The Fourier Transform on Quantum Euclidean Space

We study Fourier theory on quantum Euclidean space. A modified version of the general definition of the Fourier transform on a quantum space is used and its inverse is constructed. The Fourier transforms can be defined by their Bochner's relations and a new type of q-Hankel transforms using the...

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Дата:	2011
Автор:	Coulembier, K.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2011
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	The Fourier Transform on Quantum Euclidean Space / K. Coulembier // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 36 назв. — англ.

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spelling	irk-123456789-1471672019-02-14T01:26:07Z The Fourier Transform on Quantum Euclidean Space Coulembier, K. We study Fourier theory on quantum Euclidean space. A modified version of the general definition of the Fourier transform on a quantum space is used and its inverse is constructed. The Fourier transforms can be defined by their Bochner's relations and a new type of q-Hankel transforms using the first and second q-Bessel functions. The behavior of the Fourier transforms with respect to partial derivatives and multiplication with variables is studied. The Fourier transform acts between the two representation spaces for the harmonic oscillator on quantum Euclidean space. By using this property it is possible to define a Fourier transform on the entire Hilbert space of the harmonic oscillator, which is its own inverse and satisfies the Parseval theorem. 2011 Article The Fourier Transform on Quantum Euclidean Space / K. Coulembier // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 36 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B37; 81R60; 33D50 DOI:10.3842/SIGMA.2011.047 http://dspace.nbuv.gov.ua/handle/123456789/147167 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We study Fourier theory on quantum Euclidean space. A modified version of the general definition of the Fourier transform on a quantum space is used and its inverse is constructed. The Fourier transforms can be defined by their Bochner's relations and a new type of q-Hankel transforms using the first and second q-Bessel functions. The behavior of the Fourier transforms with respect to partial derivatives and multiplication with variables is studied. The Fourier transform acts between the two representation spaces for the harmonic oscillator on quantum Euclidean space. By using this property it is possible to define a Fourier transform on the entire Hilbert space of the harmonic oscillator, which is its own inverse and satisfies the Parseval theorem.
format	Article
author	Coulembier, K.
spellingShingle	Coulembier, K. The Fourier Transform on Quantum Euclidean Space Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Coulembier, K.
author_sort	Coulembier, K.
title	The Fourier Transform on Quantum Euclidean Space
title_short	The Fourier Transform on Quantum Euclidean Space
title_full	The Fourier Transform on Quantum Euclidean Space
title_fullStr	The Fourier Transform on Quantum Euclidean Space
title_full_unstemmed	The Fourier Transform on Quantum Euclidean Space
title_sort	fourier transform on quantum euclidean space
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/147167
citation_txt	The Fourier Transform on Quantum Euclidean Space / K. Coulembier // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 36 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT coulembierk thefouriertransformonquantumeuclideanspace AT coulembierk fouriertransformonquantumeuclideanspace
first_indexed	2025-07-11T01:31:00Z
last_indexed	2025-07-11T01:31:00Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 047, 30 pages The Fourier Transform on Quantum Euclidean Space? Kevin COULEMBIER Gent University, Galglaan 2, 9000 Gent, Belgium E-mail: Coulembier@cage.ugent.be URL: http://cage.ugent.be/~coulembier/ Received November 19, 2010, in final form April 21, 2011; Published online May 11, 2011 doi:10.3842/SIGMA.2011.047 Abstract. We study Fourier theory on quantum Euclidean space. A modified version of the general definition of the Fourier transform on a quantum space is used and its inverse is constructed. The Fourier transforms can be defined by their Bochner’s relations and a new type of q-Hankel transforms using the first and second q-Bessel functions. The behavior of the Fourier transforms with respect to partial derivatives and multiplication with variables is studied. The Fourier transform acts between the two representation spaces for the harmonic oscillator on quantum Euclidean space. By using this property it is possible to define a Fourier transform on the entire Hilbert space of the harmonic oscillator, which is its own inverse and satisfies the Parseval theorem. Key words: quantum Euclidean space; Fourier transform; q-Hankel transform; harmonic analysis; q-polynomials; harmonic oscillator 2010 Mathematics Subject Classification: 17B37; 81R60; 33D50 1 Introduction There has been a lot of interest in formulating physics on noncommutative space-times, see e.g. [3, 5, 14, 15, 23, 32]. In particular, since non-commutativity implies a quantized space- time, quantum field theories on such spaces should be well-behaved in the ultraviolet-limit, see e.g. [3]. The infinities of the commutative, continuous theories could appear as poles in the q-plane with q a deformation parameter. In such theories quantum groups replace Lie groups in the description of the symmetries. An important concept in this theory is integration and Fourier theory on quantum spaces, see e.g. [4, 23, 32, 34, 35, 36]. The Fourier kernel is defined in [29, Definition 4.1]. In this paper we study the Fourier theory on quantum Euclidean space, which has symmetry group Oq(m). The deformation parameter q is always assumed to satisfy 0 < q < 1. The Fourier transform is studied from the point of view of harmonic analysis on quantum Euclidean space, see e.g. [5, 15, 20, 31, 33]. This is captured in the Howe dual pair (Oq(m),Uq(sl2)). The quantum algebra Uq(sl2) is generated by the Oq(m)-invariant norm squared and Laplace operator on quantum Euclidean space. The Fourier transform was defined in an abstract Hopf-algebraic setting in [23]. In this article, the Fourier transform on quantum Euclidean space is studied analytically. This leads to explicit formulae for the behavior of the Fourier transform with respect to partial derivatives. The definition of the Fourier transform is also extended from spaces of polynomials weighted with Gaussians to an appropriate Hilbert space, which was a problem left open in [23]. A general theory of Gaussian-induced integration on quantum spaces was developed in [23]. We use this procedure on quantum Euclidean space for the two types of calculus defined in [6]. One of the two integrations we obtain corresponds to the result in [14, 34]. Both types of ?This paper is a contribution to the Special Issue “Relationship of Orthogonal Polynomials and Spe- cial Functions with Quantum Groups and Integrable Systems”. The full collection is available at http://www.emis.de/journals/SIGMA/OPSF.html mailto:Coulembier@cage.ugent.be http://cage.ugent.be/~coulembier/ http://dx.doi.org/10.3842/SIGMA.2011.047 http://www.emis.de/journals/SIGMA/OPSF.html 2 K. Coulembier integration can be written as a combination of integration over the quantum sphere, see [34], and radial Jackson integration, see e.g. [17, Section 1.11]. Each one of the integrations satisfies Stokes’ theorem for both types of calculus. It turns out that Fourier theory is defined more naturally using the Fourier kernel for one calculus combined with the Gaussian-induced inte- gration for the other calculus. This implies we use a generalized Gaussian-induced integration compared to [23]. This was also done implicitly for the analytical approach to Fourier theory on the braided line in [26]. We calculate the quantum sphere integral of spherical harmonics weighted with the Fourier kernel, which yields a q-deformed Bessel function. This function is known as the first q-Bessel function, see [21]. As a side result we obtain a Funk–Hecke theorem on quantum Euclidean space. This allows us to construct the reproducing kernel for the spheri- cal harmonics. The reproducing kernel can be expressed as a q-Gegenbauer polynomial in terms of the generalized powers of the inner product constructed in [32]. Because of the appearance of q-Bessel functions, the combination of radial integration with the spherical integration and the exponential leads to new q-deformed Hankel transforms. In [27] the q-Hankel transforms corresponding to the so-called third q-Bessel functions were defined and studied. In the current paper the q-Hankel transforms for the first and second q-Bessel functions are introduced. It is proven that they are each other’s inverse by applying the theory of the q- Laguerre polynomials, see [30]. Then the inverse of the Fourier transform on quantum Euclidean space is defined by its Bochner’s relations in terms of the second q-Hankel transform. The fact that the Fourier transforms can be expressed in terms of Bochner’s relations is an immediate consequence of their Oq(m)-invariance. It is proven that the transforms behave canonically with respect to partial derivatives and multiplication with variables. Furthermore, we extend the domain of the Fourier transforms from spaces corresponding to polynomials weighted with a Gaussian to the Hilbert space structure of [14]. This Hilbert space has two representations in function spaces; the Fourier transform and its inverse act between these spaces. The first and second q-Bessel function can be connected by a substitution q ↔ q−1. The first q-Bessel function has a finite domain of analyticity contrary to the second one. This implies that the inverse Fourier transform is better suited to generalize to a Hilbert space. By composing this Fourier transform with the projection operators corresponding to the two dual representations of the Hilbert space, we obtain a Fourier transform which can be defined on the entire Hilbert space. This transform is its own inverse and satisfies a Parseval theorem. In [10] the theory of the q-Dirac and q-Laplace operator on undeformed Euclidean space was developed. The q-Laplace operator is O(m)-invariant and generates Uq(sl2) together with the classical norm squared. This implies that q-harmonic analysis on Euclidean space corresponds to the Howe dual pair (O(m),Uq(sl2)), i.e. there is no spherical deformation and the radial de- formation corresponds to that of quantum Euclidean space. Therefore, the q-Hankel transforms in the current paper can also be used to construct an O(m)-invariant q-Fourier transform on Euclidean space, connected to the q-Dirac operator. The paper is organized as follows. First an introduction to q-calculus, quantum Euclidean space and Fourier theory on quantum spaces is given. Then two q-Hankel transforms are defined. By studying their behavior with respect to the q-Laguerre polynomials it is proven that the two transforms act as each other’s inverse. Then the integration on quantum Euclidean space is studied. The Fourier transform of a spherical harmonic weighted with a radial function can be expressed as the first q-Hankel transform of the radial function. The inverse Fourier transform is therefore defined by its Bochner’s relations. Next, the behavior of the Fourier transforms with re- spect to derivatives and multiplication with variables is studied. The previous results allow a con- struction of a Funk–Hecke theorem and reproducing kernels for the spherical harmonics on quan- tum Euclidean space. Then the Fourier transforms are connected with the harmonic oscillator which makes it possible to extend the Fourier transform to the Hilbert space defined for this har- monic oscillator. Finally the q-Fourier transform on undeformed Euclidean space is considered. The Fourier Transform on Quantum Euclidean Space 3 2 Preliminaries 2.1 q-calculus We give a short introduction to q-derivatives, q-integration and q-special functions, see [17, 22, 25, 26]. The report [25] that will be referred to often is also included in the book [24]. For u a number, and q the deformation parameter, 0 < q < 1, we define the q-deformation of u as [u]q = qu − 1 q − 1 . It is clear that lim q→1 [u]q = u. We also define (u; q)k = (1− u)(1− qu) · · · ( 1− qk−1u ) and (u; q)∞ = ∞∏ k=0 ( 1− uqk ) . The q-derivative of a function f(t) is defined by ∂qt (f(t)) = f(qt)− f(t) (q − 1)t . (2.1) This operator satisfies the generalized Leibniz rule ∂qt (f1(t)f2(t)) = ∂qt (f1(t))f2(t) + f1(qt)∂qt (f2(t)). (2.2) The q-integration on an interval [0, a] with a ∈ R is given by∫ a 0 f(t) dqt = (1− q)a ∞∑ k=0 f ( qka ) qk. The infinite q-integral can be defined in several ways, determined by a parameter γ ∈ R\{0},∫ γ·∞ 0 f(t)dqt = (1− q)γ ∞∑ k=−∞ f ( qkγ ) qk = lim l→+∞ ∫ q−lγ 0 dqtf(t). (2.3) The positive (γ ∈ R+) or negative (γ ∈ R−) infinite integral is a function of γ, however from the definition it is clear that ∫ γ·∞ 0 = ∫ qγ·∞ 0 . So ∂qγ ∫ γ·∞ 0 dqt = 0 holds which means the integral is a q-constant. The integral is the inverse of differentiation,∫ a 0 [∂qt f(t)] dqt = f(a)− f(0). (2.4) The q-factorial of an integer k is given by [k]q! = [k]q[k − 1]q · · · [1]q and satisfies [k]q! = (q; q)k/(1 − q)k. This can be generalized to the q-Gamma function Γq(t) for t > 0 satisfying Γq(t+ 1) = [t]qΓq(t), see e.g. [17, formula (1.10.1)]. The q-exponentials are defined as eq(t) = ∞∑ j=0 tj [j]q! and Eq(t) = eq−1(t) = ∞∑ j=0 q 1 2 j(j−1) tj [j]q! . Note that a different notation for the exponentials is used compared to [10]. The relation eq(t)Eq(−t) = 1 holds and the derivatives are given by ∂qt eq(t) = eq(t) and ∂qtEq(t) = Eq(qt). (2.5) 4 K. Coulembier For q < 1 the series Eq(t) converges absolutely and uniformly everywhere and eq(t) in the area \|t\| < 1 1−q . The function eq(t) can be analytically continued to C\{ q −k 1−q} as 1/(Eq(−t)). The zeroes of the q-exponential Eq are Eq ( − q−k 1− q ) = 0 for k ∈ N. (2.6) This follows from the infinite product representation Eq( t 1−q ) = (−t; q)∞, see [17, formula (1.3.16)]. This implies that the relation∫ 1√ 1−q ·∞ 0 dqtf(t)Eq2 ( −t2 1 + q ) = ∫ 1√ 1−q 0 dqtf(t)Eq2 ( −t2 1 + q ) (2.7) holds. The q-Hermite polynomials are given by Hq k(t) = bk/2c∑ j=0 (−1)j [k]q! [k − 2j]q![j]q2 ! qj(j+1)((q + 1)t)k−2j (2.8) and related to the discrete q-Hermite I polynomials hk(x; q) in [25, Section 3.28] by Hq k(t) = qk ( 1 + q 1− q )k/2 hk ( q−1 √ 1− q2t; q ) . We introduce the q-Laguerre polynomials for α > −1 in the normalization of [10, p. 24], L(α) j ( u\|q−2 ) = q−j(j+1+2α) j∑ i=0 q2i(i+α) (−u)i [j − i]q2 ![i]q2 ! (q2i+2α+2; q2)(j−i) (1− q2)j−i . (2.9) They can also be defined using the q-Gamma function since (q2i+2α+2;q2)(j−i) (1−q2)j−i = Γq2 (j+α+1) Γq2 (i+α+1) holds. They are connected with the q-Laguerre polynomials L (α) j (u; q) from [25, Section 3.21] by L(α) j ( u\|q−2 ) = q−j(j+1+2α)L (α) j (( 1− q2 ) u; q2 ) and to the q-Laguerre polynomials in [30] by the same formula with a substitution (1−q2)u→ u in the right hand side of the formula. The substitution q → q−1 yields, L(α) j ( u\|q2 ) = j∑ i=0 q(j−i)(j−i+1) (−u)i [j − i]q2 ![i]q2 ! (q2i+2α+2; q2)(j−i) (1− q2)j−i . (2.10) These polynomials are related to the little q-Laguerre polynomials (the Wall polynomials) pj(u; a\|q), see [25, Section 3.20], by L(α) j ( u\|q2 ) = qj(j+1) (q2α+2; q2)j (1− q2)j [j]q2 ! pj (( q−2 − 1 ) u; q2α\|q2 ) . The q-Laguerre polynomials in equation (2.10) satisfy the orthogonality relation∫ 1 1−q2 0 dq2uu αL(α) j ( u\|q2 ) L(α) k ( u\|q2 ) Eq2(−u) = δjkq 2(j+1)(α+1+j) Γq2(j + α+ 1) [j]q2 ! , The Fourier Transform on Quantum Euclidean Space 5 see [25, equation (3.20.2)]. Using the calculation rules in [10, Lemma 10], this can be rewritten as∫ 1√ 1−q 0 dqr r 2α+1L(α) j ( r2 1 + q \|q2 ) L(α) k ( r2 1 + q \|q2 ) Eq2 ( −r2 1 + q ) = δjk q2(j+1)(j+α+1)Γq2(j + α+ 1)(1 + q)α [j]q2 ! . (2.11) One of the orthogonality relations for q-Laguerre polynomials in equation (2.9) is∫ γ·∞ 0 dqt t 2α+1L(α) j ( q2t2 1 + q \|q−2 ) L(α) k ( q2t2 1 + q \|q−2 ) eq2 ( −q2t2 1 + q ) = δjk q−(j+α)(j+α+1)Γq2(j + α+ 1)(1 + q)α [j]q2 ! q(j+1)(j+2α+2) d ( γ√ 1 + q , α ) , (2.12) see [25, equation (3.21.3)], with d ( γ√ 1 + q , α ) = q(α+1)(α+2) (1 + q)αΓq2(α+ 1) ∫ γ·∞ 0 dqt t 2α+1 eq2 ( −q2t2 1 + q ) . The function d(λ, α) therefore satisfies d (√ γ, α ) = qα(α+1) Γq2(α+ 1) ∫ γ·∞ 0 dq2uu αeq2(−u). Partial integration implies that this function satisfies d(λ, α + 1) = d(λ, α) for α > −1. The explicit expression for d can be found from [25, equation (3.21.3)]. Remark 1. The q-Laguerre polynomials do not form a complete orthogonal system for the Hilbert space corresponding to the measure in equation (2.12). In [8] the compliment of the basis is constructed. The corresponding functions derived in [8, Section 4] (with a suitable renormalization) will therefore be annihilated by the q-Hankel transform Hq,γν in the subsequent Definition 2. This follows from the same calculation that leads to the subsequent equation (3.2). The q-Gegenbauer polynomials, see [16, equation (2.19)], are given by Cλn(q; t) = bn 2 c∑ j=0 (−1)jqj(j−1) [j]q2 ![n− 2j]q! (q2λ; q2)n−j (1− q2)n−j ((1 + q)t)n−2j . (2.13) They are big q-Jacobi polynomials on [−1, 1] with the two parameters equal to λ − 1 2 , see [16, equation (2.26)]. For ν > −1, the first and second q-Bessel function, introduced by Jackson, see [21, 22], are given by J (1) ν ( x\|q2 ) = ( x 1 + q )ν ∞∑ i=0 (−1)i [i]q2 !Γq2(i+ ν + 1) ( x 1 + q )2i for \|x\| < 1 1− q (2.14) and J (2) ν ( x\|q2 ) = qν 2 ( x 1 + q )ν ∞∑ i=0 q2i(i+ν)(−1)i [i]q2 !Γq2(i+ ν + 1) ( x 1 + q )2i . (2.15) 6 K. Coulembier J (1) ν (x\|q2) is analytical in the area \|x\| < 1 1−q and J (2) ν (x\|q2) is analytical on R+. The first q-Bessel function can be analytically continued by the relation J (1) ν ( x\|q2 ) = 1 qν2 eq2 ( −1− q 1 + q x2 ) J (2) ν ( x\|q2 ) , see [17, Exercise 1.24], which is defined for all x. Since x−νJ (2) ν (x\|q2) is an entire function the formula above implies that x−νJ (1) ν (x\|q2) is analytic on C outside the poles {±iq−k(1− q)−1\|k ∈ N}. Therefore x−νJ (1) ν (x\|q2) is well-defined and analytic for x ∈ R. These q-Bessel functions are related to the J (i) ν (x; q) in [21, (1.13) and (1.17)] or [17, Exer- cise 1.24] by J (1) ν ( x\|q2 ) = J (1) ν ( 2(1− q)x; q2 ) , J (1) ν ( x\|q2 ) = qν 2 J (1) ν ( 2(1− q)x; q2 ) . The generating functions for the q-Laguerre polynomials are given by J (1) α ( rt\|q2 ) = ( rt 1 + q )α ∞∑ j=0 L(α) j ( r2 1+q \|q 2 ) Γq2(α+ j + 1) t2j (1 + q)j eq2 ( − q2t2 1 + q ) , (2.16) J (2) α ( qrt\|q2 ) = ( rt 1 + q )α ∞∑ j=0 q(j+α)(j+1+α)L(α) j ( q2t2 1+q \|q −2 ) Γq2(α+ j + 1) r2j (1 + q)j Eq2 ( − r2 1 + q ) . (2.17) This follows from direct calculations, they are equivalent to [25, formulas (3.20.11) and (3.21.13)]. 2.2 The Howe dual pair and harmonic oscillator on Rm q Quantum spaces are spaces where the variables have braid statistics. The commutation relations are generalizations of the bosonic or fermionic ones by an R̂-matrix. The algebra of functions on a quantum space can be seen as the algebra O of formal power series in non-commuting variables x1, . . . , xm, O = C[[x1, . . . , xm]]/I with I the ideal generated by the commutation relations of the variables. We consider quantum spaces which satisfy the Poincaré–Birkhoff–Witt property, which states that the dimension of the space of homogeneous polynomials of a certain degree is the same as in the commutative case. Superspaces, for instance, do not satisfy this property. We focus on the case of the quantum Euclidean space Rmq . The relations for the variables can e.g. be found in [15, 20, 33]. We denote by Oq the algebra of formal power series for the specific case of the quantum Euclidean space. The quantum Euclidean space can be defined by the R̂-matrix of the quantum orthogonal group Oq(m), see [5, 14]. The matrix R̂ ∈ C(m×m)×(m×m) can be expressed in terms of its projection operators as R̂ = qPS − q−1PA + q1−mP1, and is symmetric, R̂ijkl = R̂klij . The matrix R̂ depends on the parameter q and returns to the undeformed case when q → 1 (lim q→1 R̂ijkl = δjkδ i l). The antisymmetric part defines the commutation relations P ijAx⊗ x = (PA)ijkl x kxl = 0. We will always use the summation convention. The singlet part defines the metric Cij = Cij , by (P1)ijkl = CijCkl C with C = CijCij = (1 + q2−m)[m/2]q2 . The metric satisfies the relation Cjl ( R̂±1 )lk st = ( R̂∓1 )kl js Clt (2.18) The Fourier Transform on Quantum Euclidean Space 7 and is its own inverse, CijC jk = δki . The braid matrix also satisfies the relation Cij ( R̂−1 )ij kl = qm−1Ckl. (2.19) The generalized norm squared is then defined as x2 = xiCijx j . This norm squared is central in the algebra Oq and is invariant under the co-action of Oq(m). The explicit expressions for the coaction of Oq(m) or the dually related action of Uq(so(m)) can be found in e.g. [6, 14, 15, 20]. In order to obtain a Fourier transform a second set of coordinates is needed, denoted by y, which is a copy of the x coordinates. The commutation relations between the x and y coordinates are given by yixj = q−1R̂ijklx kyl, see [23, 29, 32]. The differential calculus on Rmq was developed in [6, 31], the action of the partial derivatives is determined by ∂ixj = Cij + q ( R̂−1 )ij kl xk∂l. (2.20) The Laplace operator on Rmq is given by ∆ = ∂iCij∂ j . It is central in the algebra generated by the partial derivatives and is Oq(m)-invariant. The commutation relations for the partial derivatives can be expressed using PA or as R̂ijkl∂ k∂l = q∂i∂j + 1− q2 qm−1(1 + q2−m) Cij∆, (2.21) see [31]. Formulas (2.18) and (2.19) yield ∆xj = µ∂j + q2xj∆ and ∂jx2 = µxj + q2x2∂j with µ = 1 + q2−m. (2.22) The dilatation operator is given by Λ = 1 + ( q2 − 1 ) xiCij∂ j + (q2 − 1)2 qm−2µ2 x2∆ (2.23) and satisfies Λxi = q2xiΛ. For u ∈ R, Λu is defined by Λuxi = q2uxiΛu and Λu(1) = 1. The elements of Oq corresponding to finite summations are the polynomials, the correspon- ding algebra is denoted by P. The space Pk is defined as the space of the polynomials P in P which satisfy Λ(P ) = q2kP or Λ− 1 q2 − 1 P = [k]q2P. (2.24) For f analytical in the origin, f(x2) ∈ Oq is defined by the Taylor expansion of f . Equa- tion (2.22) leads to ∂jf ( x2 ) = xjµ∂q 2 x2f ( x2 ) + f ( q2x2 ) ∂j (2.25) for general functions of x2 and the q-derivative as defined in formula (2.1). The relation ∂q 2 x2f ( x2 ) = [ 1 (1 + q)t ∂qt f ( t2 )] t2=x2 (2.26) is a useful calculation rule. There exists a second differential calculus on Rmq of partial derivatives ∂j , which is obtained from the unbarred one by replacing ∂j , q, R̂, C by ∂j , q−1, R̂−1, C. In particular, the relation ∂jf ( x2 ) = xjqm−2µ∂q −2 x2 f ( x2 ) + f ( q−2x2 ) ∂j (2.27) holds. The algebra generated by the variables and partial derivatives ∂j is denoted Diff(Rmq ). The algebra generated by the variables and partial derivatives ∂j is the same algebra. The polynomial null-solutions of ∆ are the same as those of ∆. The space of the null-solution of degree k is denoted by Sk, so Sk = Pk ∩ ker ∆. 8 K. Coulembier Definition 1. The operator E is given by E = [m2 ]q2 + qmxiCik∂ k. Using the expression for the dilation operator in formula (2.23) the operator E can be ex- pressed as E = [m 2 ] q2 + qm ( Λ− 1 q2 − 1 − qm−2 ( q2 − 1 ) x2∆ ) . Property (2.24) then implies that for Sk ∈ Sk, ESk = [m 2 + k ] q2 Sk (2.28) holds. Together with ∆ and x2, this operator E generates an algebra which is a q-deformation of the universal enveloping algebra of sl2. Theorem 1. The operators ∆/µ, x2/µ and E generate the quantum algebra Uq(sl2),[ ∆/µ,x2/µ ] q4 = E, [ E,x2/µ ] q2 = [2]q2x 2/µ, [∆/µ,E]q2 = [2]q2∆/µ. (2.29) Proof. Combining equations (2.22) and (2.19) yields equation (2.29). Equation (2.22) implies xiCij∂ jx2 = µx2 + q2x2xiCij∂ j , which leads to the second relation. The third relation is calculated in the same way. � Remark 2. As the generators of Uq(sl2) are Oq(m)-invariant, this quantum algebra and quan- tum group form the Howe dual pair (Oq(m),Uq(sl2)), or (Uq(so(m)),Uq(sl2)). The quantum algebra Uq(sl2) is equal to the one in [10]. In [10], Uq(sl2) was generated by the standard Euclidean norm squared r2 on Rm and a q-deformation of the Laplace operator ∆q. Since ∆q is still O(m)-invariant, the Howe dual pair (O(m),Uq(sl2)) appeared. Because the Oq(m)-invariant harmonic operators on quantum Euclidean space in the present paper generate the same quantum algebra we obtain an important connection between these two theories. In particular the Oq(m)-invariant Fourier transform developed in the current paper can be used to construct the O(m)-invariant q-Fourier transform on Euclidean space, as will be done in Section 8. Lemma 1 (Fischer decomposition). The space P decomposes into irreducible pieces under the action of Uq(so(m)) as (see [20, 15]) P = ∞⊕ j=0 ∞⊕ k=0 x2jSk. The operator identities in Theorem 1 yield E ( x2lSk ) = ([m 2 + k + l ] q2 + q2[l]q2 ) x2lSk and ∆(x2lSk) = µ2[l+k+ m 2 −1]q2 [l]q2x 2l−2Sk. These calculations and the previous results lead to Theorem 2 (Howe duality). The decomposition of P into irreducible representations of Uq(so(m)) is given in Lemma 1. Each space ⊕ j x2jSk is a lowest weight module of Uq(sl2) with weight vectors x2jSk, the lowest weight vector is Sk with weight [m/2 + k]q2. The Fischer decomposition of P therefore is a multiplicity free irreducible direct sum decomposition under the joint action of Uq(sl2)× Uq(so(m)). The Fourier Transform on Quantum Euclidean Space 9 The antilinear involutive antihomomorphism ∗ on Diff(Rmq ) is defined by (AB)∗ = B∗A∗, (xj)∗ = xkCkj , (∂j)∗ = −q−m∂kCkj and λ∗ = λ with λ ∈ C and · complex conjugation. This yields( x2 )∗ = x2 and ∆∗ = q−2m∆. The harmonic oscillator on quantum Euclidean space was studied in [5, 7, 14]. The two Hamiltonians (with an unimportant different normalization compared to [14]) are given by h = 1 2 ( −∆ + x2 ) , h∗ = 1 2 ( −∆∗ + x2 ) . (2.30) Both operators have the same eigenvalues. 2.3 Integration and Fourier theory on quantum spaces In [23] a method was prescribed to generalize q-integration to higher dimensions in the context of quantum spaces. Gaussian-induced integration for general R̂-matrices is defined assuming there is a matrix η ∈ Rm×m and a solution gη ∈ O of the equation −ηij∂jgη = xigη. (2.31) Integration ∫ on the space Pgη, with P the polynomials on the quantum space is then uniquely defined by demanding ∫ ◦∂i = 0, i = 1, . . . ,m. For f ∈ P the integral ∫ fgη is of the form∫ fgη = Z[f ]I(gη), (2.32) with I(gη) = ∫ gη and Z a functional on P ⊂ O. Superspace with purely bosonic and fermionic coordinates can be seen as a limit of braided spaces, typically for q → −1, see e.g. [13]. From this point of view it is interesting to note that the Berezin integral can also be constructed in this setting. In [12, 9] this led to integration over the supersphere and a new interpretation of the Berezin integral. An explicit example of this construction was already defined on quantum Euclidean space, see [14]. In [34] it was shown that this integration can be defined and generalized using integra- tion over the quantum Euclidean sphere. In Section 4.1 we will show how this approach follows from harmonic analysis on quantum Euclidean space. In [23] a general procedure to construct a Fourier transform on quantum spaces was developed. First the appropriate Gaussian-induced integration ∫ should be constructed and the exponential or Fourier kernel (see [29, 36]) calculated. The Fourier transform on a braided-Hopf algebra B with left dual Hopf algebra B∗ is a map F : B → B∗. The co-ordinates for B are denoted by x and for B∗ by y, the Fourier transforms are given by F [f(x)](y) = ∫ x f(x) expR̂(x\|y), F∗[f(y](x) = ∫ ∗ y f(y) expR̂(x\|y). These Fourier transforms are each others inverse, F∗F = VolS, with S the antipode on B. As an explicit example we consider the braided line B = C[x], with braided-Hopf algebra structure as introduced in [28] given by ∆xk = k∑ j=0 [k]q! [k − j]q![j]q! xj ⊗ xk−j , Sxk = (−1)kq k(k−1) 2 xk, εxk = δk0. 10 K. Coulembier The dually-paired Hopf algebra B∗ is the same Hopf algebra with variable y. The relation xy = qyx holds. The exponential is exp(x\|y) = ∞∑ k=0 xkyk [k]q ! and satisfies ∂qx exp(x\|y) = exp(x\|y)y. The Fourier transforms take the form F [f ](y) = ∫ γ·∞ −γ·∞ dqxf(x) exp(x\|y) and F∗[f ](x) = ∫ δ·∞ −δ·∞ dqyf(y) exp(x\|y). (2.33) The theory of [23] then implies F∗F [f ](x) = Sf(x)Volγ,δ. We can rewrite this in a way that will be more closely related to our approach of the Fourier transform on Rmq . Define g(y) = 1 Volγ,δ F [f ](y), then the definition of the antipode implies f(−x) = ∫ δ·∞ −δ·∞ dqyg(y) ∞∑ k=0 q k(k+1) 2 ykxk [k]q! . In this equation and in the first equation of (2.33), there are no coordinates which have to be switched before integration. This implies that we can assume x and y commute and write the equations above as g(y) = 1 Volγ,δ ∫ γ·∞ −γ·∞ dqx f(x) eq(xy) and f(x) = ∫ δ·∞ −δ·∞ dqy g(y)Eq(−qyx). In [26] a closely related analytical approach was given to the one dimensional Fourier trans- form above. Consider real commuting variables x and y. Using the orthogonality relations and the generating function of the Hermite polynomials in equation (2.8), it is possible to prove 1 2Γq2(1 2) ∫ 1√ 1−q − 1√ 1−q dqx [ Hq k ( x√ 1 + q ) Eq2 ( − x2 1 + q )] eq(−ixy) = (q + 1) k−1 2 ik q 1 2 (k+1)(k+2)ykeq2 ( − q 2y2 1 + q ) , (2.34) which is equivalent to [26, equation (8.7)]. For every δ ∈ R+, 1 Cδ ∫ δ·∞ −δ·∞ dqy [ ykeq2 ( − q 2y2 1 + q )] Eq(iqyx) = ik q 1 2 (k+1)(k+2)(1 + q) k−1 2 Hq k ( x√ 1 + q ) Eq2 ( − x2 1 + q ) (2.35) holds for Cδ some constant depending on δ, see [26, equation (8.21)]. So the two Fourier transforms as defined in equations (2.34) and (2.35) can be regarded as each others inverse, which was to be expected from the theory of [23]. There is however one difference between the explicit Fourier transform in [26] and the abstract theory in [23]. While the inverse Fourier transform remains unchanged, the integration for the Fourier transform is limited to a finite interval. This will be explained in the subsequent Lemma 6. However, using property (2.7) the integral can be replaced by ∫ 1√ 1−q ·∞ − 1√ 1−q ·∞ . So the analytical approach of [26] recovers the theory from [23] with an imposed limitation on γ. For other γ, the theory from [23] would still hold, but the constant Volγ,δ is infinite. The Fourier Transform on Quantum Euclidean Space 11 3 The q-Hankel transforms In this section we define two q-Hankel transforms using the first and second q-Bessel function. These transforms will act as each others inverse. This is a generalization of the result in for- mulas (2.34) and (2.35). By evaluating the Fourier transform on the appropriate functions in Section 4.2 the first of these q-Hankel transforms will appear. We will calculate the q-Hankel transforms of the q-Laguerre and little q-Laguerre polynomials weighted with a q-Gaussian. The undeformed Fourier–Gauss transform of these polynomials was already studied in [1]. There also exists a third q-Bessel function besides the ones in equations (2.14) and (2.15). We will not explicitly need the third type, but it is interesting to note that in [27, 2] the corresponding q-Hankel transforms were constructed. These Hankel transforms could also be used to define an Oq(m)-invariant Fourier transform on Rmq . This would have the advantage that the Fourier transform is its own inverse. That Fourier transform would however not behave well with respect to the derivatives on Rmq . This is already the case for the Fourier transform on the braided line, as is proven in [27]. The braided line corresponds to Rmq for m = 1. In anticipation of the connection with the Fourier transform on quantum Euclidean space we will scale the q-Hankel transforms in the following definition with µ, see equation (2.22), although at this stage any constant could be used. The reason for the appearance of unfixed constants β, γ will become apparent in Sections 4.2, 5 and 7. Definition 2. For β, γ ∈ R+ and ν ≥ −1 2 , the q-Hankel transforms are given by Hq,βν [f(r)](t) = 1 + q µ ∫ √ µ (1−q2)β 0 dqr J (1) ν (1+q µ rt\|q2 ) (rt)ν r2ν+1[f(r)] and Hq,γν [f(t)](r) = 1 + q µ ∫ γ·∞ 0 dqt J (2) ν ( q 1+q µ rt\|q2 ) (rt)ν t2ν+1[f(t)]. In this definition it is not specified on which function spaces the q-Hankel transforms act. At the moment we define them on functions for which the expression exists. In order to connect the Fourier transform on Rmq with these q-Hankel transforms we define the following transformations, Fν q,β [ψ] ( t2 ) = Hq,βν [ψ ◦Υ](t) and Fq,γν [ψ] ( r2 ) = Hq,γν [ψ ◦Υ](r), with Υ(u) = u2. In order to prove the properties of the q-Hankel transforms we will need some identities of the q-Bessel functions in equations (2.14) and (2.15). These are summarized in the following lemma. Lemma 2. The first and second q-Bessel functions satisfy (i) ∂qu J (1) ν (u\|q2) uν = −u J (1) ν+1(u\|q2) uν+1 , ∂q −1 u J (2) ν (qu\|q2) uν = −qu J (2) ν+1(qu\|q2) uν+1 , (ii) J (1) ν+1(u\|q2) + J (1) ν−1(u\|q2) uν−1 = [2ν]q J (1) ν (qu) (qu)ν , J (2) ν+1(qu\|q2) + J (2) ν−1(qu\|q2) uν−1 = [2ν]q J (2) ν (u) (qu)ν , (iii) ∂quJ (1) ν (u\|q2)uν = J (1) ν−1(u\|q2)uν and ∂quJ (2) ν (u\|q2)uν = qνJ (2) ν−1(qu\|q2)uν . 12 K. Coulembier Proof. The right-hand side of the second property (for the first q-Bessel function) can be calculated using [ν]q2q 2i = [ν + i]q2 − [i]q2 , [ν]q2(1 + q) J (1) ν (qu) (qu)ν = 1 (1 + q)ν−1 ∞∑ i=0 ( (−1)i [i]q2 !Γ(i+ ν) ( u 1 + q )2i − (−1)i [i− 1]q2 !Γ(i+ ν + 1) ( u 1 + q )2i ) = J (1) ν−1(u\|q2) + J (1) ν+1(u\|q2) uν−1 . The first and the third property follow from a direct calculation. The left-hand sides of the properties can also be obtained from [17, Exercise 1.25]. � Corollary 1. The second q-Bessel functions satisfy the following relation: ∂quu ν+1J (2) ν−1 ( u\|q2 ) = [2ν]qu νJ (2) ν−1 ( u\|q2 ) − qν+1uν+1J (2) ν ( qu\|q2 ) . Proof. This is a direct consequence of the second formula in Lemma 2(i). � Combining generating function (2.16) and orthogonality relation (2.11) yields∫ 1√ 1−q 0 dqr J (1) ν (rt\|q2) (rt)ν r2ν+1 [ L(ν) j ( r2 1 + q \|q2 ) Eq2 ( −r2 1 + q )] = q2(j+1)(j+ν+1) [j]q2 ! t2j (1 + q)j eq2 ( − q2t2 1 + q ) . (3.1) Generating function (2.17) and orthogonality relation (2.12) lead to∫ γ·∞ 0 dqt J (2) ν (qrt\|q2) (rt)ν t2ν+1 [ L(ν) j ( q2t2 1 + q \|q−2 ) eq2 ( −q2t2 1 + q )] = d ( γ√ 1 + q , ν ) q−(j+1)(j+2ν+2) [j]q2 ! r2j (1 + q)j Eq2 ( − r2 1 + q ) . (3.2) The following expansion of a monomial in terms of the q-Laguerre polynomials is a direct consequence of equation (2.17), t2j (1 + q)j [j]q2 ! = j∑ i=0 (−1)i(q2i+2ν+2; q2)(j−i) [j − i]q2 !(1− q2)j−i q(j−i)(j−i+1)+(i+1)(i+2ν+2) q2(j+1)(j+ν+1) L(ν) i ( q2t2 1 + q \|q−2 ) . Applying this yields∫ γ·∞ 0 dqt J (2) ν (qrt\|q2) (rt)ν t2ν+1 [ t2j (1 + q)j [j]q2 ! eq2 ( −q2t2 1 + q )] = j∑ i=0 (−1)i(q2i+2ν+2; q2)(j−i) [j − i]q2 !(1− q2)j−i q(j−i)(j−i+1) q2(j+1)(j+ν+1) d ( γ√ 1 + q , ν ) 1 [i]q2 ! r2i (1 + q)i Eq2 ( − r2 1 + q ) = d ( γ√ 1 + q , ν ) q−2(j+1)(j+ν+1)L(ν) j ( r2 1 + q \|q2 ) Eq2 ( − r2 1 + q ) . These calculations imply the following relations for the q-Hankel transforms in Definition 2: Hq,βν [ L(ν) j ( β r2 µ \|q2 ) Eq2 ( −β r 2 µ )] (t) = 1 βν+1+j Cjt 2jeq2 ( −q 2t2 µβ ) for β ∈ R+, The Fourier Transform on Quantum Euclidean Space 13 Hq,γν [ Cjt 2jeq2 ( −αq 2t2 µ )] (r) = d (√αγ√ µ , ν ) αν+1+j L(ν) j ( r2 αµ \|q2 ) Eq2 ( − r 2 αµ ) for α, γ ∈ R+, with Cj = q2(j+1)(j+ν+1) [j]q2 !µj . By considering the case β = 1/α we obtain Theorem 3. For each α, γ ∈ R+, the inverse of the q-Hankel transform Hq,γν acting on R[t2]⊗ eq2(−α q 2t2 µ ), Hq,γν : R [ t2 ] ⊗ eq2 ( −αq 2t2 µ ) → R [ r2 ] ⊗ Eq2 ( − r 2 αµ ) is given by 1 d (√αγ√ µ , ν )Hq,1/αν : R [ r2 ] ⊗ Eq2 ( − r 2 αµ ) → R [ t2 ] ⊗ eq2 ( −αq 2t2 µ ) . This theorem for ν = −1 2 and ν = 1 2 is identical to the results in equations (2.34) and (2.35). In order to prove the behavior of the Fourier transform on Rmq we need the following properties of the q-Hankel transforms. The exact function spaces on which they act is again not specified, the properties hold if all the terms are well-defined. In particular these lemmata hold for the function spaces in Theorem 3. Lemma 3. The first q-Hankel transform satisfies the following properties: (i) t2Hq,βν+1[f(r)](t) +Hq,βν−1 [ r2f(r) ] (t) = µ[ν]qH q,β ν [f(r)](qt), (ii) ∂qtH q,β ν [f ](t) = −1 + q µ tHq,βν+1[f ](t) and (iii) Hq,βν [ 1 r ∂q −1 r f ] (t) = −q1 + q µ Hq,βν−1[f ](t) if f ( q−1 √ µ (1− q2)β ) = 0. The second q-Hankel transform satisfies the following properties: (i) r2Hq,γν+1[f(t)](r) +Hq,γν−1 [ t2f(t) ] (r) = µ[ν]q q2ν Hq,γν [f(t)] ( q−1r ) , (ii) ∂q −1 r Hq,γν [f ](r) = −q1 + q µ rHq,γν+1[f ](r) and (iii) Hq,γν [ 1 t ∂qt f ] (r) = −1 + q µ Hq,γν−1[f ](r). Proof. The first property is a direct consequence of Lemma 2(ii), the second a direct conse- quence of Lemma 2(i). Property (iii) is calculated using formulas (2.2) and (2.4) and Lem- ma 2(iii) Hq,βν [ 1 r ∂q −1 r f ] (t) = ∫ √ µ (1−q2)β 0 dqr J (1) ν (1+q µ rt\|q2 ) tν rν∂q −1 r f(r) = ∫ √ µ (1−q2)β 0 dqr J (1) ν (1+q µ rt\|q2 ) tν rν∂qrf ( q−1r ) q = −q ∫ √ µ (1−q2)β 0 dqr ∂qr J (1) ν (1+q µ rt\|q2 ) tν rν  f(r) 14 K. Coulembier = −q1 + q µ ∫ √ µ (1−q2)β 0 dqr J (1) ν−1 (1+q µ rt\|q2 ) tν−1 rνf(r). Property (iii) for the second q-Hankel transform is calculated similarly. � Lemma 4. The relation Hq,γν−1 [t∂qt f(t)] (r) = 1 + q µ r2Hq,γν [f ](r)− [2ν]q q2ν Hq,γν−1[f ] ( q−1r ) holds for the second q-Hankel transform. Proof. The left-hand side is calculated using Corollary 1, Hq,γν−1 [t∂qt f(t)] (r) = −1 + q µ 1 qν+1 ∫ γ·∞ 0 dqt∂ q t J (2) ν−1 (1+q µ rt\|q2 ) tν+1 rν−1  f(t) = ∫ γ·∞ 0 dqt r2 J (2) ν ( q 1+q µ rt\|q2 ) tν+1 rν f(t)− 1 + q µ [2ν]q qν+1 J (2) ν−1 (1+q µ rt ) tν rν−1 f(t)  , which proves the lemma. � 4 Integration and Fourier transform on Rm q 4.1 Integration over the quantum sphere and induced integration on Rm q First we show how the Howe dual pair (Oq(m),Uq(sl2)) uniquely characterizes the integration over the quantum sphere from [34]. Theorem 4. The unique (up to a multiplicative constant) linear functional on P invariant under the co-action of Oq(m) and satisfying ∫ Sm−1 q x2R = ∫ Sm−1 q R is given by the Pizzetti formula ∫ Sm−1 q R = ∞∑ k=0 2 ( Γq2(1 2) )m µ2k[k]q2 !Γq2(k + m 2 ) ( ∆kR ) (0) for R ∈ P. Proof. The Fischer decomposition in Lemma 1 implies that the integration on P is uniquely determined if it is determined on each of the blocks x2lSk. Since these blocks are irreducible Uq(so(m))-representations (or irreducible Oq(m)-corepresentations) the integration should be zero on each such block which is not one dimensional. This implies that the integration can only have non-zero values on the elements x2l. The second property then implies ∫ Sm−1 q x2l = ∫ Sm−1 q 1, which shows that the integration is uniquely determined up to the constant ∫ Sm−1 q 1. It is easily checked that the Pizzetti formula satisfies the conditions. � We chose the normalization such that ∫ Sm−1 q 1 = 2(Γq2 ( 1 2 )) m Γq2 (m 2 ) . This quantum sphere integration can be expressed symbolically using the first q-Bessel-function (2.14), ∫ Sm−1 q R = 2 ( Γq2 ( 1 2 ))mJ (1) m 2 −1 ( 1+q µ √ −∆\|q2 ) (√ −∆/µ )m 2 −1 R  (0). (4.1) The following lemma will be important for the sequel. The Fourier Transform on Quantum Euclidean Space 15 Lemma 5. The Fischer decomposition (see Lemma 1) of xjSk(x) with Sk ∈ Sk is given by xjSk = ( xjSk − x2 1 µ[k +m/2− 1]q2 ∂jSk ) + x2 ( 1 µ[k +m/2− 1]q2 ∂jSk ) with ( xjSk − x2 1 µ[k +m/2− 1]q2 ∂jSk ) ∈ Sk+1 and ( 1 µ[k +m/2− 1]q2 ∂jSk ) ∈ Sk−1. Proof. Since ∆ and ∂j commute ( 1 µ[k+m/2−1]q2 ∂jSk ) ∈ Sk−1. Equations (2.22), (2.29) and (2.28) yield ∆ ( xjSk − x2 1 µ[k +m/2− 1]q2 ∂jSk ) = µ∂jSk − µ2E 1 µ[k +m/2− 1]q2 ∂jSk = 0. This can also be calculated by using the projection operator from Pk onto Sk developed in [20]. � Before we use the quantum sphere integration to construct integration on Rmq along the lines of [23] and [14], the following lemma should be considered. Lemma 6. For the q-integration in equation (2.3) with γ ∈ R+, the expression∫ γ·∞ 0 Eq(−t)dqt is infinite unless γ = qj 1 1−q for some j ∈ Z. In that case it reduces to ∫ 1 1−q 0 Eq(−t)dqt. Proof. Property (2.5) can be rewritten as Eq(t) = Eq(qt) [1 + (1− q)t], which implies Eq ( −q−kγ ) = Eq ( −q1−kγ )[ 1− (1− q)q−kγ ] . First we assume Eq(−q1−kγ) is never zero for k ∈ Z. The equation above then yields∣∣Eq(−q−kγ) ∣∣ > ∣∣Eq(−q1−kγ) ∣∣ q and Sign(Eq(−q−kγ)) 6= Sign(Eq(−q1−kγ)) for k ≥ N ∈ N with N > ln ( 1+q γ(1−q) ) ln(1/q) . This implies that ∞∑ k=N Eq(−q−kγ)q−k will not converge, therefore the integral (2.3) will not exist. If Eq(−q1−kγ) = 0 holds for some k ∈ Z, the unicity of the zeroes in equation (2.6) implies that γ = qk−1−l 1−q for some k ∈ Z and l ∈ N, or γ = qj 1−q for some j ∈ Z. The integration then reduces to the proposed expression for the same reason as in equation (2.7). � Now we can construct integration on Rmq from the quantum sphere integration according to the principle of Gaussian-induced integration. The two differential calculi lead to different Gaussians and integrations. For constants α, β ∈ R+, the relations ∂jeq2 ( −αx2 ) = −αµxjeq2 ( −αx2 ) and ∂jEq2 ( −βx2 ) = −βqm−2µxjEq2 ( −βx2 ) imply that eq2(−αx2) and Eq2(−βx2) are Gaussians according to equation (2.31) for the un- barred and barred calculus respectively. Define the following two integrations for γ, λ ∈ R+ on polynomials weighted with undetermined radial functions,∫ γ·Rmq Rk(x)f ( x2 ) = ∫ γ·∞ 0 dqr r m+k−1f ( r2 ) ∫ Sm−1 q Rk(x) and 16 K. Coulembier∫ Bmq (λ) Rk(x)f ( x2 ) = ∫ λ 0 dqr r m+k−1f ( r2 ) ∫ Sm−1 q Rk(x). The Taylor expansion of the function f in the origin is assumed to converge on R+ for the first integration and on [0, λ] for the second integration. These integrations correspond to the Gaussian-induced integrations. Theorem 5. The integrations defined above satisfy∫ γ·Rmq ∂j = 0 on P ⊗ eq2 ( −αx2 ) for α, γ ∈ R+ and∫ Bmq ( 1√ (1−q2)β ) ∂j = 0 on P ⊗ Eq2 ( −βx2 ) for β ∈ R+. Proof. The first property is well-known, see e.g. [14, 34]. In order to prove the second property we consider the expression∫ Bmq (λ) ∂jf ( x2 ) Sk(x) = qm−2µ ∫ Bmq (λ) ( ∂q −2 x2 f ( x2 )) xjSk(x) + ∫ Bmq (λ) f ( q−2x2 ) ∂jSk(x). for a spherical harmonic Sk ∈ Sk. If k 6= 1 the right-hand side is always zero, because of Lemma 5 and the expression for ∫ Sm−1 q in Theorem 4. In case k = 1 the spherical harmonics are the monomials xi, i = 1, . . . ,m. Therefore we calculate using equation (2.26), Leibniz rule (2.2) and equation (2.4)∫ Bmq (λ) ∂jf ( x2 ) xi = qm−2µ ∫ Bmq (λ) ( ∂q −2 x2 f ( x2 )) xjxi + ∫ Bmq (λ) f ( q−2x2 ) Cji = 2 ( Γq2(1 2) )m Cji Γq2(1 + m 2 ) [ qm−2 ∫ λ 0 dqrr m+1 1 (1 + q−1)r ∂q −1 r f ( r2 ) + [m 2 ] q2 ∫ λ 0 dqrr m−1f ( q−2r2 )] = 2 ( Γq2(1 2) )m Cji Γq2(1 + m 2 )(1 + q) [ qm ∫ λ 0 dqrr m∂qrf ( q−2r2 ) + [m]q ∫ λ 0 dqrr m−1f ( q−2r2 )] = 2 ( Γq2(1 2) )m Cji Γq2(1 + m 2 )(1 + q) [ λmf ( q−2λ2 ) − lim r→0 rmf ( q−2r2 )] . When we substitute f(x2) = x2lEq2(−βx2) with l ∈ N and β ∈ R+ and use equation (2.6) we obtain∫ Bmq ( 1√ (1−q2)β ) ∂j x2lSk(x)Eq2 ( −βx2 ) = 0 ∀ k, l. This proves the second part of the theorem because of the Fischer decomposition in Lemma 1. � Remark 3. The properties∫ γ·Rmq ∂j = 0 on P ⊗ eq2 ( −αx2 ) for α, γ ∈ R+ and∫ Bmq ( 1√ (1−q2)β ) ∂j = 0 on P ⊗ Eq2 ( −βx2 ) for β ∈ R+ The Fourier Transform on Quantum Euclidean Space 17 also hold. They do not correspond to Gaussian-induced integration for those calculi in the strict sense. However it is a straightforward generalization, the generalized Gaussian satisfies the relation −ηij∂jgη(x) = xigη(qx) (or −ηij∂jgη(x) = xigη(q −1x)) in stead of equation (2.31). Since ∫ γ·Rmq Eq2(−βx2) will be infinite for general γ, similar to Lemma 6, the finite integration needs to be used on P⊗Eq2(−βx2). As in equation (2.7) this integration corresponds to a specific infinite integration,∫ Bmq ( 1√ (1−q2)β ) = ∫( 1√ (1−q2)β ) ·Rmq on P ⊗ Eq2 ( −βx2 ) for β ∈ R+. Theorem 5 therefore shows that for integration on P⊗eq2(−αx2) there is a bigger choice since γ does not depend on α. However, it is straightforward to calculate∫ γ·Rmq x2lSkeq2 ( −αx2 ) = δk0 Γq2(m2 + l) Γq2(m2 )αlq 1 2 l(l+m 2 −1) ∫ γ·Rmq eq2 ( −αx2 ) , which implies the integrals for different γ on P ⊗ eq2(−αx2) are proportional to each other. In the sense of equation (2.32) the only difference between the integrations for different γ is the value I(gη) while the functional Z does not depend on γ. For strict Gaussian-induced integration in the unbarred case, only one choice gives an I(gη) which is finite. 4.2 Bochner’s relations for the Fourier transform on Rm q The exponential expR̂(x\|y) on a quantum space satisfies ∂jx expR̂(x\|y) = expR̂(x\|y)yj , (4.2) see [23, 29, 36]. From now on expR̂(x\|y) stands for the exponential on Rmq . It is uniquely determined from equation (4.2) and the normalization expR̂(0\|y) = 1. In order to define the Fourier transform according to [23] the Fourier kernel needs to be combined with the Gaussian induced integrations for the unbarred calculus in Theorem 5 and evaluated on P ⊗ eq2(−αx2). It turns out that the Fourier transform defined by the generalized Gaussian-induced integration for the unbarred calculus in Remark 3 will lead to interesting properties, see Section 5. This choice also corresponds to the one dimensional theory in equation (2.34). The Fourier transform can be defined on each space P ⊗ Eq2(−β x2 µ ). First we will extend this space. Define Vβ = P ⊗ Eq2 ( −βx2 µ ) , β ∈ R+ and Vα = P ⊗ eq2 ( −αq 2x2 µ ) , α ∈ R+. The different spaces Vα (or Vβ) are not necessarily disjunct since equation (2.5) implies Vq−2jα ⊂ Vq−2j−2α and Vq2jβ ⊂ Vq2j+2β for j ∈ N. Therefore we define V[α] = ∪∞j=0Vq−2jα and V [β] = ∪∞j=0Vq2jβ for α, β ∈ R+. Since, for j ∈ N, Vq2jα ⊂ Vα, V[α] can also be identified with ∪∞j=−∞Vq−2jα and V [β] with ∪∞j=−∞Vq2jβ. 18 K. Coulembier Definition 3. The first Fourier transform F±Rmq on Rmq is a map V [β] → V[1/β] for each β ∈ R+, F±Rmq [f ](y) = 1 + q 2µ m 2 ( Γq2(1 2) )m ∫ 1√ (1−q2)β ·Rmq f(x) expR̂(±ix\|y). In the subsequent Corollary 2, it will be proven that the Fourier transform does indeed map elements of V [β] to V[1/β]. First we need the following technical lemma. For a polynomial P , we define [P ]jl by the equation ∑ l[P ]jl ∂ l x = ∂jxP − [ ∂jxP ] in Diff(Rmq ). So in the undeformed Euclidean case, [P ]jl = Pδjl . Lemma 7. For Sk ∈ Sk the relation ∑ j,l[∂jSk(x)]jlx l = [k]q2Sk(x) holds with ∂j = Cjk∂ k. Proof. The lemma is proven by calculating the expression ∑ j [∂ j x[∂jSk(x)]x2] in two different ways. The expression is equal to∑ j,l [ [∂jSk(x)]jl ∂ l xx 2 ] = ∑ j,l [∂jSk(x)]jlµx l. It can also be calculated using formulas (2.22) and (2.28)∑ j [ ∂jxx 2[∂jSk(x)] ] = ∑ j µxj [∂jSk(x)] + ∑ j q2x2 [ ∂jx[∂jSk(x)] ] = µ[k]q2Sk(x), which proves the lemma. � Remark 4. Using the same techniques for a general polynomial Pk ∈ Pk yields∑ j,l [∂jPk(x)]jl x l = [k]q2Pk(x) + (q2 − 1) µ2 x2∆Pk(x). Equations (4.1) and (4.2) imply that the quantum sphere integral of the Fourier kernel will yield the first q-Bessel function. This result can be generalized by introducing spherical har- monics in the integration. Theorem 6. For Sk ∈ Sk, the following relation holds, ∫ Sm−1 q ,x Sk(x) expR̂(ix\|y) = 2µm/2−1 ( Γq2 (1 2 ))m ik J (1) m 2 +k−1 (1+q µ √ y2\|q2 ) (√ y2 )m 2 +k−1 Sk(y). Proof. First we prove that the relation ∆k+l x Sk(x) expR̂(ix\|y) = ik+2l [k + l]q2 !µk [l]q2 ! expR̂(ix\|y)y2lSk(y) + · · · (4.3) holds, where · · · stands for terms of the form P (x) expR̂(ix\|y)q2(y) with P ∈ ⊕j>0Pj . In case k = 0, equation (4.2) implies that ∆l x expR̂(ix\|y) = (−1)l expR̂(ix\|y)y2l holds. Now we proceed by induction on k. Assuming equation (4.3) holds for k− 1 we calculate, using equations (2.28) and (2.22) ∆k+l x Sk(x) expR̂(ix\|y) = 1 [k]q2 ∆k+l x xj [∂xjSk(x)] expR̂(ix\|y) = 1 [k]q2 q2k+2lxj∆k+l x [∂jSk(x)] expR̂(ix\|y) The Fourier Transform on Quantum Euclidean Space 19 + [k + l]q2 [k]q2 µ∆k+l−1 x [∂jSk(x)]jt ∂ t x expR̂(ix\|y) = · · ·+ ik−1+2l [k + l]q2 ! [l]q2 ! µk expR̂(ix\|y)y2l 1 [k]q2 [∂jSk(y)]jl iy l. Lemma 7 then proves that equation (4.3) holds for every k. We also used the fact that for Sp ∈ Sp, [Sp] j l ∈ Sp which follows from the fact that ∂jx and ∆ commute. Equation (4.3) then yields∫ Sm−1 q ,x Sk(x) expR̂(ix\|y) = ∞∑ j=0 2 ( Γq2(1 2) )m µ2j+2k[j + k]q2 !Γq2(j + k + m 2 ) ( ∆j+k x Sk(x) expR̂(ix\|y) ) (x = 0) = ik 2 ( Γq2(1 2) )m µk ∞∑ j=0 (−1)j [j]q2 !Γq2(j + k + m 2 ) y2j µ2j Sk(y). Comparing this with equation (2.14) proves the theorem. � This theorem allows to calculate the Fourier transform of an element of P⊗Eq2(−β x2 µ ) inside one of the irreducible blocks of its Uq(so(m))-decomposition. Corollary 2. The Fourier transform in Definition 3 of a function Sk(x)ψ ( x2 ) ∈ P ⊗ Eq2 ( −βx2 µ ) ⊂ V [β] with Sk ∈ Sk is given by F± [ Sk(x)ψ ( x2 )] (y) = (±i)kSk(y)Fq,βm 2 +k−1[ψ] ( y2 ) = (±i)k 1 + q µ ∫ √ µ (1−q2)β 0 dqr r m+2k−1ψ ( r2 )J (1) m 2 +k−1 (1+q µ rt\|q2 ) (rt) m 2 +k−1  t2=y2 Sk(y) with r and t two real commuting variables. These formulae are the Bochner’s relations for the Fourier transform on Rmq . For the classical Bochner’s relations, see e.g. [19]. This corresponds exactly to the one dimensional case for β = 1. Since lim m→1 µ = 1 + q, Corollary 2 for m = 1 and k = 0 reduces to √ 1 + q 2Γq2(1 2) ∫ 1√ 1−q − 1√ 1−q dqxψ ( x2 ) eq(ixt) = ∫ 1√ 1−q 0 dqrψ(r2) J (1) − 1 2 (rt\|q2) (rt)− 1 2  for t2 < 1 1− q . Corollary 2 for m = 1, k = 1 and S1 = x is √ 1 + q 2Γq2(1 2) ∫ 1√ 1−q − 1√ 1−q dqxxψ ( x2 ) eq(ixt) = i ∫ 1√ 1−q 0 dqrr 2ψ(r2) J (1) 1 2 (rt\|q2) (rt) 1 2  t for t2 < 1 1− q . This agrees with eq(iu) = Γq2 (1 2 )( u 1 + q )(1/2) [ J (1) − 1 2 ( u\|q2 ) + iJ (1) 1 2 ( u\|q2 )] , which can be easily calculated. 20 K. Coulembier 5 Properties of the Fourier transform The Fourier transform is determined by its Bochner’s relations, see Corollary 2. The second Fourier transform is immediately defined here by its Bochner’s relations. Definition 4. The second Fourier transform F±Rmq on Rmq is a map V[α] → V [1/α] for each α ∈ R+. For a function Sk(y)ψ(y2) ∈ P ⊗ Eq2 ( − y2 µα ) the transform is given by F±Rmq [ Sk(y)ψ(y2) ] (x) = (±i)kSk(x) 1 c( √ αγ) Fq,γm 2 +k−1 [ψ] ( x2 ) , for an arbitrary γ ∈ R+ with c (√ αγ ) = q(m 2 −1)m 2 Γq2(m2 ) ∫ αq2γ2 µ ·∞ 0 dq2uu m 2 −1Eq2(−u) = d ( γ √ α µ , m 2 − 1 ) = d γ√q−2jα µ , m 2 + k − 1  for j, k ∈ N. The second Fourier transform does not depend on the choice of γ as can be seen from the expressions in the subsequent Theorem 7. From the properties of c( √ αγ) it is clear that the definition does not depend on which Vα the element of V[α] is chosen to be in. Although the Fourier transform on each space V[α] is denoted by the same symbol, each Fourier transform should be regarded as an independent operator. In Section 7 these different transforms will be combined in order to construct the Fourier transform on the Hilbert space corresponding to the harmonic oscillator. Theorem 7. The Fourier transforms in Definitions 3 and 4 are each others inverse, i.e. for each α, β ∈ R+ F∓Rmq ◦ F ± Rmq = idV[α] and F∓Rmq ◦ F ± Rmq = idV [β] . Proof. This is a consequence of Corollary 2 and Theorem 3. It can also be obtained directly from the relations, F±Rmq [ L(m 2 +k−1) j ( x2 αµ \|q2 ) Sk(x)Eq2 ( − x2 αµ )] (y) = (±i)kα m 2 +k+jCjy 2jSk(y)eq2 ( −αq 2y2 µ ) , F∓Rmq [ α m 2 +k+jCjy 2jSk(y)eq2 ( −αq 2y2 µ )] (x) = (∓i)kL(m 2 +k−1) j ( x2 αµ \|q2 ) Sk(x)Eq2 ( − x2 αµ ) , with Cj = q2(j+1)(j+m2 +k) [j]q2 !µj . These follow immediately from the calculations before Theorem 3. � Partial derivatives and multiplication with variables are operations which are defined on Vα and Vβ and therefore also on V[α] and V[β]. In this section we investigate how they interact with the Fourier transforms. The Fourier Transform on Quantum Euclidean Space 21 Theorem 8. For f ∈ V [β] and g ∈ V[α] with α, β ∈ R+, the following relations hold: (i) F±Rmq [xjf(x)](y) = ∓i∂jyF ± Rmq [f(x)](y), (ii) F±Rmq [yjg(y)](x) = ∓i qm ∂jxF±Rmq [g(y)](x), (iii) F±Rmq [ ∂jxf(x) ] (y) = ∓iqmyjF±Rmq [f(x)] (y), (iv) F±Rmq [ ∂jyg(y) ] (x) = ∓ixjF±Rmq [g(y)] (x). Proof. In order to prove (i) we choose f(x) of the form Sk(x)ψ(x2) ∈ P ⊗ Eq2 ( −β x2 µ ) for an arbitrary β ∈ R+. We use the Fischer decomposition of xjSk(x) in Lemma 5 and define ψ′(r2) = r2ψ(r2). Then, F±Rmq [ xjSk(x)ψ ( x2 )] (y) = (±i)k+1yjSk(y)Fq,βm 2 +k[ψ] ( y2 ) + (±i)k−1 y2∂jSk(y) µ[m2 + k − 1]q2 Fq,βm 2 +k[ψ] ( y2 ) + (±i)k−1 ∂jSk(y) µ[m2 + k − 1]q2 Fq,βm 2 +k−2[ψ′] ( y2 ) holds. Applying Lemma 3(i) then yields F±Rmq [ xjSk(x)ψ ( x2 )] (y) = (±i)k+1yjSk(y)Fq,βm 2 +k[ψ] ( y2 ) + (±i)k−1 [ ∂jSk(y) ] Fq,βm 2 +k−1[ψ] ( q2y2 ) . Now we calculate the right-hand side, using equations (2.25) and (2.26) and Lemma 3(ii), ∓i∂jyF ± Rmq [ Sk(x)ψ ( x2 )] (y) = −(±i)k+1yjSk(y)µ∂q 2 y2F q,β m 2 +k−1 [ψ] ( y2 ) + (±i)k−1 [ ∂jSk(y) ] Fq,βm 2 +k−1 [ψ] ( q2y2 ) = (±i)k+1yjSk(y)Fq,βm 2 +k [ψ] ( y2 ) + (±i)k−1 [ ∂jSk(y) ] Fq,βm 2 +k−1 [ψ] ( q2y2 ) . This proves property (i). Property (ii) can be calculated using the exact same techniques. Consider Sk(y)ψ(y2) ∈ P ⊗ eq2 ( −α q 2y2 µ ) . Combining Lemma 5 for the barred calculus with Lemma 3(i) yields c (√ αγ ) F±Rmq [ yjSk(y)ψ ( y2 )] (x) = (±i)k+1xjSk(x)Fq,γm 2 +k[ψ] ( x2 ) + (±i)k−1 qm [ ∂jSk(x) ] Fq,γm 2 +k−1[ψ] ( q−2x2 ) . Equation (2.27) and Lemma 3(ii) imply that this expression is equal to c( √ αγ) ∓i qm ∂jF±Rmq [f(y)](x). Properties (iii) and (iv) follow from properties (i) and (ii) and Theorem 7. As an illustration we calculate property (iv) directly. Consider Sk(y)ψ(y2) ∈ P ⊗ eq2 ( −α q 2y2 µ ) for an arbitrary α ∈ R+. Using equation (2.25) in the left-hand side of the second property yields F±Rmq [ ∂jySk(y)ψ ( y2 )] (x) = F±Rmq [ µyj ( ∂q 2 y2ψ ( y2 )) Sk(y) ] (x) + F±Rmq [ (∂jSk)(y)ψ ( q2y2 )] (x). The property ∫ γ·∞ 0 dqtf(qt) = 1 q ∫ γ·∞ 0 dqtf(t) yields c (√ αγ ) F±Rmq [ (∂jSk)(y)ψ ( q2y2 )] (x) = (±i)k−1 qm+2k−2 (∂jSk)(x)Fq,γm 2 +k−2 [ψ] ( q−2x2 ) . 22 K. Coulembier For Sk+1 ∈ Sk+1, Lemma 3(iii) implies c( √ αγ)µF±,γRmq [ Sk+1(y)∂q 2 y2ψ ( y2 )] (x) = (±i)k−1Sk+1(y)Fq,γm 2 +k−1[ψ] ( x2 ) and for Sk−1 ∈ Sk−1 Lemma 4 yields c (√ αγ ) µF±Rmq [ Sk−1(y)y2 ( ∂q 2 y2ψ ( y2 ))] (x) = (±i)k−1x2Sk−1(x)Fq,γm 2 +k−1[ψ] ( x2 ) − [m2 + k − 1]q2 qm+2k−2 µ(±i)k−1Sk−1(x)Fq,γm 2 +k−2[ψ] ( q−2x2 ) . From these calculations combined with Lemma 5 and taking Sk+1 and Sk−1 as defined by xjSk = Sk+1 + x2Sk−1, we obtain c (√ αγ ) F±Rmq [ ∂jySk(y)ψ ( y2 )] (x) = (±i)k−1 [ Sk+1(y)Fq,γm 2 +k−1[ψ] ( x2 ) + x2Sk−1(x)Fq,γm 2 +k−1[ψ] ( x2 )] = (±i)k−1xjSk(x)Fq,γm 2 +k−1[ψ] ( x2 ) . This proves property (iv). � The behavior of the Fourier transforms with respect to the Laplacian and norm squared can be calculated from Theorem 8. For F±Rmq acting on each space V [β] and for F±Rmq acting on each space V[α], the relations F±Rmq ◦ x2 = −∆y ◦ F ± Rmq , F±Rmq ◦∆∗x = −y2 ◦ F±Rmq , F±Rmq ◦ y2 = −∆∗x ◦ F±Rmq , F±Rmq ◦∆y = −x2 ◦ F±Rmq hold. This implies that the Fourier transforms map the two Hamiltonians for the harmonic oscillator in (2.30) into each other, F±Rmq ◦ h ∗ x = hy ◦ F ± Rmq and F±Rmq ◦ hy = h∗x ◦ F±Rmq . (5.1) 6 Funk–Hecke theorem on Rm q The polynomials on the quantum sphere correspond to P/(R2 − 1) with (R2 − 1) the ideal generated by the relation R2−1. The Fischer decomposition in Lemma 1 implies that this space is isomorphic to S = ⊕∞ k=0 Sk. The inner product on the quantum sphere 〈·\|·〉 : S × S → C 〈f \|g〉 = ∫ Sm−1 q f(g)∗ (6.1) is positive definite, symmetric and Oq(m)-invariant, see [15, Proposition 14]. In particular Sk ⊥ Sl when k 6= l. The symmetry can be obtained from the results in [14] or from the subsequent Lemma 9. In [32] the polynomials of degree l in x and y, (x; y)(l) which satisfy ∂jx(x; y)(l) = [l]q−2(x; y)(l−1)yj were determined. In particular this implies that the exponential on Rmq takes the form expR̂(x\|y) = ∞∑ l=0 (x; y)(l) [l]q−2 ! . The Fourier Transform on Quantum Euclidean Space 23 For later convenience we define 〈x\|y〉(l) = [l]q ! [l]q−2 !(x; y)(l), so ∂jx〈x\|y〉(l) = [l]q〈x\|y〉(l−1)yj . These polynomials satisfy the following Funk–Hecke theorem, see [18] for the classical version. Theorem 9. For Sk ∈ Sk, the relation∫ Sm−1 q ,x Sk(x)〈x\|y〉(l) = αk,lSk(y)yl−k holds with αk,l =  2Γq2(1 2)m[l]q! µl [ l−k 2 ] q2 !Γq2 ( k+l+m 2 ) if k + l is even and l ≥ k, 0 if k + l is odd, 0 if l < k. Proof. This follows immediately from Theorem 6. � This will lead to the reproducing kernel for the spherical harmonics. First the following technical lemma is needed. Lemma 8. For l > 0 and α ∈ R, the following relation holds: l∑ j=0 (−1)jqj(j−1) ( l j ) q2 Γq2(α+ l − j) Γq2(α+ 1− j) = 0. Proof. The calculation ( l j ) q2 = ( l−1 j−1 ) q2 + q2j ( l−1 j ) q2 is straightforward. Applying this yields C[l] := l∑ j=0 (−1)j ( l j ) q2 Γq2(α+ l − j) Γq2(α+ 1− j) = l−1∑ j=0 (−1)jqj(j−1) ( l − 1 j ) q2 q2j Γq2(α+ l − j) Γq2(α+ 1− j) − l−1∑ j=0 (−1)jqj(j+1) ( l − 1 j ) q2 Γq2(α+ l − j − 1) Γq2(α− j) = l−1∑ j=0 (−1)jqj(j−1) ( l − 1 j ) q2 q2j Γq2(α+ l − 1− j) Γq2(α+ 1− j) ( [α+ l − 1− j]q2 − [α− j]q2 ) = q2α[l − 1]q2C[l − 1]. This relation shows that C[1] = 0 and by induction C[l] = 0, for l > 0. � We define the coefficients cn,λj of the q-Gegenbauer polynomials (2.13) as Cλn ( q; µ 1 + q t ) = bn 2 c∑ j=0 cn,λj tn−2j . (6.2) Theorem 10. The polynomials Fn(x\|y) = Cn bn 2 c∑ j=0 c n,m 2 −1 j x2j〈x\|y〉(n−2j)y2j 24 K. Coulembier with cn,λj as defined in equation (6.2) and Cn = [m 2 +n−1]q2Γq2 (m 2 −1) 2Γq2 ( 1 2 )m satisfy ∫ Sm−1 q Sk(x)Fn(x\|y) = δknSk(y) for Sk ∈ Sk. (6.3) Proof. If n− k is odd or n < k then the left-hand side of equation (6.3) is zero because of the expression for αk,l in Theorem 9. So we consider the case n ≥ k and n− k even. The left-hand side of equation (6.3) can then be calculated using Theorem 9 ∫ Sm−1 q Sk(x)Fn(x\|y) = Cn bn 2 c∑ j=0 c n,m 2 −1 j ∫ Sm−1 q Sk(x)〈x\|y〉(n−2j)y2j = Cn n−k 2∑ j=0 c n,m 2 −1 j αk,n−2j Sk(y)yn−k. Therefore we calculate n−k 2∑ j=0 c n,m 2 −1 j αk,n−2j = n−k 2∑ j=0 (−1)jqj(j−1) [j]q2 ![n− 2j]q! Γq2(m2 − 1 + n− j) Γq2(m2 − 1) µn−2j 2Γq2(1 2)m[n− 2j]q! µn−2j [ n−k 2 − j ] q2 !Γq2 (k+n−2j+m 2 ) = 2Γq2(1 2)m Γq2(m2 − 1) [ n−k 2 ] q2 ! n−k 2∑ j=0 (−1)jqj(j−1) (n−k 2 j ) q2 Γq2(m2 − 1 + n− j) Γq2 ( k+n+m 2 − j ) . When n − k > 0 this expression is zero because of Lemma 8 for l = (n − k)/2 and α = (k + n+m)/2− 1. This implies n−k 2∑ j=0 c n,m 2 −1 j αk,n−2j = δkn 2Γq2(1 2)m Γq2(m2 − 1) 1[ m 2 + k − 1 ] q2 = δkn Cn , which proves the theorem. � This theorem implies that for bases {S(l) k } of Sk, which are orthonormal with respect to the inner product in equation (6.1), the reproducing kernel satisfies Fk(x\|y) = dimSk∑ l=1 ( S (l) k (x) )∗ S (l) k (y). The reproducing kernel can be written symbolically as a q-Gegenbauer polynomial, keeping in mind that 〈x\|y〉j should be replaced by 〈x\|y〉(j). In [11] an overview of Gegenbauer polynomials appearing as reproducing kernels for classical, Dunkl and super harmonic analysis is given. For completeness we prove that ( S (l) k (x) )∗ is still a spherical harmonic. Lemma 9. The antilinear involutive antihomomorphism ∗ on P satisfies ( x2lSk )∗ ⊂ x2lSk. The Fourier Transform on Quantum Euclidean Space 25 Proof. It is immediately clear that for Sk ∈ Sk, ( x2lSk(x) )∗ = x2l (Sk(x))∗ holds. Induction and equation (2.28) yield (Sk(x))∗ = 1 [k]q2 ! m∑ i1,...,ik=1 xi1xi2 · · ·xik∂i1x ∂i2x · · · ∂ikx Sk(x). First we will prove the relation m∑ il,il+1,il+2,...,ik=1 ∂ilx x il+1xil+2 · · ·xik∂i1x ∂i2x · · · ∂ikx Sk(x) = 0 (6.4) for i1, . . . , il−1 ∈ {1, . . . ,m}, l = k−1, k−2, . . . , 1 by induction. This clearly holds for l = k−1, since equation (2.20) implies m∑ ik−1,ik=1 ∂ ik−1 x xik∂i1x ∂ i2 x · · · ∂ikx Sk(x) = ∂i1x ∂ i2 x · · · ∂ ik−2 x ∆Sk(x). Then we assume it holds for l + 1 and calculate m∑ il,il+1,...,ik=1 ∂ilx x il+1xil+2 · · ·xik∂i1x ∂i2x · · · ∂ikx Sk(x) = m∑ il,il+1,...,ik=1 Cilil+1 xil+2 · · ·xik∂i1x ∂i2x · · · ∂ikx Sk(x) + q m∑ il,il+1,...,ik=1 m∑ s,t=1 (R̂−1) il,il+1 s,t xs∂txx il+2 · · ·xik∂i1x ∂i2x · · · ∂ikx Sk(x). The first line after the equality is zero since Sk ∈ Sk. The second one can be simplified by using relation (2.21): m∑ s=1 xs m∑ t,il+2,...,ik=1 ∂txx il+2 · · ·xik∂i1x ∂i2x · · · ∂ il−1 x ∂sx∂ t x∂ il+2 x · · · ∂ikx Sk(x). This is zero because of the induction step. Then we use equation (2.22) to calculate ∆S∗k(x) = µ 1 [k]q2 ! m∑ i1,...,ik=1 ∂i1xi2 · · ·xik∂i1x ∂i2x · · · ∂ikx Sk(x) + q2 1 [k]q2 ! m∑ i1,...,ik=1 xi1∆xi2 · · ·xik∂i1x ∂i2x · · · ∂ikx Sk(x) which implies S∗k ∈ Sk by using equation (6.4) consecutively. � 7 The Fourier transform on the Hilbert space of the harmonic oscillator The Fourier transforms have been defined on specific spaces of polynomials weighted with Gaussians. The Fourier transform can be extended to the Hilbert space structure developed in [14]. First we repeat the basic ideas of [7, 14]. Define the functions ψ0 = eq2 ( − x2 qm/2µ ) and ψ0 = Eq2 ( −q m/2q2x2 µ ) , 26 K. Coulembier which are the ground states corresponding to the Hamiltonians in equation (2.30). A q- deformation of the raising operators is given by aj+n = bn(q) ( xj − q2−n−m 2 ∂j ) Λ− 1 4 and aj+n = bn(q) ( xj − qn−2−m 2 ∂j ) Λ 1 4 , with (for now) undetermined coefficients bn(q) and bn(q). In [14] the relation bn(q) = bn(q−1) was assumed, which we do not assume here. These operators can be used to construct the functions ψin···i1n = ain+ n · · · ai1+ 1 ψ0 and ψ in···i1 n = ain+ n · · · ai1+ 1 ψ0. These are the eigenfunctions of the Hamiltonians of the harmonic oscillator: hψin···i1n = µ 2 [n+m/2]q2 qn+m/2 ψin···i1n and h∗ψ in···i1 n = µ 2 [n+m/2]q2 qn+m/2 ψ in···i1 n . (7.1) Note that the functions ψin···i1n are not linearly independent. These two types of functions generate vector spaces, which we denote by Π(V) and Π(V). They are both representation of the abstract vector space V which consists of linear combinations of abstract elements Ψin···i1 n . The maps Π : V→ Π(V) and Π : V→ Π(V) given by Π(Ψin···i1 n ) = ψin···i1n and Π(Ψin···i1 n ) = ψ in···i1 n are isomorphisms. The vector space V is an inner product space with the inner product 〈·\|·〉 developed in [14], 〈u, v〉 = ∫ γ·Rmq ( Π(u) )∗ Π(v) + (Π(u))∗Π(v). (7.2) The value of γ is not important. The harmonic oscillatorH on V is defined such that h◦Π = Π◦H and h∗ ◦ Π = Π ◦H and is hermitian with respect to the inner product. The closure of V with respect to the topology induced by 〈·\|·〉 is denoted by H. The behavior of the Fourier transforms with respect to the Hamiltonians of the harmonic oscillator was obtained in equation (5.1). This can be refined to the raising operators. Lemma 10. For the Fourier transform F±Rmq in Definition 4, the expression F±Rmq ◦ a j+ n : V[α] → V[q/α] satisfies F±Rmq ◦ a j+ n = ±iq2−n bn(q) bn(q) aj+n ◦ F±Rmq . Proof. A direct calculation or the equations in the proof of Theorem 7 yield F±Rmq ◦ Λ−1/4 = qm/2Λ1/4 ◦ F±Rmq . Combining this with Theorem 8 then yields the lemma. � It is clear that Π(V) ⊂ V [q− m 2 ] ⊕ V [q1− m 2 ] . It can be easily checked that the sum of V [q− m 2 ] and V [q1− m 2 ] is in fact direct. This implies that the Fourier transform can be trivially defined on Π(V). The Fourier transform of φ ∈ Π(V), with the unique decomposition φ = f + g with f ∈ V [q− m 2 ] and g ∈ V [q1− m 2 ] , is defined as F±Rmq (φ) = F±Rmq (f) + F±Rmq (g) (7.3) with the right hand side given in Definition 4. Now we define the Fourier transform on V. The Fourier Transform on Quantum Euclidean Space 27 Definition 5. The Fourier transform F± : V→ V is given by F± = q− m2 4 Π −1 ◦ F±Rmq ◦Π, with F±Rmq the Fourier transform on Π(V) as in equation (7.3). Now we impose the condition bn(q) = q2−nbn(q) on the undetermined coefficients bn, bn. Theorem 11. The Fourier transform F± on V satisfies F±[Ψin···i1 n ] = (±i)nΨin···i1 n . Proof. The definition of F± shows that this statement is equivalent with F±Rmq [ψin···i1n ] = (±i)nq m2 4 ψ in···i1 n . The proof of Theorem 7 implies F±Rmq [ψ0] = q m2 4 ψ0. Lemma 10 then proves the theorem by induction. � This immediately implies the following conclusions. Corollary 3. The Fourier transform on V can be continuously extended to H and satisfies F∓ ◦ F± = idH and the Parseval theorem 〈F±(f)\|F±(g)〉 = 〈f \|g〉 for f, g ∈ H. Corollary 4. The Fourier transform on H can be written symbolically as F± = exp ±iπ 2 arcsinh ( 1−q2 µ H ) ln ( 1 q ) − m 2  . Proof. This identity follows from evaluating the expression on Ψin···i1 n using equation (7.1). � The Fourier transform F±Rmq on Π(V) can therefore be written as F±Rmq = q m2 4 Π ◦Π−1 ◦ exp ±iπ 2 arcsinh ( 1−q2 2µ [ x2 −∆ ]) ln ( 1 q ) − m 2  . The Parseval theorem in Corollary 3 and the inner product (7.2) imply the following relation for f, g ∈ Π(V): q m2 4 ∫ γ·Rmq ( Π ◦Π−1(f) )∗ g + (f)∗Π ◦Π−1(g) = ∫ γ·Rmq ( F±Rmq (f) )∗ Π ◦Π −1 (F±Rmq (g)) + ( Π ◦Π −1 (F±Rmq (f)) )∗F±Rmq (g). The results in [14] show that this can be refined to q m2 4 ∫ γ·Rmq ( Π ◦Π−1(f) )∗ g = ∫ γ·Rmq ( F±Rmq (f) )∗ Π ◦Π −1 (F±Rmq (g)). 28 K. Coulembier 8 The q-Fourier transform on Euclidean space The techniques developed in this paper can also be applied to the theory of the q-Dirac operator on undeformed Euclidean space Rm, see [10]. We consider the polynomials in m commuting variables: C[x1, . . . , xm]. The classical Laplace operator and norm squared are given by ∆ = m∑ j=1 ∂2 xj and r2 = m∑ j=1 x2 j . The spherical harmonics are the homogeneous null-solutions of the Laplace operator, Hk = R[x1, . . . , xm]k ∩ Ker ∆. The q-Fourier transforms of a function Hk(x)ψ(r2) with Hk ∈ Hk are given by F±q [ Hk(x)ψ ( r2 )] (y) = (±i)kHk(y) ∫ 1√ 1−q 0 dqr J (1) m 2 +k−1(rry\|q2) (rry) m 2 +k−1 rm+2k−1ψ ( r2 ) , F±q [ Hk(y)ψ ( r2 y )] (x) = (±i)kHk(x) 1 d ( γ√ 1+q , m2 ) ∫ γ·∞ 0 dqry J (2) m 2 +k−1(qrry\|q2) (rry) m 2 +k−1 rm+2k−1 y ψ(ry). The fact that these transforms are each others inverse when evaluated on the appropriate func- tion spaces follows immediately from equations (3.1) and (3.2). The operators Dj , j = 1, . . . ,m are defined in [10, equation (25) and Definition 2]. They are a q-deformation of the partial derivatives on Rm. Similarly to equation (2.25) for the partial derivatives on q-Euclidean space the q-derivatives on Euclidean space satisfy Dif(r2) = (1 + q)xi ( ∂q 2 r2 f ( r2 )) + f ( q2r2 ) Di, see Lemma 6 in [10]. The q-Laplace operator on Rm can be defined as ∆q = m∑ j=1 D2 j . The polynomial null-solution of this q-Laplace operator correspond to the classical spacesHk. This shows that the q-deformation is purely radial. This can also be seen from the fact that the Howe dual pair of this construction is (O(m),Uq(sl2)). This implies that there is no spherical deformation and a radial deformation identical to the one from quantum Euclidean space. Using the results from the previous sections and [10] it is straightforward to prove that DjF ± q [f(x)] = ±iF±q [xjf(x)] holds. 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Fields 37 (2004), 379–389, hep-th/0401113. http://dx.doi.org/10.1063/1.530326 http://arxiv.org/abs/hep-th/9302076 http://dx.doi.org/10.1016/0022-247X(81)90048-2 http://dx.doi.org/10.1007/BF00398308 http://arxiv.org/abs/hep-th/9205003 http://dx.doi.org/10.1063/1.531136 http://arxiv.org/abs/hep-th/9409132 http://dx.doi.org/10.1063/1.531658 http://arxiv.org/abs/q-alg/9506020 http://dx.doi.org/10.1140/epjc/s2003-01397-7 http://arxiv.org/abs/hep-th/0206083 http://dx.doi.org/10.1140/epjc/s2004-01999-5 http://arxiv.org/abs/hep-th/0401113 1 Introduction 2 Preliminaries 2.1 q-calculus 2.2 The Howe dual pair and harmonic oscillator on Rmq 2.3 Integration and Fourier theory on quantum spaces 3 The q-Hankel transforms 4 Integration and Fourier transform on Rmq 4.1 Integration over the quantum sphere and induced integration on Rmq 4.2 Bochner's relations for the Fourier transform on Rmq 5 Properties of the Fourier transform 6 Funk-Hecke theorem on Rmq 7 The Fourier transform on the Hilbert space of the harmonic oscillator 8 The q-Fourier transform on Euclidean space References

The Fourier Transform on Quantum Euclidean Space

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