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 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.


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 spacetime, 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 O q (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 (O q (m), U q (sl 2 )). The quantum algebra U q (sl 2 ) is generated by the O q (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 2 Preliminaries

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 It is clear that lim q→1 [u] q = u. We also define The q-derivative of a function f (t) is defined by This operator satisfies the generalized Leibniz rule The q-integration on an interval [0, a] with a ∈ R is given by f q k a q k .
The infinite q-integral can be defined in several ways, determined by a parameter γ ∈ R\{0}, 3) The positive (γ ∈ R + ) or negative (γ ∈ R − ) infinite integral is a function of γ, however from the definition it is clear that . So ∂ q γ γ·∞ 0 d q t = 0 holds which means the integral is a q-constant. The integral is the inverse of differentiation, 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 Γ q (t) for t > 0 satisfying Γ q (t + 1) = [t] q Γ q (t), see e.g. [17, formula (1.10.1)]. The q-exponentials are defined as Note that a different notation for the exponentials is used compared to [10]. The relation e q (t)E q (−t) = 1 holds and the derivatives are given by ∂ q t e q (t) = e q (t) and ∂ q t E q (t) = E q (qt). (2.5) For q < 1 the series E q (t) converges absolutely and uniformly everywhere and e q (t) in the area |t| < 1 1−q . The function e q (t) can be analytically continued to C\{ q −k 1−q } as 1/(E q (−t)). The zeroes of the q-exponential E q are E q − q −k 1 − q = 0 for k ∈ N. (2.6) This follows from the infinite product representation E q ( t 1−q ) = (−t; q) ∞ , see [17, formula (1.3.16)]. This implies that the relation holds.
The q-Hermite polynomials are given by q j(j+1) ((q + 1)t) k−2j (2.8) and related to the discrete q-Hermite I polynomials h k (x; q) in [25,Section 3.28] by We introduce the q-Laguerre polynomials for α > −1 in the normalization of [10, p. 24], They can also be defined using the q-Gamma function since Γ q 2 (i+α+1) holds. They are connected with the q-Laguerre polynomials L (α) j (u; q) from [25,Section 3.21] by and to the q-Laguerre polynomials in [30] by the same formula with a substitution (1 − q 2 )u → u in the right hand side of the formula.
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 H q,γ ν in the subsequent Definition 2. This follows from the same calculation that leads to the subsequent equation (3.2).
For ν > −1, the first and second q-Bessel function, introduced by Jackson, see [21,22], are given by and (2.15) The generating functions for the q-Laguerre polynomials are given by This follows from direct calculations, they are equivalent to [25, formulas (3.20.11) and (3.21.13)]. 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 R m q . The relations for the variables can e.g. be found in [15,20,33]. We denote by O q the algebra of formal power series for the specific case of the quantum Euclidean space. The quantum Euclidean space can be defined by thê R-matrix of the quantum orthogonal group O q (m), see [5,14]. The matrixR ∈ C (m×m)×(m×m) can be expressed in terms of its projection operators asR = qP S − q −1 P A + q 1−m P 1 , and is symmetric,R ij kl =R kl ij . The matrixR depends on the parameter q and returns to the undeformed case when q → 1 (lim q→1R ij kl = δ j k δ i l ). The antisymmetric part defines the commutation relations

The
We will always use the summation convention. The singlet part defines the metric C ij = C ij , by (P 1 ) ij The metric satisfies the relation and is its own inverse, C ij C jk = δ k i . The braid matrix also satisfies the relation The generalized norm squared is then defined as x 2 = x i C ij x j . This norm squared is central in the algebra O q and is invariant under the co-action of O q (m). The explicit expressions for the coaction of O q (m) or the dually related action of U q (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 y i x j = q −1R ij kl x k y l , see [23,29,32]. The differential calculus on R m q was developed in [6,31], the action of the partial derivatives is determined by The Laplace operator on R m q is given by ∆ = ∂ i C ij ∂ j . It is central in the algebra generated by the partial derivatives and is O q (m)-invariant. The commutation relations for the partial derivatives can be expressed using P A or aŝ The dilatation operator is given by and satisfies Λx i = q 2 x i Λ. For u ∈ R, Λ u is defined by Λ u x i = q 2u x i Λ u and Λ u (1) = 1. The elements of O q corresponding to finite summations are the polynomials, the corresponding algebra is denoted by P. The space P k is defined as the space of the polynomials P in P which satisfy For f analytical in the origin, f (x 2 ) ∈ O q is defined by the Taylor expansion of f . Equation (2.22) leads to for general functions of x 2 and the q-derivative as defined in formula (2.1). The relation is a useful calculation rule. There exists a second differential calculus on R m q of partial derivatives ∂ j , which is obtained from the unbarred one by replacing ∂ j , q,R, C by ∂ j , q −1 ,R −1 , C. In particular, the relation holds. The algebra generated by the variables and partial derivatives ∂ j is denoted Diff(R m q ). 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 S k , so S k = P k ∩ ker ∆. Definition 1. The operator E is given by E = [ m 2 ] q 2 + q m x i C ik ∂ k . Using the expression for the dilation operator in formula (2.23) the operator E can be expressed as Property (2.24) then implies that for S k ∈ S k , holds. Together with ∆ and x 2 , this operator E generates an algebra which is a q-deformation of the universal enveloping algebra of sl 2 .
Theorem 1. The operators ∆/µ, x 2 /µ and E generate the quantum algebra U q (sl 2 ), which leads to the second relation. The third relation is calculated in the same way.

Remark 2.
As the generators of U q (sl 2 ) are O q (m)-invariant, this quantum algebra and quantum group form the Howe dual pair (O q (m), U q (sl 2 )), or (U q (so(m)), U q (sl 2 )).
The quantum algebra U q (sl 2 ) is equal to the one in [10]. In [10], U q (sl 2 ) was generated by the standard Euclidean norm squared r 2 on R m and a q-deformation of the Laplace operator ∆ q . Since ∆ q is still O(m)-invariant, the Howe dual pair (O(m), U q (sl 2 )) appeared. Because the O q (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 O q (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 U q (so(m)) as (see [20,15]) The operator identities in Theorem 1 yield These calculations and the previous results lead to Theorem 2 (Howe duality). The decomposition of P into irreducible representations of U q (so(m)) is given in Lemma 1. Each space j x 2j S k is a lowest weight module of U q (sl 2 ) with weight vectors x 2j S k , the lowest weight vector is S k with weight [m/2 + k] q 2 . The Fischer decomposition of P therefore is a multiplicity free irreducible direct sum decomposition under the joint action of U q (sl 2 ) × U q (so(m)).
The antilinear involutive antihomomorphism * on Diff(R m q ) is defined by (AB) * = B * A * , (x j ) * = x k C kj , (∂ j ) * = −q −m ∂ k C kj and λ * = λ with λ ∈ C and · complex conjugation. This yields 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 Both operators have the same eigenvalues.

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 generalR-matrices is defined assuming there is a matrix η ∈ R m×m and a solution g η ∈ O of the equation Integration on the space Pg η , with P the polynomials on the quantum space is then uniquely defined by demanding with I(g η ) = g η 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 integration 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 These Fourier transforms are each others inverse, F * F = Vol S, 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 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 x k y k [k]q! and satisfies ∂ q x exp(x|y) = exp(x|y)y. The Fourier transforms take the form and The theory of [23] then implies We can rewrite this in a way that will be more closely related to our approach of the Fourier transform on R m q . Define g(y) = 1 Vol γ,δ F[f ](y), then the definition of the antipode implies 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 In [26] a closely related analytical approach was given to the one dimensional Fourier transform 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 which is equivalent to [26, equation (8.7)]. For every δ ∈ R + , holds for C δ 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 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 Vol γ,δ is infinite.

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 formulas (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 O q (m)-invariant Fourier transform on R m q . 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 R m q . This is already the case for the Fourier transform on the braided line, as is proven in [27]. The braided line corresponds to R m q 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 and 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 R m q with these q-Hankel transforms we define the following transformations, 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 Proof . The right-hand side of the second property (for the first q-Bessel function) can be calculated using 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: 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 Generating function (2.17) and orthogonality relation (2.12) lead to The following expansion of a monomial in terms of the q-Laguerre polynomials is a direct consequence of equation (2.17), Applying this yields These calculations imply the following relations for the q-Hankel transforms in Definition 2: By considering the case β = 1/α we obtain Theorem 3. For each α, γ ∈ R + , the inverse of the q-Hankel transform H q,γ ν acting on 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 R m q 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: The second q-Hankel transform satisfies the following properties: Proof . The first property is a direct consequence of Lemma 2(ii), the second a direct consequence of Lemma 2(i). Property (iii) is calculated using formulas (2.2) and (2.4) and Lemma 2(iii) Property (iii) for the second q-Hankel transform is calculated similarly.

Lemma 4. The relation
holds for the second q-Hankel transform.
Proof . The left-hand side is calculated using Corollary 1, which proves the lemma.
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 x 2l S k . Since these blocks are irreducible U q (so(m))-representations (or irreducible O q (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 x 2l . The second property then implies S m−1 q x 2l = S m−1 q 1, which shows that the integration is uniquely determined up to the constant S m−1 q 1. It is easily checked that the Pizzetti formula satisfies the conditions. We chose the normalization such that S m−1 . This quantum sphere integration can be expressed symbolically using the first q-Bessel-function (2.14), The following lemma will be important for the sequel.
Lemma 5. The Fischer decomposition (see Lemma 1) of x j S k (x) with S k ∈ S k is given by Proof . Since ∆ and ∂ j commute This can also be calculated by using the projection operator from P k onto S k developed in [20].
Before we use the quantum sphere integration to construct integration on R m q along the lines of [23] and [14], the following lemma should be considered.

Lemma 6. For the q-integration in equation
is infinite unless γ = q j 1 1−q for some j ∈ Z. In that case it reduces to Proof . Property (2.5) can be rewritten as E q (t) = E q (qt) [1 + (1 − q)t], which implies First we assume E q (−q 1−k γ) is never zero for k ∈ Z. The equation above then yields E q (−q −k γ) > E q (−q 1−k γ) q and Sign(E q (−q −k γ)) = Sign(E q (−q 1−k γ)) for k ≥ N ∈ N with N > ln 1+q γ(1−q) ln(1/q) . This implies that ∞ k=N E q (−q −k γ)q −k will not converge, therefore the integral (2.3) will not exist.
If E q (−q 1−k γ) = 0 holds for some k ∈ Z, the unicity of the zeroes in equation (2.6) implies that γ = q k−1−l 1−q for some k ∈ Z and l ∈ N, or γ = q j 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 R m q 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 ∂ j e q 2 −αx 2 = −αµx j e q 2 −αx 2 and imply that e q 2 (−αx 2 ) and E q 2 (−βx 2 ) are Gaussians according to equation (2.31) for the unbarred and barred calculus respectively. Define the following two integrations for γ, λ ∈ R + on polynomials weighted with undetermined radial functions, 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 Proof . The first property is well-known, see e.g. [14,34]. In order to prove the second property we consider the expression for a spherical harmonic S k ∈ S k . If k = 1 the right-hand side is always zero, because of When we substitute f (x 2 ) = x 2l E q 2 (−βx 2 ) with l ∈ N and β ∈ R + and use equation (2.6) we obtain This proves the second part of the theorem because of the Fischer decomposition in Lemma 1.

Remark 3. The properties
γ·R m q ∂ j = 0 on P ⊗ e q 2 −αx 2 for α, γ ∈ R + and 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 Since γ·R m q E q 2 (−βx 2 ) will be infinite for general γ, similar to Lemma 6, the finite integration needs to be used on P ⊗E q 2 (−βx 2 ). As in equation (2.7) this integration corresponds to a specific infinite integration, Theorem 5 therefore shows that for integration on P ⊗ e q 2 (−αx 2 ) there is a bigger choice since γ does not depend on α. However, it is straightforward to calculate which implies the integrals for different γ on P ⊗ e q 2 (−αx 2 ) 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(g η ) while the functional Z does not depend on γ. For strict Gaussian-induced integration in the unbarred case, only one choice gives an I(g η ) which is finite.

Bochner's relations for the Fourier transform on R m q
The exponential expR(x|y) on a quantum space satisfies ∂ j x expR(x|y) = expR(x|y)y j , (4.2) see [23,29,36]. From now on expR(x|y) stands for the exponential on R m q . It is uniquely determined from equation (4.2) and the normalization expR(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 ⊗ e q 2 (−αx 2 ). 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 ⊗ E q 2 (−β x 2 µ ). First we will extend this space. Define The different spaces V α (or V β ) are not necessarily disjunct since equation (2.5) implies Therefore we define 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 , . So in the undeformed Euclidean case, [P ] j l = P δ j l .
Proof . The lemma is proven by calculating the expression It can also be calculated using formulas (2.22) and (2.28) which proves the lemma.
Remark 4. Using the same techniques for a general polynomial P k ∈ P k yields j,l 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 harmonics in the integration. Theorem 6. For S k ∈ S k , the following relation holds, Proof . First we prove that the relation holds, where · · · stands for terms of the form P (x) expR(ix|y)q 2 (y) with P ∈ ⊕ j>0 P j . In case k = 0, equation (4.2) implies that ∆ l x expR(ix|y) = (−1) l expR(ix|y)y 2l 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) Lemma 7 then proves that equation (4.3) holds for every k. We also used the fact that for S p ∈ S p , [S p ] j l ∈ S p which follows from the fact that ∂ j x and ∆ commute. Equation (4.3) then yields Comparing this with equation (2.14) proves the theorem.
This theorem allows to calculate the Fourier transform of an element of P ⊗ E q 2 (−β x 2 µ ) inside one of the irreducible blocks of its U q (so(m))-decomposition.

Corollary 2. The Fourier transform in Definition 3 of a function
with S k ∈ S k is given by with r and t two real commuting variables.
These formulae are the Bochner's relations for the Fourier transform on R m q . 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 Corollary 2 for m = 1, k = 1 and S 1 = x is This agrees with e q (iu) = Γ q 2 1 2 which can be easily calculated.

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. [1/α] for each α ∈ R + . For a function S k (y)ψ(y 2 ) ∈ P ⊗ E q 2 − y 2 µα the transform is given by

Definition 4. The second Fourier transform
for an arbitrary γ ∈ R + with 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 V α 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.
Proof . This is a consequence of Corollary 2 and Theorem 3. It can also be obtained directly from the relations, with C j = q 2(j+1)(j+ m 2 +k) [j] q 2 !µ j . These follow immediately from the calculations before Theorem 3.
Partial derivatives and multiplication with variables are operations which are defined on V α and V β and therefore also on V [α] and V [β] . In this section we investigate how they interact with the Fourier transforms.
Property (ii) can be calculated using the exact same techniques. Consider S k (y)ψ(y 2 ) ∈ P ⊗ e q 2 −α q 2 y 2 µ . Combining Lemma 5 for the barred calculus with Lemma 3(i) yields

Equation (2.27) and Lemma 3(ii) imply that this expression is equal to
Properties (iii) and (iv) follow from properties (i) and (ii) and Theorem 7. As an illustration we calculate property (iv) directly. Consider S k (y)ψ(y 2 ) ∈ P ⊗ e q 2 −α q 2 y 2 µ for an arbitrary α ∈ R + . Using equation (2.25) in the left-hand side of the second property yields From these calculations combined with Lemma 5 and taking S k+1 and S k−1 as defined by This proves property (iv).
hold. This implies that the Fourier transforms map the two Hamiltonians for the harmonic oscillator in (2.30) into each other, The polynomials on the quantum sphere correspond to P/(R 2 − 1) with (R 2 − 1) the ideal generated by the relation R 2 − 1. The Fischer decomposition in Lemma 1 implies that this space is isomorphic to S = ∞ k=0 S k . The inner product on the quantum sphere ·|· : is positive definite, symmetric and O q (m)-invariant, see [15,Proposition 14]. In particular S k ⊥ S l when k = 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 ∂ j x (x; y) (l) = [l] q −2 (x; y) (l−1) y j were determined. In particular this implies that the exponential on R m q takes the form For later convenience we define x|y (l) = [l]q!
These polynomials satisfy the following Funk-Hecke theorem, see [18] for the classical version.
Theorem 9. For S k ∈ S k , the relation 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.
We define the coefficients c n,λ j of the q-Gegenbauer polynomials (2.13) as C λ n q; c n,λ j t n−2j . x 2j x|y (n−2j) y 2j with c n,λ j as defined in equation (6.2) and C n = 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 α k,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 Therefore we calculate 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 which proves the theorem.
This theorem implies that for bases {S (l) k } of S k , which are orthonormal with respect to the inner product in equation (6.1), the reproducing kernel satisfies 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 which are the ground states corresponding to the Hamiltonians in equation (2.30). A qdeformation of the raising operators is given by with (for now) undetermined coefficients b n (q) and b n (q). In [14] the relation b n (q) = b n (q −1 ) was assumed, which we do not assume here. These operators can be used to construct the functions ψ in···i 1 n = a in+ n · · · a i 1 + 1 ψ 0 and ψ in···i 1 n = a in+ n · · · a i 1 + 1 ψ 0 .
These are the eigenfunctions of the Hamiltonians of the harmonic oscillator: and V [q 1− 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 [q 1− m 2 ] , is defined as with the right hand side given in Definition 4. Now we define the Fourier transform on V.
Definition 5. The Fourier transform F ± : V → V is given by with F ± R m q the Fourier transform on Π(V) as in equation (7.3). Now we impose the condition b n (q) = q 2−n b n (q) on the undetermined coefficients b n , b n .
Theorem 11. The Fourier transform F ± on V satisfies Proof . The definition of F ± shows that this statement is equivalent with F ±   Proof . This identity follows from evaluating the expression on Ψ in···i 1 n using equation (7.1).
The Fourier transform F ± R m q on Π(V) can therefore be written as The Parseval theorem in Corollary 3 and the inner product (7.2) imply the following relation for f, g ∈ Π(V): The results in [14] show that this can be refined to 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 R m , see [10]. The fact that these transforms are each others inverse when evaluated on the appropriate function spaces follows immediately from equations (3.1) and (3.2). The operators D j , j = 1, . . . , m are defined in [10, equation (25) and Definition 2]. They are a q-deformation of the partial derivatives on R m . Similarly to equation (2.25) for the partial derivatives on q-Euclidean space the q-derivatives on Euclidean space satisfy D i f (r 2 ) = (1 + q)x i ∂ q 2 r 2 f r 2 + f q 2 r 2 D i , see Lemma 6 in [10].
The q-Laplace operator on R m can be defined as ∆ q = m j=1 D 2 j . The polynomial null-solution of this q-Laplace operator correspond to the classical spaces H k . 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), U q (sl 2 )). 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 holds. Also the theory of the harmonic oscillator and the corresponding Hilbert space can be trivially translated to this setting.