Translation-Invariant Clifford Operators

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 2 2 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 2 2 translation-invariant submodules of L (R , H) and its bounded linear translation-invariant operators. The Cliﬀord-Fourier characterisation of d Hardy-type spaces on R is also explored.

1. 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. Com- plex 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 verti- cal 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 spectacu- larly 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 (espe- cially for purposes of compression, but also for interpolation), for although

2010 Mathematics Subject Classification. Primary 42B10; Secondary 15A66. Key words and phrases. Fourier transform, Clifford analysis, quaternions, Hilbert transform, monogenic functions. 48 TRANSLATION-INVARIANT CLIFFORD OPERATORS 49 the intra-channel redundancy may have been reduced, cross-channel redun- dancies 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 d said about multichannel functions defined on R (d ≥ 2) and taking values in the associated Clifford algebrs Rd.

2. Clifford and quaternionic analysis 2.1. 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. d Let {e1, e2, . . . , ed} be an orthonormal basis for R . The associative Clif- d ford algebra Rd is the 2 -dimensional algebra generated by the collection {eA; A ⊂ {1, 2, . . . , d}} with algebraic properties 2 e∅ = e0 = 1 (identity), ej = −1, and ejek = −ekej = e{j,k} if j, k ∈ {1, 2, . . . , d} and j 6= k. Notice that for convenience we write ej1j2···jk = e{j1,j2,...,jk} = ej1 ej2 ··· ejk . In particular we have X Rd = { xAeA; xA ∈ R}. A d The canonical mapping of the euclidean space R into Rd maps the vector d Pd (x1, x2, . . . , xd) ∈ R to j=1 xjej ∈ Rd. For this reason, elements of Rd of Pd the form j=1 xjej are also known as vectors. Notice that Rd decomposes as P Rd = Λ0 ⊕ Λ1 ⊕ · · · ⊕ Λd, where Λj = { |A|=j xAeA; xA ∈ R}. In particular, Λ0 is the collection of scalars while Λ1 is the collection of vectors. Given P x ∈ Rd of the form x = xAeA and 0 ≤ p ≤ d we write [x]p to mean the PA “Λp-part” of x, i.e, [x]p = |A|=p xAeA. 2 2 It is a simple matter to show that if x, y ∈ Rd are vectors, then x = −|x| (a scalar) and their Clifford product xy may be expressed as

xy = −hx, yi + x ∧ y ∈ Λ0 ⊕ Λ2. Here hx, yi is the usual dot product of x and y while x ∧ y is their wedge product. The linear involutionu ¯ of u ∈ Rd is determined by the rulesx ¯ = −x if x ∈ Λ1 while uv =v ¯u¯ for all u, v ∈ Rd. 50 JEFF HOGAN AND ANDREW J. MORRIS

As examples, note that R1 is identified algebraically with the field of complex numbers C while R2, which has basis {e0, e1, e2, e12} and whose typical element has the form q = a + be1 + ce2 + de12 (with a, b, c, d ∈ R) is identifiable with the associative algebra of quaternions H. d 2.2. The Dirac operator. We consider functions f :Ω ⊂ R → Rd and define the Dirac operator D acting on such functions by d X ∂f Df = e . j ∂x j=1 j d+1 If f :Σ ⊂ R → Rd we define a Dirac operator ∂ by d ∂f X ∂f ∂f = + e . ∂x j ∂x 0 j=1 j d d+1 We say f is left monogenic on Ω ⊂ R (respectively Σ ⊂ R ) if Df = 0 2 (respectively ∂f = 0). If d = 1 and f :Σ ⊂ R → R1 ≡ C, then f is left monogenic if and only if f(x, y) = u(x, y) + e1v(x, y) is complex-analytic, or equivalently, if and only if u and v satisfy the Cauchy-Riemann equations. 2 When d = 2, then f = f0 + f1e1 + f2e2 + f12e12 :Ω ⊂ R → R2 ≡ H is monogenic if and only if f0, f1, f2, f12 satisfy the generalised Cauchy- Riemann equations  0 − ∂ − ∂ 0      ∂x1 ∂x2 f0 0  ∂ 0 0 ∂  ∂x1 ∂x2  f1  0  ∂ ∂    =   .  0 0 −  f2 0  ∂x2 ∂x1      0 − ∂ ∂ 0 f12 0 ∂x2 ∂x1 3 When d = 3, the monogenicity of f = E + iH with E :Ω ⊂ R → Λ1 3 and H :Ω ⊂ R → Λ2 (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) = −∆2 + k2. P If u = A⊂{1,2,...,d} uAeA ∈ Rd, we define its even and odd parts ue and P P uo to be ue = |A| even uAeA and uo = |A| odd uAeA. 2.3. The Clifford Fourier transform. The Clifford-Fourier transform d (CFT) on R was introduced by Brackx, De Schepper and Sommen in [1] as the exponential of a differential operator, much in the same manner that the classical Fourier transform can be defined as exp(i(π/2)Hd) where Hd is the 2 1 2 Pd ∂ Hermite operator Hd = (−∆+|x| −d). Here ∆ = j=1 2 is the Lapla- 2 ∂xj d cian on R . Defining the angular momentum operators Lij (1 ≤ i, j ≤ d) by ∂ ∂ Lij = xi − xj and the angular Dirac operator Γ by ∂xj ∂xi XX (1) Γ = − eiejLij 1≤i

± (with ei, ej Clifford units in Rd), two Clifford-Hermite operators Hd are defined by ± Hd = Hd ± (Γ − d/2) ± and corresponding CFT’s Fd are defined by the operator exponentials ± ± (2) Fd = exp(−i(π/2)Hd ). In [2], Brackx, DeSchepper and Sommen compute a closed form for the two-dimensional Clifford-Fourier kernels, namely ± ±x∧y (3) K2 (x, y) = e = cos(|x ∧ y|) ± e12 sin(|x ∧ y|) which act on the left via Z ± ± F2 f(y) = K2 (y, x)f(x) dx. R2 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 Λ0 ⊕ Λ2 ⊂ H while the latter takes scalar values only. Performing the QFT by integration + against the kernel K2 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 ∈ 2 2 R R 2 L ( , ), let hf, gi = 2 f(x)¯g(x) dx so that kfk2 = hf, fi = 2 |f(x)| dx. R H R R 2 Given y ∈ R , let τy be the translation operator τyf(x) = f(x − y) and My x∧y be the modulation operator Myf(x) = e f(x). Given a > 0, let Da be −1 the dilation operator Daf(x) = a f(x/a). Finally, if R is a rotation on the −1 plane, let σR be the rotation operator σRf(x) = f(R x). Then the QFT has the following properties: + + 2 1. Parseval indentity: hF2 f, F2 gi = 4π hf, gi. + + 2. F2 τy = M−yF2 + + 3. F2 My = τyF2 + + 4. F2 σR = σRF2 + + 5. F2 Da = Da−1 F2 . 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. d Given a function f : R → Rd, we define its parity matrix [f(y)] to be the matrix-valued function f(y) f(y) [f(y)] = e o . f(−y)o f(−y)e n If P = (pij)i,j=1 is an n × n matrix with quaternionic entries, then P de- n n termines a mapping TP : H → H which acts by left multiplication by P :

TP q = P q 52 JEFF HOGAN AND ANDREW J. MORRIS

T n (with q = (q1, q2, . . . , qn) ∈ H ) which is right H-linear on the H-module 1/2 n Pn 2 H . With kqk = j=1 |qj| , the norm of P is the operator norm of

TP , i.e., kP k = sup06=q∈Hn kP qk/kqk and its Fr¨obenius norm is deﬁned to be 1/2 Pn Pn 2 kP kF = i=1 j=1 |pij| . Then kP k ≤ kP kF and if P,Q are n × n matrices with quaternionic entries, then kPQk ≤ kP kkQk. 2 A crucial property of the parity matrices of functions A : R → H is outlined in the following result.

2 x∧y Lemma 1. For ﬁxed x ∈ R , the parity matrix of the function e(y) = e commutes with all parity matrices.

Proof. Note that if s1, s2 ∈ Λ0 ⊕ Λ2 and v ∈ Λ1 then s1s2 = s2s1 and 2 2 vs1 = s1v. Let A(y) = s(y) + v(y) with s : R → Λ0 ⊕ Λ2 and v : R → Λ1. Then s(y) v(y) ex∧y 0 [A(y)][ex∧y] = v(−y) s(−y) 0 e−x∧y s(y)ex∧y v(y)e−x∧y = v(−y)ex∧y s(−y)e−x∧y ex∧ys(y) ex∧yv(y) = e−x∧yv(−y) e−x∧ys(−y) ex∧y 0 s(y) v(y) = = [ex∧y][A(y)] 0 e−x∧y v(−y) s(−y) and the proof is complete. 1 d Given f, g ∈ L (R , Rd), the convolution of f and g is the function f ∗g ∈ L1( d, ) deﬁned by f ∗ g(x) = R f(y)g(x − y) dy. Note that in general R Rd Rd f ∗ g 6= g ∗ f. In the case d = 2 we have the following generalization of the Fourier convolution theorem [7].

1 2 Theorem 2 (Convolution theorem). Let f, g ∈ L (R , H). Then the parity matrix of the QFT of the convolution f ∗ g factorises as + + + (4) [F2 (f ∗ g)(y)] = [F2 f(y)][F2 g(y)]. ihx,yi d The classical Fourier kernel e (x, y ∈ R ) is an eigenfunction of the ∂ partial differential operators (1 ≤ j ≤ d). Consequently the classi- ∂xj ∂ cal Fourier transform intertwines and multiplication by xj, and more ∂xj generally intertwines constant coefficient differential operators with mutli- plication by polynomials. The Clifford analogue is provided by the following result. ± Theorem 3. Let D be the d-dimensional Dirac operator and Kd be the Clifford-Fourier kernels. Then ± ∓ DxKd (x, y) = ∓Kd (x, y)y TRANSLATION-INVARIANT CLIFFORD OPERATORS 53

1 d or equivalently, if f, Df ∈ L (R , Rd), then ± ∓ Fd Df(y) = ∓Fd f(y)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. + 1 d Theorem 4 (Inversion theorem). Suppose f, F f ∈ L (R , Rd). Then Z 1 + + d Kd (x, y)Fd f(y) dy = f(x). (2π) Rd 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 Qf(x) = xf(x).

Lemma 5. Let the angular momentum operators Lij (1 ≤ i, j ≤ d) be as above. Then XX X (5) eiej xkekLij = 0. i

Proof. Given 1 ≤ a < b < c ≤ d, we note that the term in the sum (5) that involves the eaebec is

xaeabcLbc + xbebacLac + xcecabLab = eabc(xaLbc − xbLac + xcLab). However,

xaLbcf − xbLacf + xcLabf ∂f ∂f ∂f ∂f ∂f ∂f = xa xb − xc − xb xa − xc + xc xa − xb ∂xc ∂xb ∂xc ∂xa ∂xb ∂xa ∂f ∂f ∂f ∂f ∂f ∂f = xaxb − + xaxc − + xbxc − = 0 ∂xc ∂xc ∂xb ∂xb ∂xa ∂xa and the proof is complete. Proposition 6. Let Γ, Q be as above. Then (6) ΓQ = (d − 1)Q − QΓ.

Proof. By direct computation we ﬁnd that XX ΓQf(x) = − eiejLij(xf) i

Proof. By iterating the result of Proposition 6, we have for each non-negative integer j, ΓjQ = Q[(d − 1)I − Γ]j. Consequently, ∞ ∞ X (it)j X (it)j exp(itΓ)Q = ΓjQ = Q[(d − 1)I − Γ]j j! j! j=0 j=0 = Q exp(it[(d − 1)I − Γ]) = eit(d−1)Q exp(−itΓ) where we have used the fact that (d − 1)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. d Given a space X of functions deﬁned on R , we say an operator T : X → X is translation-invariant if it commutes with the translation operator, i.e., if T τy = τyT d for all y ∈ R . It is well-known that T is a bounded, linear, translation- 2 d 2 d invariant operator on L (R ) = L (R , C) if and only if there is a bounded d measurable function m : R → C (a multiplier) such that

FdT f(y) = m(y)Fdf(y) d for a.e. y ∈ R . Here Fd 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 opera- 2 2 tor on L (R , H) if and only if there exists a uniformly bounded H-valued 2 function A(y) deﬁned on R for which + + (9) [F2 (T f)(y)] = [A(y)][F2 f(y)] 2 2 2 for all f ∈ L (R , H) and a.e. y ∈ R . TRANSLATION-INVARIANT CLIFFORD OPERATORS 55

Proof. Suppose first that T is bounded, right H-linear and translation- 1 2 2 2 invariant. We may choose a radial real-valued function ϕ ∈ L (R )∩L (R ) dt such that F +ϕ(y) = 0 for |y| < 1 and R ∞ |F +ϕ(ty)|2 = 1 for all y ∈ 2. 2 0 2 t R 2 2 Then any f ∈ L (R , H) may be decomposed via the Calderònreproducing formula Z ∞ ∗ dt f(x) = ϕt ∗ ϕt ∗ f(x) 0 t −2 ∗ where ϕt(x) = t ϕ(x/t) and ϕ (x) =ϕ ¯(−x). Since T is right H-linear and translation-invariant, we have Z ∞ ∗ dt (10) T f(x) = (T ϕt) ∗ ϕt ∗ f(x) . 0 t Taking the QFT of both sides of (10) and applying the convolution theorem (Theorem 2) gives Z ∞ + + + ∗ dt + [F2 (T f)(y)] = [F2 (T ϕt)(y)][F2 ϕ(ty)] [F2 f(y)]. 0 t Our first task is to show that the integral defining Z ∞ + + ∗ dt [A(y)] = [F2 (T ϕt)(y)][F2 ϕ(ty)] 0 t 2 is convergent for a.e. y ∈ R . Let Z 1 + + ∗ dt [A1(y)] = [F2 (T ϕt)(y)][F2 ϕ(ty)] , 0 t Z ∞ + + ∗ dt [A2(y)] = [F2 (T ϕt)(y)][F2 ϕ(ty)] 1 t so that [A(y)] = [A1(y)] + [A2(y)]. Then by the Minkowski and Cauchy- Schwarz inequalities, Z Z Z ∞ + + dt k[A2(y)]k dy ≤ k[F2 (T ϕt)(y)]kk[F2 ϕ(ty)]k dy R2 R2 1 t Z ∞ Z 1/2 Z 1/2 + 2 + 2 dt ≤ k[F2 (T ϕt)(y)]k dy k[F2 ϕ(ty)]k dy . 1 R2 R2 t 2 2 However, for every g ∈ L (R , H), its parity matrix [g(y)] satisfies 2 2 2 2 k[g(y)]k ≤ k[g(y)]kF = |g(y)| + |g(−y)| so that R k[g(y)]k2 dy ≤ 2kgk2. An application of the Plancherel identity R2 2 for the QFT now yields Z Z ∞ d/2 + dt k[A2(y)]k dy ≤ 2(2π) kT ϕtk2kF2 ϕ(t·)k2 R2 1 t Z ∞ d/2 + dt ≤ 2(2π) kT k kϕtk2kF2 ϕ(t·)k2 1 t Z ∞ d 2 dt d 2 ≤ 2(2π) kT kkϕk2 3 = (2π) kT kkϕk2. 1 t 56 JEFF HOGAN AND ANDREW J. MORRIS

+ Hence the integral defining [A2(y)] is convergent for a.e. y. Since F2 ϕ(y) = −α + 2 2 0 for |y| < 1, we have |y| F2 ϕ ∈ L (R , H) for all α ≥ 0, and Z Z 1 Z −2 + + dy dt |y| k[A1(y)]k dy ≤ k[F2 (T ϕt)(y)]k[F ϕ(ty)]k 2 R2 0 R2 |y| t Z 1 Z 1/2 Z 1/2 + 2 + 2 dy dt ≤ k[F2 (T ϕt)(y)]k dy k[F2 ϕ(ty)]k 4 0 R2 R2 |y| t Z 1 Z 1/2 d/2 + 2 dy d 2 ≤ 2(2π) kT kkϕk2 t dt |F2 ϕ(y)| 4 ≤ (2π) kT kkϕk2 0 R2 |y| so that the integral defining [A1(y)] is convergent for a.e. y. We conclude that the integral defining [A(y)] is convergent for a.e. y. Suppose now that T satisfies (9) for some uniformly bounded function A. 2 2 Then the Plancherel theorem immediately gives the L (R , H)-boundedness 2 2 of T . Suppose a ∈ H and f1, f2 ∈ L (R , H). We denote by [a] the matrix a a [a] = s v . Then av as

+ + [F2 (T (f1a + f2))(y)] = [A(y)][F2 (f1a + f2)(y)] + + = [A(y)][F2 f1(y)a + F2 (y)] + + = [A(y)]([F2 f1(y)][a] + [F2 f2(y)]) + + = [A(y)][F2 f1(y)][a] + [A(y)][F2 f2(y)] + + = [F2 (T f1)(y)a] + [F2 (T f2)(y)] + + = [F2 (T f1)(y)a + F2 (T f2)(y)] + = [F2 ((T f1)a + T f2)(y)] from which we conclude that T (f1a + f2) = (T f1)a + T f2, i.e., T is right H-linear. Furthermore, with an application of Lemma 1 we have + + x∧y + [F2 (T τxf)(y)] = [A(y)][F2 (τxf)(y)] = [A(y)][e F2 f(y)] x∧y + x∧y + = [A(y)][e ][F2 f(y)] = [e ][A(y)][F2 f(y)] x∧y + x∧y + + = [e ][F2 (T f)(y)] = [e F2 (T f)(y)] = [F2 (τxT f)(y)] from which we conclude that T τx = τxT , i.e., T is translation-invariant.

2 d A submodule Y ⊂ L (R ,X) is said to be translation-invariant when d τyf ∈ Y for all f ∈ Y and y ∈ R . In the classical case, it is well-known 2 d 2 d that the only closed translation-invariant subspaces of L (R ) = L (R , C) are of the form

2 d Y = YE = {f ∈ L (R ); Fdf(y) = 0 if y∈ / E} d where E is a measurable subset of R . Equivalently, Y is a closed translation- 2 d invariant subspace of L (R ) if and only if there is a measurable subset d d E ⊂ R for which Fdf(y) = χE(y)Fdf(y) for all f ∈ Y and a.e. y ∈ R . The 2 2 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. TRANSLATION-INVARIANT CLIFFORD OPERATORS 57

s(y) v(y) The (Hermitian) adjoint of a parity matrix [A(y)] = v(−y) s(−y) s¯(y) −v(−y) is [A(y)]∗ = . The parity matrix [A(y)] is said to be −v(y)s ¯(−y) self-adjoint if [A(y)]∗ = [A(y)] and idempotent if [A(y)]2 = [A(y)]. 2 2 Theorem 9. Y ⊂ L (R , H) is a closed translation-invariant right H-linear 2 2 submodule of L (R , H) if and only if there exists an idempotent self-adjoint parity matrix [A(y)] such that for all f ∈ Y , + + 2 (11) [F2 f(y)] = [A(y)][F2 f(y)] for a.e. y ∈ R .

2 2 Proof. Suppose Y is a closed translation-invariant submodule of L (R , H). ⊥ 2 If f ∈ Y , g ∈ Y and x ∈ R , then the unitarity of the translation τx gives

hf, τxgi = hτ−xf, gi = 0 ⊥ since τ−xf ∈ Y . Let PY be the orthogonal projection onto Y , If g ∈ Y then PY τxg − τxPY g = 0 since PY Y ⊥ = 0. Further, if f ∈ Y then, because of the translation-invariance of Y we have

PY τxf − τxPY f = PY τxPY f − PY τxPY f = 0 from which we conclude that PY is a translation-invariant operator. We now apply Theorem 8 to conclude that PY is a mulitplier operator, i.e., + there exists a bounded parity matrix [A(y)] for which [F2 (PY f)(y)] = + 2 2 [A(y)][F2 f(y)] for all f ∈ L (R , H). When f ∈ Y , this equation becomes (11). The idempotence and self-adjointness of PY now gives the idempotence and self-adjointness of the multiplier matrix [A(y)]. Conversely, suppose [A(y)] is a bounded, idempotent, self-adjoint parity matrix, and let 2 2 + + 2 Y = {f ∈ L (R , H); [F2 f(y)] = [A(y)][F2 f(y)] for all y ∈ R }. 2 2 It is a simple matter to see that Y is a right H-linear submodule of L (R , H). The boundedness of [A(y)] ensures the closedness of Y . To see that Y is 2 2 d translation-invariant, suppose that f ∈ L (R , H) and x ∈ R . Then + x∧y + x∧y + x∧y + [F2 (τxf)(y)] = [e F2 f(y)] = [e ][F2 f(y)] = [e ][A(y)][F2 f(y)] x∧y + x∧y + + = [A(y)][e ][F2 f(y)] = [A(y)][e F2 f(y)] = [A(y)][F2 (τxf)(y)], i.e., τxf ∈ Y . This concludes the proof. As a consequence of Theorem 9, we see that there are many more examples 2 2 of closed (right H-linear) translation-invariant submodules of L (R , H) than 2 2 there are closed translation-invariant subspaces of L (R , C). To see this, 2 let E ⊂ R be measurable with characteristic function χE(y). The parity matrix associated with E is χE(y) 0 [χE(y)] = . 0 χE(−y)

Then [χE(y)] is idempotent and self-adjoint and its associated translation- 2 2 + invariant subspace is Y = YE = {f ∈ L (R , H); F2 f(y) = 0 if y∈ / E}. d In the classical case, given a measurable subset E ⊂ R and associated 2 d closed translation-invariant space Y = YE of L (R , C) as above, we we 58 JEFF HOGAN AND ANDREW J. MORRIS

2 2 0 have the orthogonal decomposition L (R , H) = YE ⊕ YE0 where E is the d complement of E in R . The most celebrated example on the line is that of 2 the Hardy spaces H±(R) where

2 2 H+(R) = Y[0,∞),H−(R) = Y(−∞,0).

2 2 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 R ∞ 2 supy>0 −∞ |F (x + iy)| dx < ∞. Here C+ is the upper half-plane C+ = {z = x + iy ∈ C; y > 0}. The multiplier associated with the the Hardy spaces is of course m+(y) = χ[0,∞)(y) and m−(y) = χ(−∞,0)(y) which may 1 y be written as m (y) = 1 ± . ± 2 |y| The situation in the quaternionic case is quite diﬀerent. Deﬁne parity matrices [A±(y)] by

1 1 ∓y/|y| [A (y)] = . ± 2 ±y/|y| 1

Then [A±(y)] are uniformly bounded, idempotent, self-adjoint and, by The- 2 2 orem 9, generate closed translation-invariant submodules X± of L (R , H). The fact that [A+(y)][A−(y)] = [A−(y)][A+(y)] = 0 shows that X+ and X− are orthogonal subspaces. Also, since [A+(y)] + [A−(y)] = I, we have the orthogonal decomposition

2 2 L (R , H) = X+ ⊕ X−.

Its important to note that [A±(y)] is the parity matrix of the function 1 y A (y) = 1 ∓ which, in this the quaternionic setting, is not a char- ± 2 |y| acteristic function. In the next section it is shown how X± may be realized as the boundary values of Cliﬀord monogenic functions and that the characterization extends to arbitrary dimensions. 3 2 2 3 As a ﬁnal example, let c(y) = 4y1 − 3y1|y| , d(y) = 3y2|y| − 4y2, and  c(y)e + d(y)e  1 1 2 1 3 [A(y)] =  |y|  . −c(y)e − d(y)e  2  1 2 1  |y|3

Then [A(y)] is uniformly bounded, idempotent and self-adjoint and there- fore generates a closed translation-invariant right H-linear submodule of 2 2 L (R , H) which also does not arise from a characteristic function.

4. Hilbert transforms and monogenic extensions d Consider the Green’s functions G(x, t)(x ∈ R , t ∈ R) deﬁned by 1 x + t G(x, t) = d+1 σd−1 |x + t| TRANSLATION-INVARIANT CLIFFORD OPERATORS 59

2π(d+1)/2 where σ = is the surface area of the unit sphere in d. By d−1 Γ((d + 1)/2) R direct calculation we find that ∂G(x, t) (12) D G(x, t) − = 0. x ∂t 2 d The Cauchy transform Cf of a function f ∈ L (R , Rd) is defined by the convolution Z Cf(y + t) = G(x − y, t)f(x) dx. Rd d+1 d Then by (12), Cf(y − t) is left monogenic on R \ R . 2 d Consider operators P± defined on L (R , Rd) by

(13) P+f(x) = lim Cf(x + t); P−f(x) = − lim Cf(x − t). t↓0 t↓0 2 d Then [3] P± are bounded projections on L (R , Rd) and we define Hardy 2 d spaces H±(R ) by 2 d H±(R ) = Ran (P±). We aim to give a Clifford-Fourier characterization of these spaces which complements the classical Fourier characterization of [3]. 2 d 2 d Theorem 10. Let H±(R ) be defined as above and f ∈ L (R , Rd). Then 1 (14) f ∈ H2 ( d) ⇐⇒ [F +f(y)] = [1 ∓ y/|y|][F +f(y)]. ± R d 2 d

Proof. Observe that 1 (15) Cf(x + t) = − φt ∗ f(x) σd−1 x − 1 where φ(x) = and φ is the L1-normalized dilate of φ given by |x − 1|d+1 t −d φt(x) = t φ(x/t). Also, Z e−ihx,yi π(d+1)/2 dx = e−|y| 2 (d+1)/2 Rd (1 + |x| ) Γ((d + 1)/2) and consequently, Z x e−ihx,yi π(d+1)/2 ∂ π(d+1)/2 y j dx = i e−|y| = −i j e−|y|. 2 (d+1)/2 Rd (1 + |x| ) Γ((d + 1)/2) ∂yj Γ((d + 1)/2) |y| From this we see that the classical FT of φ is given by π(d+1)/2 y (16) F φ(y) = − 1 + i e−|y|. d Γ((d + 1)/2) |y| 1 d Now if f, g ∈ C (R , Rd) and g is radial, then Γ(fg) = (Γf)g. Further, y y Γ = (d − 1) , so combining (15) and (16) gives F Cf(·, t)(y) = |y| |y| d 1 y 1 + i e−t|y|F f(y) and letting t → 0 gives 2 |y| d 1 y F P f(y) = 1 + i F f(y). d + 2 |y| d 60 JEFF HOGAN AND ANDREW J. MORRIS

Hence, with an application of Corollary 7 we have eiπd/4 y F +P f(y) = e−i(π/2)Γ F f(y) + i F f(y) d + 2 d |y| d 1 i = F +f(y) + eiπd/4 e−i(π/2)ΓQF f(y) 2 d |y| d 1 iy = F +f(y) + e−iπd/4 ei(π/2)ΓF f(y) 2 d |y| d 1 y = F +f(y) − F −f(y) . 2 d |y| d + 2 + − 2 However, since (Fd ) = I (the identity) and Fd Fd = Fd = τ, where τ is − + the inversion τf(x) = f(−x), we see that Fd f(y) = Fd f(−y), so that 1 y F +P f(y) = F +f(y) − F +f(−y) . d + 2 d |y| d 1 y Similarly we may show that F +P f(y) = F +f(y) + F +f(−y) . To d − 2 d |y| d 2 d prove the Clifford-Fourier characterization (14) of the Hardy spaces H+(R ), note that if u ∈ Rd and y ∈ Λ1, then (yu)e = yuo and (yu)o = yue, and consequently 2 f ∈ H± ⇐⇒ P±f = f + + ⇐⇒ Fd P±f = Fd f(y) 1 y (17) ⇐⇒ F +f(y) = F +f(y) ∓ F +f(−y) . d 2 d |y| d Splitting each side of (17) into even and odd parts gives (14). 1 y 1 1 ∓y/|y| Note that the parity matrices [χ (y)] = 1∓ = ± 2 |y| 2 ±y/|y| 1 satisfy 2 2 [χ+(y)] = [χ−(y)] , [χ+(y)][χ−(y)] = [χ−(y)][χ+(y)] = 0 2 d from which we see that L (R , Rd) decomposes orthogonally as 2 d 2 d 2 d L (R , Rd) = H+(R ) ⊕ H−(R ). 2 d Define now the Clifford-Hilbert transform H which acts on L (R , Rd) by the principal value singular integral 1 Z y − x (18) Hf(y) = lim d+1 f(x) dx. ε→0 σd ε<|x−y|<1/ε |y − x| Theorem 11 below provides a characterization of the Hardy spaces in terms of the action of the Clifford-Hilbert transform. 1 x For each 0 < ε < 1, let kε(x) = d+1 χ<ε<|x|<1/ε(x). Then kε ∈ σd |x| 1 d L (R , Rd) and Hf(x) = limε→0 kε ∗ f(x). Consequently,

FdHf = lim(Fdkε)(Fdf). ε→0 TRANSLATION-INVARIANT CLIFFORD OPERATORS 61

∂ −ihx,yi We aim ﬁrst to compute m(y) = limε→0 Fdkε(y). Since e = ∂yj −ihx,yi −ixje , we have

d Z 1 X xj −ihx,yi m(y) = ej lim e dx σ ε→0 |x|d+1 d j=1 ε<|x|<1/ε d i X ∂ Z e−ihx,yi = ej lim dx. σ ε→0 ∂y |x|d+1 d j=1 j ε<|x|<1/ε Conversion of the integral on the right hand side to spherical co-ordinates R −ihω,ξi d/2 1−d/2 and applying the fact that Sd−1 e dω = (2π) |ξ| Jd/2−1(|ξ|) (with Jd/2−1 the Bessel function of the ﬁrst kind [6]) yields

d Z 1/ε Z i X ∂ 1 −irhω,yi m(y) = ej lim 2 e dω dr σ ε→0 ∂y r d−1 d j=1 j ε S d Z 1/ε i X ∂ d/2 1−d/2 dr = ej lim (2π) (r|y|) Jd/2−1(r|y|) . σ ε→0 ∂y r2 d j=1 j ε

However, the Bessel functions Jν satisfy the diﬀerential recurrence relation 1 d (t−νJ (t)) = −t−ν−1J (t), so t dt ν ν+1

∂ 1−d/2 d 1−d/2 ∂ [(r|y|) Jd/2−1(r|y|)] = [t Jd/2−1(t)] (r|y|) ∂yj dt t=r|y| ∂yj ry y = − j (t1−d/2J (t)) = −r2−d/2 j J (r|y|) d/2 d/2 d/2 |y| t=r|y| |y| and as a consequence,

d/2 d Z 1/ε i(2π) X yj −d/2 m(y) = − ej lim r Jd/2(r|y|) dr σ |y|d/2 ε→0 d j=1 ε d/2 Z ∞ i(2π) y −d/2 (19) = − s Jd/2(s) ds. σd |y| 0 ν The Gegenbauer polynomials Cn and Bessel functions Jµ are related via the Fourier transform as follows: Z 1 1−ν n 2 ν−1/2 iax ν π2 i Γ(2ν + n) −ν (1 − x ) e Cn(x) dx = a Jν+n(a) −1 n!Γ(ν) ν whenever <ν > −1/2 (see equation 7.321 of [6]). However C0 ≡ 1 for all ν, so Z 1 1−ν 2 ν−1/2 iax π2 Γ(2ν) −ν (1 − x ) e dx = a Jν(a) −1 Γ(ν) or equivalently, 1−ν Z ∞ 2 ν−1/2 1 π2 Γ(2ν) −ν (1 − x ) = a Jν(a) da. 2π Γ(ν) 0 62 JEFF HOGAN AND ANDREW J. MORRIS

Γ(ν)2ν−1 Putting x = 0 gives R ∞ a−νJ (a) da = so that with an application 0 ν Γ(2ν) 22x−1 of the Gamma function doubling formula Γ(2x) = √ Γ(x)Γ(x+1/2) (see π [6]) equation (19) becomes i(2π)d/2 y Γ(d/2)2d/2−1 i y m(y) = − = − . σd |y| Γ(d) 2 |y| Consequently, i y (20) F +(Hf)(y) = − eiπd/4e−i(π/2)ΓQf(y) = − F +(−y). d 2|y| 2|y| d Comparing (20) with (17) gives the following result. 2 Theorem 11. Let H± be the Hardy spaces defined above and H be the Clifford Hilbert transform defined in (18). Then 2 f ∈ H± ⇐⇒ Hf = ∓f. Acknowledgements Hogan gratefully acknowledges the support of the University of Newcas- tle’s Centre for Computer Assisted Research in Mathematics and its Ap- plications (CARMA). He also wishes to thank Wayne Lawton and Philip Taylor for their insights. Thanks, Roy. Thanks, HG.

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Jeff Hogan: School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2289, Australia E-mail address: [email protected]

Andrew J. Morris: School of Mathematical and Physical Sciences, Univer- sity of Newcastle, Callaghan, NSW 2289, Australia E-mail address: [email protected]