The Cylindrical Fourier Transform

The Cylindrical Fourier Transform Fred Brackx, Nele De Schepper, and Frank Sommen Abstract In this paper we devise a so-called cylindrical Fourier transform within the Clifford analysis context. The idea is the following: for a fixed vector in the image space the level surfaces of the traditional Fourier kernel are planes perpendicular to that fixed vector. For this Fourier kernel we now substitute a new Clifford-Fourier kernel such that, again for a fixed vector in the image space, its phase is constant on co-axial cylinders w.r.t. that fixed vector. The point is that when restricting to dimension two this new cylindrical Fourier transform coincides with the earlier introduced Clifford-Fourier transform. We are now faced with the following situation: in dimension greater than two we have a first Clifford-Fourier transform with elegant properties but no kernel in closed form, and a second cylindrical one with a kernel in closed form but more complicated calculation formulae. In dimension two both transforms coincide. The paper concludes with the calculation of the cylindrical Fourier spectrum of an L -basis consisting of generalized Clifford-Hermite functions. Introduction The Fourier transform is by far the most important integral transform. Since its introduction by Fourier in the early 8s it has remained an indispensible and stimulating mathematical concept that is at the core of the highly evolved branch of mathematics called Fourier analysis. The second player in this paper is Clifford analysis, an elegant and powerful higher dimensional generalization of the theory of holomorphic functions, which is moreover closely related but complementary to harmonic analysis. Clifford analysis also offers the possibility to generalize one-dimensional mathematical analysis to higher Clifford Research Group, Department of Mathematical Analysis, Ghent University, Galglaan, 9 Gent, Belgium; e-mail: fb@cage.ugent.be Fred Brackx, nds@cage.ugent.be Nele De Schepper, fs@cage.ugent.be Frank Sommen.

Fred Brackx, Nele De Schepper, and Frank Sommen dimension in a rather natural way by encompassing all dimensions at once, as opposed to the usual tensorial approaches. It is precisely this last qualification which has been exploited in [ and [3 to construct a genuine multidimensional Fourier transform within the context of Clifford analysis. This so-called Clifford-Fourier transform is briefly discussed in Section 3. In this paper see Section 4 we devise a new, so-called cylindrical Fourier transform by substituting for the standard inner product in the classical exponential Fourier kernel a wedge product as argument. Finally, our aim is to calculate the cylindrical Fourier spectrum of an L -basis consisting of generalized Clifford-Hermite functions. To make the paper self-contained, we have also included a section Section on Clifford analysis. The Clifford analysis toolkit Clifford analysis see e.g. [ offers a function theory which is a higher dimensional analogue of the theory of the holomorphic functions of one complex variable. The functions considered are defined in R m m > and take their values in the Clifford algebra R,m or its complexification C m = R,m C. If e,...,e m is an orthonormal basis of R m, then a basis for the Clifford algebra R,m or C m is given by all possible products of basis vectors e A : A {,...,m} where e / = is the identity element. The non-commutative multiplication in the Clifford algebra is governed by the rules: e j e k + e k e j = δ j,k j,k =,...,m. Conjugation is defined as the anti-involution for which e j = e j j =,...,m. In case of C m, the Hermitean conjugate of an element λ = A λ A e A λ A C is defined by λ = A λ c A e A, where λ c A denotes the complex conjugate of λ A. This Hermitean conjugation leads to a Hermitean inner product and its associated norm on C m given respectively by λ,µ = [λ µ and λ = [λ λ = λ A, A where [λ denotes the scalar part of the Clifford element λ. The Euclidean space R m is embedded in the Clifford algebras R,m and C m by identifying the point x,...,x m with the vector variable x given by x = m j= e jx j. The product of two vectors splits up into a scalar part the inner product up to a minus sign and a so-called bivector part the wedge product: where x. y = < x,y > = m j= x y = x. y + x y, x j y j and x y = m m i= j=i+ e i e j x i y j x j y i.

The Cylindrical Fourier Transform 3 Note that the square of a vector variable x is scalar-valued and equals the norm squared up to a minus sign: x = < x,x > = x. The central notion in Clifford analysis is the notion of monogenicity, a notion which is the multidimensional counterpart to that of holomorphy in the complex plane. A function Fx,...,x m defined and continuously differentiable in an open region of R m and taking values in R,m or C m, is called left monogenic in that region if x [F =. Here x is the Dirac operator in R m : x = m j= e j x j, an elliptic, rotation-invariant, vector differential operator of the first order, which may be looked upon as the square root of the Laplace operator in R m : m = x. This factorization of the Laplace operator establishes a special relationship between Clifford analysis and harmonic analysis in that monogenic functions refine the properties of harmonic functions. In the sequel the monogenic homogeneous polynomials will play an important rôle. A left monogenic homogeneous polynomial P k of degree k k in R m is called a left solid inner spherical monogenic of order k. The set of all left solid inner spherical monogenics of order k will be denoted by M l + k. The dimension of k is given by M + l The set dim m + k M l + k = m = m + k! m! k! φ s,k, j x = m/4 γ s,k / H s,k x P j k x e x / s,k N, j dim M l + k, constitutes an orthonormal basis for the space L R m of square integrable functions. Here { P j k x; j dim M l + k} denotes an orthonormal basis of M l + k and γ s,k a real constant depending on the parity of s. The polynomials H s,k x are the so-called generalized Clifford-Hermite polynomials introduced by Sommen; they are a multidimensional generalization to Clifford analysis of the classical Hermite polynomials on the real line. Note that H s,k x is a polynomial of degree s in the variable x with real coefficients depending on k. Furthermore H s,k x only contains even powers of x and is hence scalar-valued, while H s+,k x only contains odd ones and is thus vector-valued. A result which will be frequently used in subsection 4.3 is the following generalization of the classical Funk-Hecke theorem. Theorem. [Funk-Hecke theorem in space Let S k be a spherical harmonic of degree k and η a fixed point on the unit sphere S m in R m. Denote < ω,η > = cos ω,η = t η for ω S m. Then gr f t η S k ω dv x R m + = A m gr r m dr f t t m 3/ P k,m t dt S k η,.

4 Fred Brackx, Nele De Schepper, and Frank Sommen where dv x denotes the Lebesgue measure on R m, P k,m t the Legendre polynomial of degree k in m-dimensional Euclidean space and A m = πm / area of the unit sphere S m in R m. Γ m the surface As the Legendre polynomials are even or odd according to the parity of k, we can also state the following corollary. Corollary. Let S k be a spherical harmonic of degree k and η a fixed point on the unit sphere S m. Denote < ω,η > = t η for ω S m, then the 3D-integral is zero whenever f is an odd function and k is even f is an even function and k is odd. R m gr f t η S k ω dv x 3 The Clifford-Fourier transform In [ a new multidimensional Fourier transform in the framework of Clifford analysis, the so-called Clifford-Fourier transform, is introduced. The idea behind its definition originates from an alternative representation for the standard tensorial multidimensional Fourier transform given by F [ f ξ = π m/ R m e i<x,ξ > f x dv x. It is indeed so that this classical Fourier transform can be seen as the operator exponential F = e i π/ H = i π k H k k! where H is the scalar-valued differential operator H = m + r m. Note that due to the scalar character of the standard Fourier kernel, the Fourier spectrum inherits its Clifford algebra character from the original signal, without any interaction with the Fourier kernel. So in order to genuinely introduce the Clifford analysis character in the Fourier transform, the idea occurred to us to replace the scalar-valued operator H in the operator exponential by a Clifford algebra-valued one. To that end we aimed at factorizing the operator H, making use of the factorization of the Laplace operator by the Dirac operator. Splitting H into a sum of Clifford algebra-valued second order operators, leads in a natural way to a pair of transforms F H ±, the harmonic average of which is precisely the standard Fourier transform F : F = F H + F H. The two-dimensional case of this Clifford-Fourier transform is special in that we are able to find a closed form for the kernel of the integral representation. Indeed, k=

The Cylindrical Fourier Transform 5 the two-dimensional Clifford-Fourier transform takes the form F H ±[ f ξ = ±ξ x f x dv x. π R e This closed form enables us to generalize the well-known results for the standard Fourier transform both in the L and in the L -context see [3. Note that we have not succeeded yet in obtaining such a closed form in arbitrary dimension. 4 The cylindrical Fourier transform 4. Definition The cylindrical Fourier transform is obtained by taking the multidimensional generalization of the two-dimensional Clifford-Fourier kernel. Definition. The cylindrical Fourier transform of a function f is given by with e x ξ = r= [ f ξ = x ξ r r!. π m/ R m ex ξ f x dv x The integral kernel of this cylindrical Fourier transform can be rewritten in terms of the cosine and the sinc function, which also reveals its form of a scalar plus a bivector, i.e. a so-called parabivector. Proposition. The kernel of the cylindrical Fourier transform can be rewritten as where sincx := sinx x e x ξ = cos x ξ + x ξ sinc x ξ is the unnormalized sinc function. Proof. Splitting the defining series expansion of e x ξ into its even and odd part and taking into account that x ξ = x ξ yields e x ξ = l= x ξ l l l! + x ξ l= x ξ l l l +! = cos x ξ + x ξ sinc x ξ. Let us now explain why we have chosen the name cylindrical for our new Fourier transform. From x ξ = x ξ < x,ξ > = x ξ cos x,ξ = x ξ sin x,ξ

6 Fred Brackx, Nele De Schepper, and Frank Sommen Fig. In case of the cylindrical Fourier transform, for fixed ξ, the phase x ξ is constant on co-axial cylinders. Fig. In case of the classical Fourier transform, for fixed ξ, the phase < x,ξ > is constant on planes perpendicular to ξ. x x cosx, it is clear that for ξ fixed, the phase x ξ is constant if and only if x sin x,ξ is constant. In other words, for a fixed vector ξ in the image space, the phase x ξ is constant on co-axial cylinders w.r.t. that fixed vector see Figure. For comparison, for a fixed vector ξ in the image space the level surfaces of the traditional Fourier kernel are planes perpendicular to that fixed vector, since < x,ξ > = x ξ cos x,ξ see Figure. 4. Properties The cylindrical Fourier transform is well-defined for each integrable function. Theorem. Let f L R m. Then [ f L R m C R m and moreover Fcyl [ f m/ f π. Proof. Taking into account Proposition, we have that e x ξ x ξ = cos x ξ + sin x ξ x ξ cos x ξ + sin x ξ

The Cylindrical Fourier Transform 7 which leads to the desired result. Although the cylindrical Fourier transform has a simple integral kernel, it satisfies calculation formulae which are more complicated than those of the multidimensional Clifford-Fourier transform see [4. For example, we state the differentiation and multiplication rule which nicely show that the two-dimensional case, in which the cylindrical Fourier transform and the Clifford-Fourier transform coincide, is special. Proposition differentiation and multiplication rule. Let f,g L R m. The cylindrical Fourier transform satisfies: i the differentiation rule [ x [ f x m ξ = ξ [ f x ξ + π m/ ξ sinc x ξ f x dv x Rm with sincx := sinx x the unnormalized sinc function; ii the multiplication rule [ [x f xξ = ξ Fcyl [ f x ξ m + π m/ sinc x ξ x f x dv x. Rm 4.3 Spectrum of the L -basis consisting of generalized Clifford- Hermite functions Finally our aim is to calculate the cylindrical Fourier spectrum of the L -basis. As these basis elements belong to the space of rapidly decreasing functions S R m L R m, their cylindrical Fourier image should be a bounded and continuous function. The calculation method is based on the Funk-Hecke theorem in space see Theorem and the following cylindrical Fourier kernel decomposition see Proposition e x ξ = cos rρ t η where we have introduced spherical co-ordinates rρ t η sinc rρ tη rρ η ω sinc rρ tη x = rω, ξ = ρη, r = x, ρ = ξ, ω, η S m and the notation t η = < ω,η >. For convenience s sake, we denote the three terms in the decomposition by A, B and C. As a first example, let us now calculate the cylindrical Fourier transform of the basis function φ,k, j x which is given, up to constants, by P k x e x / with P k a left solid inner spherical monogenic of order k. By Corollary it is obvious that we must make a distinction between k even and odd.,

8 Fred Brackx, Nele De Schepper, and Frank Sommen A k even In the case where k is even, as a consequence of Corollary the integrals containing the B- and C-term of the kernel decomposition reduce to zero. Furthermore, applying the Funk-Hecke theorem in space see Theorem we have that [e x / P k x ξ = π m/ rρ tη P k ωdv x R m e r / r k cos = A + m π m/ P kη e r/ r k+m dr cos rρ t t m 3/ P k,m t dt Taking into account the series expansion of the cosine function, this result becomes. [e x / k! m 3! A m P k x ξ = k + m 3! π m/ P kη l l= l! ρl + e r/ r l+k+m dr t l+m 3/ C m / t dt k, 3 where we have also used the expression k! m 3! P k,m t = k + m 3! Cm / k t of the Legendre polynomials in R m in terms of the Gegenbauer polynomials Ck λ t. As these Gegenbauer polynomials Ck λ are orthogonal on,[ w.r.t. the weight function t λ / λ >, it is easily seen that for l k holds t l t m 3/ C m / k t dt =. Moreover, combining the integral formula see [7, p. 86, formula 4 with α = β t α Ck λ t dt = α+ Γ α + Γ k + λ k! Γ λ Γ α + 3F k,k + λ,α + ;λ +,α + ;, where Reα > and 3 F a,b,c;d,e;z denotes the generalized hypergeometric series, with Watson s theorem see for e.g. [6 3F a,b,c; a + b +,c; = π Γ c + Γ a+b+ Γ a b+c Γ a+ Γ b+ Γ a+c Γ b+c

The Cylindrical Fourier Transform 9 results into = t α Ck λ t dt π α+ Γ α + Γ k + λ Γ α + 3 Γ λ + Γ α λ+3 k! Γ λ Γ α + Γ k+ Γ k+λ+ Γ. α+3+k Γ α λ+3 k 4 Applying the above result and taking into account that Γ z = π / z Γ z Γ z + /, equation 3 can be simplified to [e x / P k x ξ k/ π = Γ k+ Γ k+m P k ξ l l l! Γ l+m l=k/ l! Γ ξ l k l+ k = F m ; k + ξ ; e ξ / P k ξ with F a;c;z Kummer s function, also called confluent hypergeometric function. B k odd For k odd, the integral containing the A-term of the kernel decomposition is zero, as a consequence again of Corollary. By means of the Funk-Hecke theorem in space we obtain [e x / P k x ξ = ρ A m sinc rρ + π m/ P kη e r/ r k+m dr t t m 3/ P k+,m t tp k,m t dt. Now, taking into account the Gegenbauer recurrence relation we have that which in its turn yields k + λ t C λ k t k + Cλ k+ t = λ t C λ+ k t, k! m! P k+,m t tp k,m t = k + m! t C m/ k t,

Fred Brackx, Nele De Schepper, and Frank Sommen [e x / P k x ξ = + e r/ r k+m dr sinc k! m! A m k + m! π m/ ρ P kη rρ t t m / C m/ k t dt Next, applying consecutively the series expansion of the sinc function, the orthogonality of the Gegenbauer polynomials and expression 4, we find [e x / P k x ξ = F m ; k + ξ ; e ξ / P k ξ. So note that the cylindrical Fourier transform reproduces the Gaussian times the spherical monogenic up to a Kummer s function factor. A second example is provided by the cylindrical Fourier transform of the basis function φ,k, j which is given, up to constants, by e x / x P k x. Its calculation runs along similar lines. Making again a distinction between k even and k odd, we find A k even [e x / x P k x ξ = k + m k + F m ; k + 3 ξ ; e ξ / ξ P k ξ B k odd [e x / x P k x ξ = F m ; k + ξ ; e ξ / ξ P k ξ. showing again the reproducing property up to a Kummer s function factor. For the calculation of the cylindrical Fourier spectrum of a general basis element φ s,k, j we refer to [5. 5 Concluding remarks In the foregoing section we established the image under the cylindrical Fourier transform of an L -basis for the space of all L -functions in R m. Using density arguments, these results may be used to approximate the cylindrical Fourier image of various types of functions and distributions in R m. But for certain types of functions or distributions, direct calculation methods are available on top of this approximation. A

The Cylindrical Fourier Transform Fig. 3 The real part of the cylindrical Fourier spectrum of the characteristic function of a geodesic triangle on S. typical example is the case of distributions concentrated on the unit sphere of the form Fx = δr f ω, x = rω, r = x [, [, ω S m. The corresponding cylindrical Fourier transform is given by [Fξ = [ f ξ = π m/ expω ξ f ω dsω. Sm Hereby ξ still belongs to the whole space R m, while the data f ω are defined on the unit sphere, a codimension one surface of R m. It is hence expected that the data f ω are already determined by the cylindrical Fourier image restricted to a suitable codimension one surface as well, typical examples being: i ξ = η S m, leading to an integral transform from S m to S m, ii ξ = ξ e +... + ξ m e m + e m, i.e. ξ belongs to the affine subspace given by ξ m =. To evaluate the cylindrical Fourier transform explicitly it suffices in both cases to express the function f ω as a series of spherical monogenics and to apply a Funk- Hecke argument on the spherical monogenics. This may lead to correspondences between function spaces on S m and isomorphisms between them including inversion methods. The establishment of direct inversion formulae remains an independent and interesting problem for future research. Fig. 4 The e e -component of the cylindrical Fourier spectrum of the characteristic function of a geodesic triangle on S.

Fred Brackx, Nele De Schepper, and Frank Sommen As an example see Figure 3, 4, 5 and 6 we have computed directly the cylindrical Fourier image of the characteristic function of a geodesic triangle on the two sphere S that may be expressed in spherical co-ordinates by the integral π 3/ π/ π/ expω ξ sinθ dθ dφ with ω = sinθcosφe + sinθsinφe + cosθe 3 and ξ = ae + be + e 3. Fig. 5 The e e 3 -component of the cylindrical Fourier spectrum of the characteristic function of a geodesic triangle on S. Fig. 6 The e e 3 -component of the cylindrical Fourier spectrum of the characteristic function of a geodesic triangle on S. References. Brackx, F., Delanghe, R., Sommen, F.: Clifford analysis. Pitman Publishers, Boston - London - Melbourne 98. Brackx, F., De Schepper, N., Sommen, F.: The Clifford-Fourier Transform. J. Fourier Anal. Appl. 5 doi:.7/s4-5-479-9 3. Brackx, F., De Schepper, N., Sommen, F.: The Two-Dimensional Clifford-Fourier Transform. J. Math. Imaging Vision 6 doi:.7/s85-6-365-y 4. Brackx, F., De Schepper, N., Sommen, F.: The Fourier Transform in Clifford Analysis to appear in Advances in Imaging & Electron Physics 5. Brackx, F., De Schepper, N., Sommen, F.: The Cylindrical Fourier Spectrum of an L -basis consisting of Generalized Clifford-Hermite Functions, submitted for: Proceedings of ICNAAM 8, Kos, Greece, 8 6. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions. McGraw-Hill, New York 953 7. Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. Academic Press, New York - London - Toronto - Sydney - San Francisco 98