TRANSLATION-INVARIANT CLIFFORD OPERATORS

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1 TRANSLATION-INVARIANT CLIFFORD OPERATORS JEFF HOGAN AND ANDREW J. MORRIS Abstract. This paper is concerned with quaternion-valued functions on the plane and operators which act on such functions. In particular, we investigate the space L R, H) of square-integrable quaternion-valued functions on the plane and apply the recently developed Clifford-Fourier transform and associated convolution theorem to characterise the closed translation-invariant submodules of L R, H) and its bounded linear translation-invariant operators. The Clifford-Fourier characterisation of Hardy-type spaces on R d is also explored.. Introduction Modern signal and image processing are mainstays of the new information economy. The roots of these branches of engineering lay in the mathematical discipline of Fourier analysis and, more generally, harmonic analysis. Time series such as those arising from speech or music have been effectively treated by algorithms derived from Fourier analysis the fast) Fourier and wavelet transfoms being perhaps the most celebrated among many. Complex analysis has also played its part, contributing signal analysis tools such as the analytic signal, the Paley-Wiener theorem, Blaschke products and many others. Two-dimensional signals such as grayscale images may be dealt with by applying the one-dimensional algorithms in each of the horizontal and vertical directions. Colour images, however, pose a new set of problems. At each pixel in the image there is specified not one but three numbers the red, green and blue pixel values. This scenario does not fit the standard set-up of Fourier analysis, namely that of real- or complex-valued functions. Even greater challenges are posed by the new breed of hyperspectral sensors which create images containing hundreds sometimes thousands of channels. Fourier- and wavelet-based compression algorithms have been spectacularly successful in their ability to identify and remove redundant information from grayscale images without significant loss of fidelity. When dealing with multichannel signals, the standard practice has been to treat each channel separately, using one-channel algorithms. Natural images, however, contain very significant cross-channel correlations [5] changes in the green channel are often mirrored by changes in the blue and red channels. Any algorithm using the channel-by-channel paradigm is doomed to be sub-optimal especially for purposes of compression, but also for interpolation), for although Mathematics Subject Classification. Primary 4B; Secondary 5A66. Key words and phrases. Fourier transform, Clifford analysis, quaternions, Hilbert transform, monogenic functions. 48

2 TRANSLATION-INVARIANT CLIFFORD OPERATORS 49 the intra-channel redundancy may have been reduced, cross-channel redundancies will be unaffected. Clearly, the current model for these signals is inadequate. Electrical engineers have responded to the challenges posed by multichannel signals by developing techniques through which such a signal can be treated as an algebraic whole rather than as an ensemble of disparate, unrelated single-channel signals [8], [9]. A colour image is now viewed as a signal taking values in the quaternions, an associative, non-commutative algebra. The algebra structure gives a meaning to the pointwise product of such signals. In this paper we outline recent developments in the treatment of functions on the plane which take values in the quaternions through the development of Fourier-type transforms. Some of the consequences for the theory of quaternionic functions, such as a description of translation-invariant operators and submodules will also be given, as will an indication of what can be said about multichannel functions defined on R d d ) and taking values in the associated Clifford algebrs R d.. Clifford and quaternionic analysis.. Clifford algebra. In this section we give a quick review of the basic concepts of Clifford algebra and Clifford analysis. Although this will be given in greater generality than is necessary for the application at hand, the greater generality gives a deeper understanding of the relevant algebraic properties. The interested reader is referred to [4] for more details. Let {e, e,..., e d } be an orthonormal basis for R d. The associative Clifford algebra R d is the d -dimensional algebra generated by the collection {e A ; A {,,..., d}} with algebraic properties e = e = identity), e j =, and e j e k = e k e j = e {j,k} if j, k {,,..., d} and j k. Notice that for convenience we write e j j j k = e {j,j,...,j k } = e j e j e jk. In particular we have R d = { A x A e A ; x A R}. The canonical mapping of the euclidean space R d into R d maps the vector x, x,..., x d ) R d to d j= x je j R d. For this reason, elements of R d of the form d j= x je j are also known as vectors. Notice that R d decomposes as R d = Λ Λ Λ d, where Λ j = { A =j x Ae A ; x A R}. In particular, Λ is the collection of scalars while Λ is the collection of vectors. Given x R d of the form x = A x Ae A and p d we write [x] p to mean the Λ p -part of x, i.e, [x] p = A =p x Ae A. It is a simple matter to show that if x, y R d are vectors, then x = x a scalar) and their Clifford product xy may be expressed as xy = x, y + x y Λ Λ. Here x, y is the usual dot product of x and y while x y is their wedge product. The linear involution ū of u R d is determined by the rules x = x if x Λ while uv = vū for all u, v R d.

3 5 JEFF HOGAN AND ANDREW J. MORRIS As examples, note that R is identified algebraically with the field of complex numbers C while R, which has basis {e, e, e, e } and whose typical element has the form q = a + be + ce + de with a, b, c, d R) is identifiable with the associative algebra of quaternions H... The Dirac operator. We consider functions f : Ω R d R d and define the Dirac operator D acting on such functions by Df = d j= e j f x j. If f : Σ R d+ R d we define a Dirac operator by f = f x + d j= e j f x j. We say f is left monogenic on Ω R d respectively Σ R d+ ) if Df = respectively f = ). If d = and f : Σ R R C, then f is left monogenic if and only if fx, y) = ux, y) + e vx, y) is complex-analytic, or equivalently, if and only if u and v satisfy the Cauchy-Riemann equations. When d =, then f = f + f e + f e + f e : Ω R R H is monogenic if and only if f, f, f, f satisfy the generalised Cauchy- Riemann equations x x x x x x x x f f f f =. When d = 3, the monogenicity of f = E + ih with E : Ω R 3 Λ and H : Ω R 3 Λ where i is the imaginary unit in the complex plane) is equivalent to the pair of vector fields E, H) satisfying a form of Maxwell s equations. More generally, Dirac operators are important in mathematical physics since they factorise the Laplacian and the Helmholtz operator: D + ik)d ik) = + k. If u = A {,,...,d} u Ae A R d, we define its even and odd parts u e and u o to be u e = A even u Ae A and u o = A odd u Ae A..3. The Clifford Fourier transform. The Clifford-Fourier transform CFT) on R d was introduced by Brackx, De Schepper and Sommen in [] as the exponential of a differential operator, much in the same manner that the classical Fourier transform can be defined as expiπ/)h d ) where H d is the Hermite operator H d = + x d). Here = d j= x j is the Laplacian on R d. Defining the angular momentum operators L ij i, j d) by L ij = x i x j and the angular Dirac operator Γ by x j x i ) Γ = e i e j L ij d

4 TRANSLATION-INVARIANT CLIFFORD OPERATORS 5 with e i, e j Clifford units in R d ), two Clifford-Hermite operators H ± d defined by H ± d = H d ± Γ d/) and corresponding CFT s F ± d are defined by the operator exponentials are ) F ± d = exp iπ/)h± d ). In [], Brackx, DeSchepper and Sommen compute a closed form for the two-dimensional Clifford-Fourier kernels, namely 3) K ± x, y) = e±x y = cos x y ) ± e sin x y ) which act on the left via F R ± fy) = K ± y, x)fx) dx. Since the underlying Clifford algebra in this case is the quaternions, we call these transforms the Quaternionic Fourier transforms QFT). The main point of difference between the kernels of the QFT and the classical FT is that the former takes values in the quaternionic subalgebra Λ Λ H while the latter takes scalar values only. Performing the QFT by integration against the kernel K + of 3) mixes the channels whereas the classical FT does not. The QFT enjoys many of the properties of the classical FT. Given f, g L R, H), let f, g = R fx)ḡx) dx so that f = f, f = R fx) dx. Given y R, let τ y be the translation operator τ y fx) = fx y) and M y be the modulation operator M y fx) = e x y fx). Given a >, let D a be the dilation operator D a fx) = a fx/a). Finally, if R is a rotation on the plane, let σ R be the rotation operator σ R fx) = fr x). Then the QFT has the following properties:. Parseval indentity: F + f, F + g = 4π f, g.. F + τ y = M y F + 3. F + M y = τ y F + 4. F + σ R = σ R F + 5. F + D a = D a F +. What s missing from this list, of course, is the convolution theorem and the action of the QFT under partial differentiation. In its expected form, the convolution theorem fails due to the non-commutativity of the quaternions. However there is a replacement which is sufficient for the purpose we have in mind. Given a function f : R d R d, we define its parity matrix [fy)] to be the matrix-valued function ) fy)e fy) [fy)] = o. f y) o f y) e If P = p ij ) n i,j= is an n n matrix with quaternionic entries, then P determines a mapping T P : H n H n which acts by left multiplication by P : T P q = P q

5 5 JEFF HOGAN AND ANDREW J. MORRIS with q = q, q,..., q n ) T H n ) which is right H-linear on the H-module ) / n H n. With q = j= q j, the norm of P is the operator norm of T P, i.e., P = sup q H n P q / q and its Fröbenius norm is defined to be ) n / P F = n i= j= p ij. Then P P F and if P, Q are n n matrices with quaternionic entries, then P Q P Q. A crucial property of the parity matrices of functions A : R H is outlined in the following result. Lemma. For fixed x R, the parity matrix of the function ey) = e x y commutes with all parity matrices. Proof. Note that if s, s Λ Λ and v Λ then s s = s s and vs = s v. Let Ay) = sy) + vy) with s : R Λ Λ and v : R Λ. Then ) ) [Ay)][e x y sy) vy) e x y ] = v y) s y) e x y sy)e x y vy)e = x y ) v y)e x y s y)e x y e = x y sy) e x y ) vy) e x y v y) e x y s y) e x y = and the proof is complete. ) sy) vy) e x y v y) s y) ) = [e x y ][Ay)] Given f, g L R d, R d ), the convolution of f and g is the function f g L R d, R d ) defined by f gx) = R d fy)gx y) dy. Note that in general f g g f. In the case d = we have the following generalization of the Fourier convolution theorem [7]. Theorem Convolution theorem). Let f, g L R, H). Then the parity matrix of the QFT of the convolution f g factorises as 4) [F + f g)y)] = [F + fy)][f + gy)]. The classical Fourier kernel e i x,y x, y R d ) is an eigenfunction of the partial differential operators j d). Consequently the classical Fourier transform intertwines and multiplication by x j, and more x j x j generally intertwines constant coefficient differential operators with mutliplication by polynomials. The Clifford analogue is provided by the following result. Theorem 3. Let D be the d-dimensional Dirac operator and K ± d Clifford-Fourier kernels. Then D x K ± d x, y) = K d x, y)y be the

6 TRANSLATION-INVARIANT CLIFFORD OPERATORS 53 or equivalently, if f, Df L R d, R d ), then F ± d Dfy) = F d fy)y. The classical Fourier transform is a 4-th order operator in the sense that its fourth power is the identity. The Clifford-Fourier transform, on the other hand, is second order so that its inverse is itself. This leads to the following inversion formula. Theorem 4 Inversion theorem). Suppose f, F + f L R d, R d ). Then π) d K + d x, y)f + d fy) dy = fx). R d Of critical importance in section 4 is Proposition 6 which appears below and describes the near anti-commutation of the differential operator Γ and the multiplication operator Q which acts via Qfx) = xfx). Lemma 5. Let the angular momentum operators L ij i, j d) be as above. Then 5) e i e j x k e k L ij =. k / {i,j} Proof. Given a < b < c d, we note that the term in the sum 5) that involves the e a e b e c is However, x a e abc L bc + x b e bac L ac + x c e cab L ab = e abc x a L bc x b L ac + x c L ab ). x a L bc f x b L ac f + x c L ab f f f = x a x b x c x c x b f = x a x b f x c x c and the proof is complete. ) f f x b x a x c x c x a ) + x a x c f x b f x b Proposition 6. Let Γ, Q be as above. Then 6) ΓQ = d )Q QΓ. Proof. By direct computation we find that ΓQfx) = e i e j L ij xf) 7) ) f f + x c x a x b x b x a ) ) + x b x c f x a f x a = [ e i e j x i e j f + x f ] [ x j e i f + x f ]) x j x i = x i e i + x j e j )f e i e j xl ij f = d )Qf e i e j x i e i + x j e j + = d )Qf e i e j x i e i + x j e j )L ij k / {i,j} = x k e k )L ij f )

7 54 JEFF HOGAN AND ANDREW J. MORRIS where in the last step we have applied Lemma 5. On the other hand QΓf = x k e k e i e j L ij f k = x i e i + x j e j + x k e k )e i e j L ij f 8) = k / {i,j} e i e j x i e i + x j e j )L ij f where in the last step we have again employed Lemma 5. Comparing 7) and 8) now gives the result. Corollary 7. Let Γ, Q be as above and t R. Then expitγ)q = e itd ) Q exp itγ). Proof. By iterating the result of Proposition 6, we have for each non-negative integer j, Γ j Q = Q[d )I Γ] j. Consequently, it) j expitγ)q = Γ j Q = j! j= it) j Q[d )I Γ] j j! j= = Q expit[d )I Γ]) = e itd ) Q exp itγ) where we have used the fact that d )I and Γ commute. 3. Translation-invariance In this section we outline one of the curious differences between the function theories of the classical FT and the QFT. Given a space X of functions defined on R d, we say an operator T : X X is translation-invariant if it commutes with the translation operator, i.e., if T τ y = τ y T for all y R d. It is well-known that T is a bounded, linear, translationinvariant operator on L R d ) = L R d, C) if and only if there is a bounded measurable function m : R d C a multiplier) such that F d T fy) = my)f d fy) for a.e. y R d. Here F d is the classical Fourier transform. The corresponding result for the QFT takes the following form. Theorem 8. T is a bounded, right H-linear, translation-invariant operator on L R, H) if and only if there exists a uniformly bounded H-valued function Ay) defined on R for which 9) [F + T f)y)] = [Ay)][F + fy)] for all f L R, H) and a.e. y R.

8 TRANSLATION-INVARIANT CLIFFORD OPERATORS 55 Proof. Suppose first that T is bounded, right H-linear and translationinvariant. We may choose a radial real-valued function ϕ L R ) L R ) such that F + ϕy) = for y < and F + ϕty) t = for all y R. Then any f L R, H) may be decomposed via the Calderòn reproducing formula fx) = ϕ t ϕ t fx) t where ϕ t x) = t ϕx/t) and ϕ x) = ϕ x). Since T is right H-linear and translation-invariant, we have ) T fx) = T ϕ t ) ϕ t fx) t. Taking the QFT of both sides of ) and applying the convolution theorem Theorem ) gives [F + T f)y)] = [F + T ϕ t)y)][f + Our first task is to show that the integral defining [Ay)] = is convergent for a.e. y R. Let [A y)] = [A y)] = [F + T ϕ t)y)][f + [F + T ϕ t)y)][f + [F + T ϕ t)y)][f + ϕty)] t [F + fy)]. ϕty)] t ϕty)] t, ϕty)] t so that [Ay)] = [A y)] + [A y)]. Then by the Minkowski and Cauchy- Schwarz inequalities, [A y)] dy [F + T ϕ t)y)] [F + ϕty)] R R t dy ) / ) / [F + T ϕ t)y)] dy [F + ϕty)] dy R R t. However, for every g L R, H), its parity matrix [gy)] satisfies [gy)] [gy)] F = gy) + g y) so that R [gy)] dy g. An application of the Plancherel identity for the QFT now yields R [A y)] dy π) d/ π) d/ T T ϕ t F + ϕt ) π) d T ϕ t ϕ t F + ϕt ) t t 3 = π)d T ϕ.

9 56 JEFF HOGAN AND ANDREW J. MORRIS Hence the integral defining [A y)] is convergent for a.e. y. Since F + ϕy) = for y <, we have y α F + ϕ L R, H) for all α, and y [A y)] dy [F + T ϕ t)y)] [F + ϕty)] dy R R y t ) / [F + T ϕ t)y)] dy [F + dy ϕty)] R R y 4 π) d/ T ϕ t F + dy ϕy) R y 4 ) / t ) / π) d T ϕ so that the integral defining [A y)] is convergent for a.e. y. We conclude that the integral defining [Ay)] is convergent for a.e. y. Suppose now that T satisfies 9) for some uniformly bounded function A. Then the Plancherel theorem immediately gives the L R, H)-boundedness of T. Suppose) a H and f, f L R, H). We denote by [a] the matrix as a [a] = v. Then a v a s [F + T f a + f ))y)] = [Ay)][F + f a + f )y)] = [Ay)][F + f y)a + F + y)] = [Ay)][F + f y)][a] + [F + f y)]) = [Ay)][F + f y)][a] + [Ay)][F + f y)] = [F + T f )y)a] + [F + T f )y)] = [F + T f )y)a + F + T f )y)] = [F + T f )a + T f )y)] from which we conclude that T f a + f ) = T f )a + T f, i.e., T is right H-linear. Furthermore, with an application of Lemma we have [F + T τ xf)y)] = [Ay)][F + τ xf)y)] = [Ay)][e x y F + fy)] = [Ay)][e x y ][F + fy)] = [ex y ][Ay)][F + fy)] = [e x y ][F + T f)y)] = [ex y F + T f)y)] = [F + τ xt f)y)] from which we conclude that T τ x = τ x T, i.e., T is translation-invariant. A submodule Y L R d, X) is said to be translation-invariant when τ y f Y for all f Y and y R d. In the classical case, it is well-known that the only closed translation-invariant subspaces of L R d ) = L R d, C) are of the form Y = Y E = {f L R d ); F d fy) = if y / E} where E is a measurable subset of R d. Equivalently, Y is a closed translationinvariant subspace of L R d ) if and only if there is a measurable subset E R d for which F d fy) = χ E y)f d fy) for all f Y and a.e. y R d. The corresponding class of closed translation-invariant submodules of L R, H) is far richer than the the classical case, and is described by Theorem 9 below.

10 TRANSLATION-INVARIANT CLIFFORD OPERATORS 57 ) sy) vy) The Hermitian) adjoint of a parity matrix [Ay)] = v y) s y) ) sy) v y) is [Ay)] =. The parity matrix [Ay)] is said to be vy) s y) self-adjoint if [Ay)] = [Ay)] and idempotent if [Ay)] = [Ay)]. Theorem 9. Y L R, H) is a closed translation-invariant right H-linear submodule of L R, H) if and only if there exists an idempotent self-adjoint parity matrix [Ay)] such that for all f Y, ) [F + fy)] = [Ay)][F + fy)] for a.e. y R. Proof. Suppose Y is a closed translation-invariant submodule of L R, H). If f Y, g Y and x R, then the unitarity of the translation τ x gives f, τ x g = τ x f, g = since τ x f Y. Let P Y be the orthogonal projection onto Y, If g Y then P Y τ x g τ x P Y g = since P Y Y =. Further, if f Y then, because of the translation-invariance of Y we have P Y τ x f τ x P Y f = P Y τ x P Y f P Y τ x P Y f = from which we conclude that P Y is a translation-invariant operator. We now apply Theorem 8 to conclude that P Y is a mulitplier operator, i.e., there exists a bounded parity matrix [Ay)] for which [F + P Y f)y)] = [Ay)][F + fy)] for all f L R, H). When f Y, this equation becomes ). The idempotence and self-adjointness of P Y now gives the idempotence and self-adjointness of the multiplier matrix [Ay)]. Conversely, suppose [Ay)] is a bounded, idempotent, self-adjoint parity matrix, and let Y = {f L R, H); [F + fy)] = [Ay)][F + fy)] for all y R }. It is a simple matter to see that Y is a right H-linear submodule of L R, H). The boundedness of [Ay)] ensures the closedness of Y. To see that Y is translation-invariant, suppose that f L R, H) and x R d. Then [F + τ xf)y)] = [e x y F + fy)] = [ex y ][F + fy)] = [ex y ][Ay)][F + fy)] = [Ay)][e x y ][F + fy)] = [Ay)][ex y F + fy)] = [Ay)][F + τ xf)y)], i.e., τ x f Y. This concludes the proof. As a consequence of Theorem 9, we see that there are many more examples of closed right H-linear) translation-invariant submodules of L R, H) than there are closed translation-invariant subspaces of L R, C). To see this, let E R be measurable with characteristic function χ E y). The parity matrix associated with E is ) χe y) [χ E y)] =. χ E y) Then [χ E y)] is idempotent and self-adjoint and its associated translationinvariant subspace is Y = Y E = {f L R, H); F + fy) = if y / E}. In the classical case, given a measurable subset E R d and associated closed translation-invariant space Y = Y E of L R d, C) as above, we we

11 58 JEFF HOGAN AND ANDREW J. MORRIS have the orthogonal decomposition L R, H) = Y E Y E where E is the complement of E in R d. The most celebrated example on the line is that of the Hardy spaces H±R) where H +R) = Y [, ), H R) = Y,). It is well-known that H+R) coincides with the space of those f L R) which arise as boundary values of analytic functions F : C + C for which sup y> F x + iy) dx <. Here C + is the upper half-plane C + = {z = x + iy C; y > }. The multiplier associated with the the Hardy spaces is of course m + y) = χ [, ) y) and m y) = χ,) y) which may be written as m ± y) = y ) ±. y The situation in the quaternionic case is quite different. Define parity matrices [A ± y)] by [A ± y)] = ) y/ y. ±y/ y Then [A ± y)] are uniformly bounded, idempotent, self-adjoint and, by Theorem 9, generate closed translation-invariant submodules X ± of L R, H). The fact that [A + y)][a y)] = [A y)][a + y)] = shows that X + and X are orthogonal subspaces. Also, since [A + y)] + [A y)] = I, we have the orthogonal decomposition L R, H) = X + X. Its important to note that [A ± y)] is the parity matrix of the function A ± y) = y ) which, in this the quaternionic setting, is not a characteristic function. In the next section it is shown how X ± may be realized y as the boundary values of Clifford monogenic functions and that the characterization extends to arbitrary dimensions. As a final example, let cy) = 4y 3 3y y, dy) = 3y y 4y 3, and cy)e + dy)e [Ay)] = y 3 cy)e dy)e. y 3 Then [Ay)] is uniformly bounded, idempotent and self-adjoint and therefore generates a closed translation-invariant right H-linear submodule of L R, H) which also does not arise from a characteristic function. 4. Hilbert transforms and monogenic extensions Consider the Green s functions Gx, t) x R d, t R) defined by Gx, t) = x + t σ d x + t d+

12 TRANSLATION-INVARIANT CLIFFORD OPERATORS 59 where σ d = πd+)/ Γd + )/) is the surface area of the unit sphere in Rd. By direct calculation we find that Gx, t) ) D x Gx, t) =. t The Cauchy transform Cf of a function f L R d, R d ) is defined by the convolution Cfy + t) = Gx y, t)fx) dx. R d Then by ), Cfy t) is left monogenic on R d+ \ R d. Consider operators P ± defined on L R d, R d ) by 3) P + fx) = lim t Cfx + t); P fx) = lim t Cfx t). Then [3] P ± are bounded projections on L R d, R d ) and we define Hardy spaces H ±R d ) by H ±R d ) = Ran P ± ). We aim to give a Clifford-Fourier characterization of these spaces which complements the classical Fourier characterization of [3]. Theorem. Let H ±R d ) be defined as above and f L R d, R d ). Then 4) f H ±R d ) [F + d fy)] = [ y/ y ][F + d fy)]. Proof. Observe that 5) Cfx + t) = σ d φ t fx) where φx) = x x d+ and φ t is the L -normalized dilate of φ given by φ t x) = t d φx/t). Also, and consequently, Rd e i x,y πd+)/ + x dx = )d+)/ Γd + )/) e y Rd x j e i x,y πd+)/ + x dx = )d+)/ Γd + )/) i e y π d+)/ y j = i y j Γd + )/) y e y. From this we see that the classical FT of φ is given by 6) F d φy) = πd+)/ + i y ) e y. Γd + )/) y Now if ) f, g C R d, R d ) and g is radial, then Γfg) = Γf)g. Further, y Γ = d ) y y y, so combining 5) and 6) gives F dcf, t)y) = + i y ) e t y F d fy) and letting t gives y F d P + fy) = + i y ) F d fy). y

13 6 JEFF HOGAN AND ANDREW J. MORRIS Hence, with an application of Corollary 7 we have F + d P +fy) = eiπd/4 e iπ/)γ F d fy) + i y ) y F dfy) = ) F + i d fy) + eiπd/4 y e iπ/)γ QF d fy) = ) F + iy d fy) + e iπd/4 y eiπ/)γ F d fy) = F + d fy) y ) y F d fy). However, since F + d ) = I the identity) and F + d F d = F d = τ, where τ is the inversion τfx) = f x), we see that F d fy) = F + d f y), so that F + d P +fy) = F + d fy) y ) y F + d f y). Similarly we may show that F + d P fy) = F + d fy) + y ) y F + d f y). To prove the Clifford-Fourier characterization 4) of the Hardy spaces H+R d ), note that if u R d and y Λ, then yu) e = yu o and yu) o = yu e, and consequently 7) f H ± P ± f = f F + d P ±f = F + d fy) F + d fy) = F + d fy) y ) y F + d f y). Splitting each side of 7) into even and odd parts gives 4). Note that the parity matrices [χ ± y)] = [ y ] = ) y/ y y ±y/ y satisfy [χ + y)] = [χ y)], [χ + y)][χ y)] = [χ y)][χ + y)] = from which we see that L R d, R d ) decomposes orthogonally as L R d, R d ) = H +R d ) H R d ). Define now the Clifford-Hilbert transform H which acts on L R d, R d ) by the principal value singular integral y x 8) Hfy) = lim fx) dx. ε σ d y x d+ ε< x y </ε Theorem below provides a characterization of the Hardy spaces in terms of the action of the Clifford-Hilbert transform. For each < ε <, let k ε x) = x σ d x d+ χ <ε< x </εx). Then k ε L R d, R d ) and Hfx) = lim ε k ε fx). Consequently, F d Hf = lim ε F d k ε )F d f).

14 TRANSLATION-INVARIANT CLIFFORD OPERATORS 6 We aim first to compute my) = lim ε F d k ε y). ix j e i x,y, we have Since y j e i x,y = my) = σ d d j= x j e j lim ε ε< x </ε x d+ e i x,y dx = i σ d d j= e j lim ε y j ε< x </ε e i x,y x d+ Conversion of the integral on the right hand side to spherical co-ordinates and applying the fact that S d e i ω,ξ dω = π) d/ ξ d/ J d/ ξ ) with J d/ the Bessel function of the first kind [6]) yields my) = i σ d = i σ d d j= d j= e j lim ε e j lim ε y j y j /ε ε /ε ε r e ir ω,y dω dr S d π) d/ r y ) d/ J d/ r y ) dr r. However, the Bessel functions J ν satisfy the differential recurrence relation d t t ν J ν t)) = t ν J ν+ t), so [r y ) d/ J y d/ r y )] = d j [t d/ J d/ t)] r y ) t=r y y j = ry j y t d/ J d/ t)) = r d/ y j t=r y y d/ J d/r y ) and as a consequence, 9) my) = iπ)d/ σ d = iπ)d/ σ d d y /ε j e j lim y d/ ε j= ε y y s d/ J d/ s) ds. r d/ J d/ r y ) dr The Gegenbauer polynomials C ν n and Bessel functions J µ are related via the Fourier transform as follows: x ) ν / e iax Cnx) ν dx = π ν i n Γν + n) a ν J ν+n a) n!γν) whenever Rν > / see equation 7.3 of [6]). However C ν for all ν, so x ) ν / e iax dx = π ν Γν) a ν J ν a) Γν) or equivalently, x ) ν / = π ν Γν) π Γν) a ν J ν a) da. dx.

15 6 JEFF HOGAN AND ANDREW J. MORRIS Putting x = gives a ν J ν a) da = Γν)ν Γν) so that with an application of the Gamma function doubling formula Γx) = x π Γx)Γx +/) see [6]) equation 9) becomes Consequently, my) = iπ)d/ σ d y Γd/) d/ y Γd) = i y y. ) F + d Hf)y) = i y eiπd/4 e iπ/)γ Qfy) = y y F + d y). Comparing ) with 7) gives the following result. Theorem. Let H± be the Hardy spaces defined above and H be the Clifford Hilbert transform defined in 8). Then f H ± Hf = f. Acknowledgements Hogan gratefully acknowledges the support of the University of Newcastle s Centre for Computer Assisted Research in Mathematics and its Applications CARMA). He also wishes to thank Wayne Lawton and Philip Taylor for their insights. Thanks, Roy. Thanks, HG. References. F. Brackx, N. De Schepper and F. Sommen, The Clifford-Fourier transform, Jour. Fourier Anal. Appl. 5), p F. Brackx, N. De Schepper and F. Sommen, The two-dimensional Clifford-Fourier transform, J. Math. Imag. Vision 6 6) p J. Cnops The wavelet transform in Clifford analysis, Computat. Methods Funct. Theory ) p R. Delanghe, F. Sommen and V. Soucek, Clifford Algebra and Spinor-Valued Functions: A Function Theory for the Dirac Operator, Kluwer, Dordrecht 99). 5. B.K. Gunturk, Y. Altunbasak and R.M. Mersereau, Color plane interpolation using alternating projections, IEEE Trans. Image Proc. 9) ) p I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, 7th ed., Academic Press, J.A. Hogan and A.J. Morris, Quaternionic wavelets, Numer. Funct. Anal. Optim., 33:7 9, S.J. Sangwine, The problem of defining the Fourier transform of a colour image, Int. Conf. on Image Process. 998). 9. S.J. Sangwine and T.A. Ell, Colour image filters based on hypercomplex convolution, IEEE Proc.-Vis. Image Signal Process. 47 ) p Jeff Hogan: School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 89, Australia address: jeff.hogan@newcastle.edu.au Andrew J. Morris: School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 89, Australia address: Andrew.Morris@newcastle.edu.au