψ s f,ψ s g >=< Ψ r f,ψ r g >=< f,g >, that is p m = q m, (m = 0,,,3. emark 4.0 By applying (. Hitzer [] proved p 0 = q 0, it follows that F s = f. We have shown that p is also equal to q. One may be wondering what is the relationship between p and q (p 3 and q 3? At the moment, we do not know their relation. Since Ψ s is not left H-linear, so f + jg F s + jg s. In fact, f + gj = F s + G s j. It follows that Sc(< f,gj > = Sc(< F s,g s j > rather than Sc(< f,jg > = Sc(< F s,jg s >. However, Sc(< f,jg > = Sc(< F s,jg s > is equivalent to p = q by applying (.. 5 The quaternion linear canonical transform The linear canonical transform (LCT, as a generalization of the classical Fourier transform, has more degrees of freedom than the FT and the FFT, but with similar computation cost as the conventional FT [3]. In [4], the authors generalized the classical LCT to the quaternionic algebra and deمحned QLCTs. In this section, we will only focus on two-sided (sandwich QLCT and right-sided QLCT. 5

16 5. The SQLCT in L (,H and its extension to L (,H ( ai b i Definition 5. LetA i = be areal matrix parametersuch thatdet(a i = c i d i for i =,. The two-sided (sandwich QLCT of f L (,H is defined by K A i (x,ω f(x,x K j A (x,ω dx dx, b,b 0, d d (L s f(ω,ω := eic ω f(d ω,x K j A (x,ω dx, b = 0,b 0, d Ki A (x,ω f(x,d ω e jc d ω dx, b 0,b = 0, d d e ic d ω f(d ω,d ω e jc d ω, b = b = 0. where and K i A (x,ω := K j A (x,ω := iπb e i( a b x b x ω + d b w, for b 0 (5. jπb e j( a b x b x ω + d b w, for b 0. (5. We only consider the case of b b 0. If b i 0 for i =,, then ( ( ai b i di b = i. (5.3 c i d i c i a i Lemma 5. Let f L (,H and h(x,x = e i a x b f(x,x e j a x b, then (L s f(ω,ω = e i d w ω b (F s h(, ω e j d w b. (5.4 ib b b jb Theorem 5.3 (Inversion of SQLCT If f,l s f L (,H and g(x,x = K i A then f(x,x = g(x,x for almost every (x,x. Proof. Applying Lemma 5. and Theorem 4.4 we have f(x,x = e i a x b h(x,x e j a x b ( = e i a b x = π = K i A = g(x,x. e i (ω,x (L s f(ω,ω K j (ω A,x dω dω, (5.5 ω b x (F s h( ω e jω b x dω dω e j a b x, ω π b b b b e i( d w b +ω x b a x b (L s f(ω,ω e j( d ib jb (ω,x (L s f(ω,ω K j (ω A,x dω dω w b +ω x b a b x dω dω 6

17 The second equality holds for almost every (x,x and the last equality is a consequence of (5., (5. and (5.3. So we can deمحne the inverse SQLCT L s by right-side of (5.5. Furthermore, L s (L s can be extended to L (,H. Theorem 5.4 (SQLCT on L (,H Letf,g L (,H, h(x,x := e i a x b f(x,x e j a x b, define (Φ s f(ω,ω := e i d w b ω (Ψ s h(, ω e j d w b. ib b b jb Then the following assertions hold.. Φ s is a bijection on L (,H. Moreover, Φ s and Φ s coincide with L s and L s on L (,H L (,H respectively.. Parseval sidentity Φ s f = f holds. Furthermore, Sc(< f,g > = Sc(< Φ s f,φ s g > and Sc(i < f,g > = Sc(i < Φ s f,φ s g >. 5. The QLCT in L (,H and its extension to L (,H Now we turn our study to right-sided QLCT. ( ai b i Definition 5.5 LetA i = be areal matrix parametersuch thatdet(a i = c i d i for i =,. The right-sided QLCT of f L (,H is defined by f(x,x KA i (x,ω K j A (x,ω dx dx, b,b 0, d (L r f(ω,ω := f(d ω,x e ic d ω K j A (x,ω dx, b = 0,b 0, d f(x,d ω KA i (x,ω e jc d ω dx, b 0,b = 0, d d f(d ω,d ω e ic d ω e j c d ω, b = b = 0. We want to build relationship between QLCT and transforms deمحned in former sections so that previous results could be employed in this section. However, due to the quadratic term of kernel (5. and (5., itلاs an extravagant hope to have a good relation like (4. for L r and general H-valued functions f. The two-dimensional QLCT can be considered as a cascade of two one-dimensional transforms. For any xedمح unit pure imaginary quaternion µ, let f(t L (,H, deمحne transform F µ of f as (F µ f(ξ = f(te µtξ dt. π By making a similar but easier proof to Theorem 3.3 and 3.0, we obtain the following Lemma. 7

18 Lemma 5.6 f,f µ f L (,H, then holds for almost every t. f(t = π (F µ f(ξe µtξ dξ Furthermore, F µ can be defined for every f L (,H and F µ f = f. To investigate right-sided QLCT, we will only be concerned with the case of b b 0 in following. Theorem 5.7 (Inversion of QLCT If f,l r f L (,H and f(x,x = (L r f(ω,ω K j (ω A,x K i (ω A,x dω dω, (5.6 holds for almost every (x,x. Proof. Let g x (x = g(x,x = f(x,x e i a x b, m ω (x = f(x,x KA i (x,ω dx and n ω (x = m ω (x e i a x b. Then ( ω (L r f(ω,ω = (F j n ω e j d ω b. (5.7 jb Since f,l r f L (,H. It is easy to verify that m ω,n ω,(l r f(ω, L (,H for almost every ω by Fubiniلاs theorem. Then by invoking Lemma 5.6, n ω (x = (L r f(ω,ω e j d ω b jb e jω x b b dω holds for almost every x. Note that Thus m ω (x = (F i g x ( ω b b e j d b ω ib. (5.8 f(x,x = g x (x e i a x b ( = n ω (x e i a x b e j d ω b ib e iω x b b dω = (L r f(ω,ω K j A (ω,x K i (ω A,x dω dω holds for almost every (x,x by invoking Lemma 5.6 again. e i a b x For every f L p (,H(p =,, we can decompose it into two parts: f = f i +jf i (5.9 where f i,f i are C i -valued. If we deمحne Φ r f := Φ s f i + jφ s f i, then Φ r is well-deمحned for every f L (,H and coincides with L r on L (,H L (,H. 8

19 Theorem 5.8 For every f,g L (,H, < f,g >=< Φ r f,φ r g > holds. In particular, Φ r f = f. Proof. If f L (,H L (,H then L r f = L s f i + jl s f i. Since L s f i,l s f i L (,H, thus L r f L (,H. Applying Lemma 5.6 and noticing that (5.7 and (5.8, we have and Therefore (L r f(ω,ω dω = n ω (x dω = = = = = ( (F ω jn ω b b dω ( (F ω jn ω b b dω (F j n ω (ω dω = n ω (x dx m ω (x dω ( (F ω ig x b b dω (F i g x (ω dω = g(x,x dx. (L r f(ω,ω dω dω = n ω (x dx dω = n ω (x dω dx = g(x,x dx dx, namely, L r f = f. Since Φ r is the extension of L r on L (,H, then Φ r f = f. Since Φ r is left H-linear, applying Parsevalلاs identity to f+ig = Φ r f+iφ r g, f +jg = Φ rf +jφ r g, f +kg = Φ rf +kφ r g respectively, then < f,g >=< Φ r f,φ r g > holds for every f,g L (,H. In Theorem 4.9, we conclude that if both f and g are C i -valued, then < f,g >=< Ψ s f,ψ s g >. But the analogous result for SQLCT is not reached yet in Theorem 5.4. In fact, if f is C i -valued, we have Φ r f = Φ s f. Then by Theorem 5.8, we obtain the following corollary which is a supplement for Theorem 5.4. Corollary 5.9 Suppose that f,g L (,H and both of them are C i -valued, then < f,g >=< Φ s f,φ s g >. 9

20 6 Discussions and Conclusions Due to the non-commutativity of multiplication of quaternions, there are at least eight types of QFTs and we only consider two typical types of them. How about the rest of QFTs?. The left-sided QFT (LQFT e iω x f( follows a similar pattern to QFT, with kernel moved to left side. As left-sided QFT is right H-linear, we only need to revise the deمحnition of inner product in L (,H to be < f,g > L (,H= f(x,x g(x,x dx dx. Then the results of QFT still hold for LQFT case.. Ifi,jaresubstituted intoµ,µ respectively, whereµ,µ beanytwoperpendicular unit pure imaginary quaternion, all of above results still hold. 3. The types of e µω x e µω x f(,f( e µω x e µω x and e µω x f( e µω x (single axis types obviously easier than the types in present paper (factored types. The single axis types of QFTs have similar properties to factor types of QFTs. Moreover, the proofs will be simpler. 4. The inversion theorem and Plancherel theorem of transforms e µ ω x µ ω x f( and f( e µ ω x µ ω x (dual axis types have not been worked out yet in this paper. Each type of QLCTs corresponds to a speciمحc type of QFTs. So above four facts about QFTs are suited for QLCTs as well. In this paper, we investigate the behaviors of the QFTs (except for dual axis types on the space L (,H. They are reversible under the suitable condition (the transformed function still integrable. The QFTs on L (,H have symmetric property, for example, QFT is unitary on L (,H. We are largely exploiting the relations between different types of QFTs. These relations are helpful to study the properties of QFTs. 7 Acknowledgements The authors acknowledge nancialمح support from the National Natural Science Funds for Young Scholars (No , 5005 and University of Macau No. MYG FST, MYG099(Y-L-FST3-KKI and the Macao Science and Technology Development Fund FDCT/094/0A, FDCT/099/0/A3. 0

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