A $q$-analogue of convolution on the line

Lemma 3.4. Let $f \in \mathcal{I}^{\omega}_{\gamma,\alpha}$ and $g \in \mathcal{H}^{D}_{a}$ . Then $f _{\gamma} g \in \mathcal{H}^{D}_{a}$ . If g can be continued analytically on a starlike domain $\Omega$ centered at 0, then $f _{\gamma} g$ is well-defined and analytic on $\Omega$ .

Lemma 3.5. Let $g \in \mathcal{H}_a^D \mathcal{I}_{\gamma}^{\infty}$ for some $a > \gamma$ . Then for every $r \in (\gamma, a)$ there is a constant B > 0 such that $$\int_{-\gamma \cdot \infty}^{\gamma \cdot \infty} |x^k(\partial^e g)(x)| \, d_q x \le \left( \int_{-\gamma \cdot \infty}^{\gamma \cdot \infty} |x^k g(x)| \, d_q x + r^k B \right) \frac{2^e}{\gamma^e (1-q)^e}$$ for every e and k in $\mathbf{Z}_{\geq 0}$ . Hence $\partial^e g \in \mathcal{H}_a^{\mathrm{D}} \mathcal{I}_{\gamma}^{\infty}$ . If $g \in \mathcal{H}_a^D \mathcal{I}_{\gamma}$ but not necessarily in $\mathcal{I}_{\gamma}^{\infty}$ , the above conclusion still holds for k = 0 and $e \in \mathbf{Z}_{\geq 0}$ . Then $\partial^e g \in \mathcal{H}_a^D \mathcal{I}_{\gamma}$ .

Proposition 3.7. Let $f \in \mathcal{I}_{\gamma'}^{\omega}$ and $g \in \mathcal{H}_a^D \mathcal{I}_{\gamma}$ . Then $f _{\gamma'} g$ is well-defined and absolutely q-integrable on $L(\gamma)$ , and $f _{\gamma'} g \in \mathcal{H}_a^D \mathcal{I}_{\gamma}$ .

Corollary 3.8. Let $f \in \mathcal{I}_{\gamma'}^{\omega}$ and let $g \in \mathcal{H}_a^{\mathrm{D}}\mathcal{I}_{\gamma}$ . Then $\partial^k(f _{\gamma'} g)$ and $f _{\gamma'} \partial^k g$ are also absolutely q-integrable on $L(\gamma)$ for every $k \in \mathbf{Z}_{>0}$ .

Lemma 4.1. Let $f \in \mathcal{I}_{\gamma}^{\infty}$ such that $\partial^{j} f \in \mathcal{I}_{\gamma}^{\infty}$ for every $j \in \mathbf{Z}_{\geq 0}$ . Then $$\int_{-\gamma \cdot \infty}^{\gamma \cdot \infty} (\partial^a f)(x) \, x^b \, d_q x = \begin{cases} (-1)^a q^{-ba + \frac{a^2 - a}{2}} \frac{[b]_q!}{[b-a]_q!} \int_{-\gamma \cdot \infty}^{\gamma \cdot \infty} f(x) \, x^{b-a} \, d_q x & \text{if } b \ge a, \\ 0 & \text{otherwise.} \end{cases}$$ In particular, (4.1) $$\mu_{e+k,\gamma}(\partial^k f) = (-1)^k \frac{[e+k]_q!}{[e]_q!} \mu_{e,\gamma}(f) \quad \text{for every } k \in \mathbf{Z}_{\geq 0} ,$$ and $\mu_{l,\gamma}(\partial^k f) = 0$ if l < k.

Lemma 4.2. If $f \in \mathcal{H}^{\mathcal{D}}\mathcal{I}^{\omega}_{\gamma,\alpha}$ then $X^k \partial^j f \in \mathcal{H}^{\mathcal{D}}\mathcal{I}^{\omega}_{\gamma,\alpha}$ for every $k \in \mathbb{Z}_{>0}$ .

Lemma 4.3. Let $f \in \mathcal{H}_a^D \mathcal{I}_{\gamma}^{\omega}$ and let g be defined, together with its q-derivatives, on a domain $\Omega$ . Let $0 \neq x \in \Omega$ be such that, for some $R_x > 0$ , $|\partial^e g(x)| = O(R_x^e)$ as $e \to \infty$ . Then, for every $k \in \mathbf{Z}_{\geq 0}$ , $\partial^k (f _{\gamma} g)$ , $(\partial^k f _{\gamma} g)$ and $(f *_{\gamma} \partial^k g)$ are well defined at x and $$\partial^k (f _{\gamma} g)(x) = (\partial^k f _{\gamma} g)(x) = (f *_{\gamma} \partial^k g)(x).$$ In particular, the result holds if $g \in \mathcal{H}^D \mathcal{I}_{\gamma}$ and $x \in L(\gamma)$ , or if g is analytic on a starlike domain $\Omega$ centered at 0 and $x \in \Omega$ .

Proposition 4.4. Let $f \in \mathcal{H}^{\mathrm{D}}\mathcal{I}^{\omega}_{\gamma'}$ and $g \in \mathcal{H}^{\mathrm{D}}_{a}\mathcal{I}^{\infty}_{\gamma}$ . Then $X^{k} \partial^{l}(f *_{\gamma'} g) \in \mathcal{H}^{\mathrm{D}}_{a}\mathcal{I}^{\infty}_{\gamma}$ for all $k, l \in \mathbf{Z}_{>0}$ .

Lemma 4.5. Let $f \in \mathcal{H}^{\mathrm{D}}\mathcal{I}_{\gamma}^{\omega}$ , $g \in \mathcal{H}^{\mathrm{D}}\mathcal{I}_{\gamma'}^{\omega}$ , $k \in \mathbf{Z}_{\geq 0}$ . Then (4.2) $$\mu_{k,\gamma'}(f *_{\gamma} g) = \sum_{e=0}^{k} \begin{bmatrix} k \\ e \end{bmatrix}_{q} \mu_{e,\gamma}(f) \mu_{k-e,\gamma'}(g).$$ Thus, if $\gamma = \gamma'$ then $\int_{\gamma} (f _{\gamma} g) X^k = \int_{\gamma} (g _{\gamma} f) X^k$ for every $k \in \mathbb{Z}_{\geq 0}$ and $\int_{\gamma} (f *_{\gamma} g) = (\int_{\gamma} f) (\int_{\gamma} g)$ .

Proposition 4.6. Let $f \in \mathcal{H}^{\mathrm{D}}\mathcal{I}^{\omega}_{\gamma,\alpha}$ and $g \in \mathcal{H}^{\mathrm{D}}\mathcal{I}^{\omega}_{\gamma',\beta}$ . Then $f *_{\gamma} g \in \mathcal{H}^{\mathrm{D}}\mathcal{I}^{\omega}_{\gamma',\eta}$ with $\eta := \frac{\alpha\beta}{\alpha+\beta}$ .

Theorem 4.8. Let $f \in \mathcal{H}^D \mathcal{I}_{\gamma}^{\omega}$ and $g \in \mathcal{H}^D \mathcal{I}_{\gamma'}^{\omega}$ . Let h be defined, together with its q-derivatives, on a domain $\Omega$ and let $x \in \Omega$ be such that, for some $R_x > 0$ , $|(\partial^e h)(x)| = O(R_x^e)$ as $e \to \infty$ . Then $((f _{\gamma} g) _{\gamma'} h)(x) = (f _{\gamma} (g _{\gamma'} h))(x)$ . In particular, the equality holds for every $x \in \Omega$ if h is analytic on a starlike domain $\Omega$ centered at 0 and it holds for every $x \in L(\gamma'')$ if $h \in \mathcal{H}^D \mathcal{I}_{\gamma''}^{\infty}$ .

Corollary 4.9. The class $\mathcal{H}^{\mathcal{D}}\mathcal{I}^{\omega}_{\gamma}$ is an algebra (not necessary unital) with respect to $*_{\gamma}$ . Its subclass $\mathcal{H}^{\mathcal{S}}\mathcal{I}^{\omega}_{\gamma}$ is also an algebra (not necessary unital) and it is a left ideal of $\mathcal{H}^{\mathcal{D}}\mathcal{I}^{\omega}_{\gamma}$ .

Proposition 5.1. Let $f \in \mathcal{I}_{\gamma,\alpha}^{s\omega}$ . If $\alpha > 1$ then $|f(\pm q^{-j}\gamma)| = 0$ for every $j \in \mathbf{Z}$ ; if $\alpha = 1$ then $|f(\pm q^{-j}\gamma)| = 0$ for every $j \in \mathbf{Z}$ sufficiently large; if $0 < \alpha < 1$ then, for some c > 0, $|f(\pm q^{-j}\gamma)| = O(q^{\frac{\beta}{2}j^2}c^j)$ as $j \to \infty$ with $\beta = \frac{1}{1-\alpha}$ . Conversely, let f be a function on $L(\gamma)$ which is bounded on $\{\pm q^j \gamma \mid j \geq 1\}$ and let $\beta > 1$ . If, for some c > 0, $|f(\pm q^{-j}\gamma)| = O(q^{\frac{\beta}{2}j^2}c^j)$ as $j \to \infty$ then $f \in \mathcal{I}_{\gamma,\alpha}^{s\omega}$ with $\alpha = 1 - \beta^{-1}$ .

Lemma 5.7. Let $f \in \mathcal{H}^D\mathcal{I}_{\gamma,\alpha}^{s\omega}$ . Then, there are C, c, R > 0 such that $$\nu_{e,\gamma}(\partial_{\cdot}^{k}f) \leq Cq^{\frac{\alpha}{2}e^{2}}c^{e}R^{k}$$ for all $k, e \in \mathbf{Z}_{\geq 0}$ . In particular, $\partial^k f \in \mathcal{H}^{\mathrm{D}}\mathcal{I}_{\gamma,\alpha}^{\mathrm{s}\omega}$ .

Proposition 5.8. Let $f \in \mathcal{H}^{\mathcal{D}}\mathcal{I}^{\omega}_{\gamma,\alpha}$ and $g \in \mathcal{H}^{\mathcal{D}}\mathcal{I}^{s\omega}_{\gamma',\beta}$ . Then $f *_{\gamma} g \in \mathcal{H}^{\mathcal{D}}\mathcal{I}^{s\omega}_{\gamma',\beta}$ . Proof. For any $e \geq 0$ , $$q^{\frac{e^2+e}{2}} \int_{\gamma'} |f *_{\gamma} g| \, |X^e| \leq \sum_{k=0}^{\infty} \frac{|\mu_{k,\gamma}(f)|}{[k]_q!} q^{\frac{e^2+e}{2}} \int_{\gamma'} |X|^e \, |\partial^k g| \leq C \sum_{k=0}^{\infty} \frac{q^{\frac{\alpha}{2}k^2}b^k}{[k]_q!} R^k q^{\frac{1}{2}\beta e^2} c^e$$ by formula (1.3), Definition 3.1(b) and Lemma 5.7.

Corollary 5.9. The class $\mathcal{H}^D\mathcal{I}_{\gamma}^{s\omega}$ is a subalgebra of $\mathcal{H}^D\mathcal{I}_{\gamma}^{\omega}$ . Its subclass $\mathcal{H}^S\mathcal{I}_{\gamma}^{s\omega}$ is a left ideal of $\mathcal{H}^S\mathcal{I}_{\gamma}^{\omega}$ , $\mathcal{H}^D\mathcal{I}_{\gamma}^{\omega}$ and $\mathcal{H}^D\mathcal{I}_{\gamma}^{s\omega}$ .

Corollary 5.11. The classes $\mathcal{H}^{S}\mathcal{I}_{\gamma,>c}^{s\omega}$ (for $c \in [0,1)$ ) and $\mathcal{H}^{S}\mathcal{I}_{\gamma,c}^{s\omega}$ (for $c \in (0,1]$ ) are left ideals of $\mathcal{H}^{S}\mathcal{I}_{\gamma}^{s\omega}$ and of $\mathcal{H}^{S}\mathcal{I}_{\gamma}^{\omega}$ . Similar properties hold for $\mathcal{H}^{D}\mathcal{I}_{\gamma}^{s\omega}$ .

Corollary 5.12. Let $f, g \in \mathcal{H}^{\mathcal{D}}\mathcal{I}^{\omega}_{\gamma}$ , $h \in \mathcal{H}^{\mathcal{D}}$ . Then $(f _{\gamma} g _{\gamma} h)(x) = (g _{\gamma} f _{\gamma} h)(x)$ for every x where the product is defined. In particular, for every pair of ideals $I \subset J$ with $I, J \in \{\mathcal{H}^{\mathcal{S}}\mathcal{I}^{s\omega}_{\gamma}, \mathcal{H}^{\mathcal{S}}\mathcal{I}^{\omega}_{\gamma}, \mathcal{H}^{\mathcal{D}}\mathcal{I}^{s\omega}_{\gamma}, \mathcal{H}^{\mathcal{D}}\mathcal{I}^{s\omega}_{\gamma}\}$ , I is a left module over $J/[J, J]$ , where $[J, J]$ denotes the commutator ideal.

Lemma 6.1. Let $f \in \mathcal{H}^D \mathcal{I}^{s\omega}_{\gamma, > 1/2}$ be such that $\int_{\gamma} f X^k = 0$ for every $k \in \mathbf{Z}_{\geq 0}$ . Then f(x) = 0 for every x in some neighbourhood of zero. In particular, if $f \in \mathcal{H}^S \mathcal{I}^{s\omega}_{\gamma, > 1/2}$ , then f(x) = 0 in each point x of the strip around $\mathbf{R}$ where f is analytic.

Theorem 6.4. $\mathcal{H}^{S}\mathcal{I}_{\gamma,>c}^{s\omega}$ is a commutative algebra for every $c \in [1/2,1)$ .

Theorem 6.5. Let $g(x) := e_{q^2}(-x^2)$ (so $g \in \mathcal{H}_1^{\mathrm{S}}\mathcal{I}_{\gamma,1/2}^{s\omega}$ by Example 3.2 (a)). Let $f \in \mathcal{H}^{\mathrm{S}}\mathcal{I}_{\gamma,>1/2}^{s\omega}$ be an entire function, not identically zero. Then $(Q^{-n}f) _{\gamma} g \neq g _{\gamma} (Q^{-n}f)$ for n sufficiently large.

Proposition 8.1. Let g be a function defined on the half q-lattice $L^{\epsilon}(\gamma) = \{\epsilon q^k \gamma\}$ for some $\gamma > 0$ and $\epsilon \in \{\pm 1\}$ . Suppose that there exist constants C > 0 and $r > \gamma$ such that (8.1) $$\left| (\partial^k g) (\epsilon q^t \gamma) \right| \le \frac{Cr}{(r - q^t \gamma)} \frac{1}{r^k (1 - q)^k}$$ for every $k \geq 0$ and for every $t \in \mathbf{Z}$ for which $q^t \gamma < r$ . Then the limit $l_p := \lim_{k \to \infty} (\partial^p g) (\epsilon q^k \gamma)$ exists and is finite for every $p \in \mathbf{Z}_{\geq 0}$ , and there exists a unique analytic function $\tilde{g}$ on $\{x \in \mathbf{C} \mid |x| < r\}$ such that $\tilde{g} = g$ on $\{\epsilon q^k \gamma \mid q^k \gamma < r\}$ . If g is defined on the whole q-lattice $L(\gamma)$ and if the condition above is satisfied for every $\epsilon \in \{\pm 1\}$ and if $l_p := \lim_{k \to \infty} (\partial^p g) (\epsilon q^k \gamma)$ is independent of $\epsilon$ , then there exists a unique analytic function $\tilde{g}$ on $\{x \in \mathbf{C} \mid |x| < r\}$ such that $\tilde{g} = g$ on $\{\pm q^k \gamma \mid q^k \gamma < r\}$ . In particular, if g is a function defined on $\mathbf{R}$ such that on every q-lattice $L(\gamma)$ the above conditions are satisfied and $l_p$ is independent of $\gamma$ for every $p \in \mathbf{Z}_{\geq 0}$ then g is analytic.

Definition 1.1. Let $f \in \mathcal{I}_{\gamma}^{\infty}$ and let g be a function defined on some subset of $\mathbf{C}$ . Then the q-convolution product $f *_{\gamma} g$ is the function given by (1.3) $$(f *_{\gamma} g)(x) := \sum_{e=0}^{\infty} \frac{(-1)^{e} \mu_{e,\gamma}(f)}{[e]_{q}!} (\partial^{e} g)(x)$$ for $x \in \mathbb{C}$ such that the q-derivatives $(\partial^e g)(x)$ are well-defined for all $e \in \mathbb{Z}_{\geq 0}$ and the sum on the right converges absolutely, and with notation (1.2) being used. By the asymmetric form of our definition of $f *_{\gamma} g$ , the initial choices of function classes for f and g are quite different: for f the moments $\mu_{e,\gamma}(f)$ should behave as $O(q^{\frac{\alpha e^2}{2}}b^e)$ for $e \to \infty$ with $\alpha, b > 0$ (we call this of left type $\alpha$ ), while for g the q-derivatives $(\partial^k g)(x)$ at x should behave as $O(R^k)$ for $k \to \infty$ with R > 0. We also need functions on $L(\gamma)$ which extend to analytic functions, on a disk centered at 0, or on a strip around R. Some equivalent characterizations of function spaces derived in this paper may have independent interest (see for instance the Appendix). Commutativity of the convolution product is the hardest and most interesting issue of this paper. Both the homomorphism property and the interpretation on the (commutative) braided line suggest commutativity, but we find significant counterexamples. An explanation is that the q-Fourier transform of f only depends on the moments of $\mu_{e,\gamma}(f)$ and that $\mu_{e,\gamma}(f*_{\gamma}g)$ is symmetric in f and g, but that f in certain function classes is not completely determined by its moments. For getting commutativity we need functions f (and g) in the convolution product for which the strict moments $\nu_{e,\gamma}(f)$ behave as $O(q^{\frac{\alpha e^2}{2}}b^e)$ for $e\to\infty$ with $\alpha>1/2$ , b>0 (we call this of strict left type $\alpha>1/2$ ), and which are holomorphic on a strip around $\mathbf{R}$ . It turns out that the q-Gaussian $x \mapsto e_{q^2}(-x^2)$ , which has has strict left type 1/2, does not commute with many entire functions of strict left type > 1/2. Part of the results of this paper occurred in the recent dissertation [4] by the first author. She discusses some further aspects of q-convolution in the subsequent paper [5]. ACKNOWLEDGEMENT. We thank the referee for his careful reading of the manuscript and his constructive remarks. FURTHER NOTATIONS. We denote as usual: $$(1.4) \qquad (a;q)_k := \prod_{j=0}^{k-1} (1 - aq^j), \quad (a;q)_{\infty} := \lim_{k \to \infty} (a;q)_k, [k]_q := \frac{1-q^k}{1-q}, \quad [k]_q! := \frac{(q;q)_k}{(1-q)^k}, \quad \begin{bmatrix} k \\ j \end{bmatrix}_q := \frac{[k]_q!}{[j]_q![k-j]_q!} = \frac{(q;q)_k}{(q;q)_j(q;q)_{k-j}}.$$ A function $f: x \mapsto f(x)$ may also be denoted as f(X). This will be useful for functions like $fX^e: x \mapsto f(x)x^e$ and $e_{q^2}(-X^2): x \mapsto e_{q^2}(-x^2)$ . For q-hypergeometric series the notation of Gasper & Rahman [6] will be followed. The (bounded) q-integral of a function f on $\{\pm \gamma q^k \mid k \in \mathbf{Z}_{\geq 0}\}$ is denoted and defined by (1.5) $$\int_{-\gamma}^{\gamma} f(t)d_q t = (1-q) \sum_{k=0}^{\infty} \sum_{\epsilon=+1} q^k \gamma f(\epsilon q^k \gamma)$$ provided the summation absolutely converges. 2. Motivation of the definition of q-convolution. We will give three different motivations for our Definition 1.1 of q-convolution in the following three remarks. Remark 2.1. The braided line (see [14]) is a braided Hopf algebra $\mathcal A$ which, as an algebra, is equal to the (commutative) algebra $\mathbf C[[x]]$ of formal power series in x, and which has braiding $\Phi(x^k\otimes x^l):=q^{kl}x^l\otimes x^k$ , comultiplication $\Delta(x^k):=\sum_{j=0}^k {k\brack j}_q x^{k-j}\otimes x^j$ , counit $\varepsilon(x^k):=\delta_{k,0}$ and braided antipode $S(x^k):=(-1)^k q^{\frac{k^2-k}{2}}x^k$ . Then the q-analogue of Taylor's formula is given by (2.1) $$\Delta(f(x)) = \sum_{j=0}^{\infty} \frac{x^j}{[j]_q!} \otimes \partial^j(f(x)),$$ where $\partial$ denotes the q-derivative. The original formal definition of a convolution on any braided covector algebra A with a bosonic integral $\int: A \to \mathbf{C}$ invariant under translation, was given by Kempf & Majid [7] as follows: $$(2.2) (f*g)(x) := (\int \otimes \operatorname{id})(m \otimes \operatorname{id})(\operatorname{id} \otimes S \otimes \operatorname{id})(\operatorname{id} \otimes \Delta)(f \otimes g)(x).$$ In the case of the braided line we take for $\int$ the integral defined by (1.1). This integral is invariant under translation (see [7], and [11] and [3] for an analytic proof), but it is not bosonic. Associativity will fail for the convolution defined by (2.2). Therefore, we slightly modify (2.2) into $$(2.3) \ (f *_{\gamma} g)(x) := (\int \otimes \operatorname{id})(m \otimes \operatorname{id})(\operatorname{id} \otimes Q \otimes \operatorname{id})(\operatorname{id} \otimes S \otimes \operatorname{id})(\operatorname{id} \otimes \Delta)(f \otimes g)(x).$$ When we substitute the q-Taylor formula (2.1) into (2.3) then we formally get formula (1.3) (with (1.2) substituted), i.e., our original definition of q-convolution. We may interpret $f *_{\gamma} g$ formally as the action of a pseudo-q-differential operator $\mathcal{D}f(\partial)$ on g(x), where $\mathcal{D}f(\partial) := \mathcal{F}'_{S}(\gamma)(f)(i(1-q)\partial)$ . Here $\mathcal{F}'_{S}(\gamma)(f) := \sum_{e=0}^{\infty} \frac{(-1)^{e}\mu_{e,\gamma}(f)}{[e]_{q}!} \otimes \partial^{e}$ is the weak braided Fourier transform considered in [3] (with $q^{2}$ instead of q). Remark 2.2. The formal limit for $q \uparrow 1$ of formula (1.3) is the classical convolution product: $$\sum_{e=0}^{\infty} \left( \int_{-\infty}^{\infty} dt \, f(t) \frac{(-1)^e t^e}{e!} \right) g^{(e)}(x) \stackrel{\text{formally}}{=} \int_{-\infty}^{\infty} dt \left( f(t) \left( \sum_{e=0}^{\infty} \frac{(-1)^e t^e}{e!} g^{(e)}(x) \right) \right)$$ $$\stackrel{\text{formally}}{=} \int_{-\infty}^{\infty} dt \, f(t) g(x-t) = (f * g)(x).$$ Remark 2.3. Recall the two q-exponentials (see [6]): $$(2.4) e_q(x) := \sum_{k=0}^{\infty} \frac{x^k}{(q;q)_k} = \frac{1}{(x;q)_{\infty}}, E_q(x) := \sum_{k=0}^{\infty} \frac{q^{\frac{k(k-1)}{2}} x^k}{(q;q)_k} = (-x;q)_{\infty},$$ where |x| < 1 in the infinite sum defining $e_q(x)$ . In [11] a q-Fourier transform pair was presented as follows: $$(2.5) \ \psi(x) = \frac{1}{c_q(\gamma)b_q} \int_{-1}^1 e_q(-ixy) \, \phi(y) \, d_q y, \quad \phi(y) = \int_{-\gamma,\infty}^{\gamma,\infty} E_q(iqxy) \, \psi(x) \, d_q x,$$ where $b_q$ and $c_q(\gamma)$ are given by (2.6) $$b_q := \int_{-1}^1 E_{q^2}(-q^2x^2) \, d_q x = (1-q) \, (q, -q, -1; q)_{\infty},$$ $$(2.7) c_q(\gamma) := \int_{-\gamma \infty}^{\gamma \infty} e_{q^2}(-x^2) d_q x = \frac{2(1-q)(q^2, -q\gamma^2, -q\gamma^{-2}; q^2)_{\infty} \gamma}{(-\gamma^2, -q^2/\gamma^2, q; q^2)_{\infty}}.$$ Write the second transform in (2.5) as $\phi = \mathcal{F}_{\gamma} \psi$ . Then an immediate formal computation shows that $$(2.8) \qquad (\mathcal{F}_{\gamma}(f *_{\gamma} g))(x) = (\mathcal{F}_{\gamma} f)(x) (\mathcal{F}_{\gamma} g)(x).$$ The transform $\mathcal{F}_{\gamma}$ is essentially the weak braided Fourier transform $\mathcal{F}'_{S}(\gamma)$ (see Remark 2.1 and reference [3]). Equation (2.8) will be rigorously proved for suitable f and g in Section 7. For later use we recall the formulas (9.8), (9.14) in [11] (for $n \in \mathbb{Z}_{\geq 0}$ ): (2.9) $$\int_{-\gamma \cdot \infty}^{\gamma \cdot \infty} x^n \, e_{q^2}(-x^2) \, d_q x = c_q(\gamma) \, q^{-k^2} \, (q; q^2)_k \quad \text{if } n = 2k \quad (= 0 \text{ otherwise})$$ $$(2.10) \int_{-1 \cdot \infty}^{1 \cdot \infty} x^n E_{q^2}(-x^2) d_q x = b_q q^{2k+1} (q; q^2)_k \quad \text{if } n = 2k \quad (= 0 \text{ otherwise}).$$ 3. Good function spaces for q-convolution. In general, it is not true that $(f_{\gamma}g)(x)$ can be expanded as a (possibly formal) power series if g(x) has a convergent power series expansion. The reason is that the coefficients of an expansion in powers of x of $(f_{\gamma}g)(x)$ will be in general infinite series themselves. We want to find conditions on f and g so that, with the given definition, $(f*_{\gamma}g)(x)$ makes sense on some subset of the complex plane. One should give conditions on the growth of the moments $|\mu_{e,\gamma}(f)|$ of f and of the power series coefficients of g. For this purpose we will now introduce the class of functions of left type $\alpha$ . For convenience, we also give here some variants of this definition and corresponding notation which will be needed later in the paper. Recall the definitions of the function spaces $\mathcal{I}_{\gamma}$ and $\mathcal{I}_{\gamma}^{\infty}$ , given in the Introduction. Definition 3.1. - (a) For $\alpha>0$ the space $\mathcal{I}_{\gamma,\alpha}^{\omega}$ of functions of left type $\alpha$ on $L(\gamma)$ consists of all $f\in\mathcal{I}_{\gamma}^{\infty}$ such that, for some b>0, $|\mu_{e,\gamma}(f)|=O(q^{\frac{\alpha e^2}{2}}b^e)$ as $e\to\infty$ . The space $\mathcal{I}_{\gamma}^{\omega}$ consists of all functions of some left type $\alpha>0$ on $L(\gamma)$ . - (b) For $\alpha>0$ the space $\mathcal{I}_{\gamma,\alpha}^{s\omega}$ of functions of strict left type $\alpha$ on $L(\gamma)$ consists of all $f\in\mathcal{I}_{\gamma}^{\infty}$ such that, for some b>0, $\nu_{e,\gamma}(f)=O(q^{\frac{\alpha e^2}{2}}b^e)$ as $e\to\infty$ . The space $\mathcal{I}_{\gamma}^{s\omega}$ consists of all functions of some strict left type $\alpha>0$ on $L(\gamma)$ . - (c) For a > 0 the space $\mathcal{H}_a^D$ consists of all functions which are holomorphic on the disk $\{z \in \mathbb{C} \mid |z| < a\}$ . By $\mathcal{H}^D$ we denote the space of all functions which are holomorphic on some disk centered at 0. - (d) For a > 0 the space $\mathcal{H}_a^S$ consists of all functions which are holomorphic on the strip $\{z \in \mathbf{C} \mid |\mathrm{Im}(z)| < a\}$ . By $\mathcal{H}^S$ we denote the space of all functions which are holomorphic on some strip around $\mathbf{R}$ . The "intersection" of a $\mathcal{H}$ -space defined in (c) or (d) with one of the $\mathcal{I}$ -spaces (i.e. $\mathcal{I}_{\gamma}$ , $\mathcal{I}_{\gamma}^{\infty}$ or a space defined in (a) or (b)) will be denoted by putting the two symbols $\mathcal{H}$ and $\mathcal{I}$ behind each other. Here we mean intersection in a special sense. For instance, $\mathcal{H}^{D}\mathcal{I}_{\gamma}^{\infty}$ will denote the space of the functions on $L(\gamma)$ belonging to $\mathcal{I}_{\gamma}^{\infty}$ which coincide within some disk centered at 0 with the restriction of a (necessarily unique) holomorphic function on that disk. Note that, conversely, a function $f \in \mathcal{H}^{D}\mathcal{I}_{\gamma}^{\infty}$ is in general not uniquely determined by the function in $\mathcal{H}^{D}$ with which it has a common restriction within some disk. We will always assume for the "intersections" just defined that $\gamma$ is less than the parameter a (radius of a disk or half width of a strip) occurring in $\mathcal{H}_a^{\mathrm{D}}$ or $\mathcal{H}_a^{\mathrm{S}}$ . Clearly, this assumption is not restrictive, since $\gamma$ may be replaced by $q^k \gamma$ for arbitrary $k \in \mathbf{Z}_{\geq 0}$ . Observe that, if $f \in \mathcal{I}_{\gamma}^{\infty}$ is of left type $\alpha$ (respectively strict left type $\alpha$ ) on $L(\gamma)$ then so is $X^k f$ for for each $k \in \mathbf{Z}_{\geq 0}$ . Indeed, $\mu_{e,\gamma}(fX^k) = q^{-\frac{k^2}{2} - \frac{k}{2} - ek} \mu_{k+e,\gamma}(f)$ , and similarly for $\mu$ replaced by $\nu$ . Example 3.2. (a) By (2.9) the q-Gaussian $e_{q^2}(-X^2)$ belongs to $\mathcal{I}_{\gamma}^{\infty}$ for each $\gamma > 0$ and $$\mu_{2k,\gamma}(e_{q^2}(-X^2)) = c_q(\gamma) (q;q^2)_k q^{k^2+k}, \quad \mu_{2k+1,\gamma}(e_{q^2}(-X^2)) = 0.$$ Hence $e_{q^2}(-X^2)$ is of left type 1/2 on $L(\gamma)$ . It is also of strict left type 1/2 on $L(\gamma)$ . Indeed, if $f \in \mathcal{I}_{\gamma}^{\infty}$ is an even and nonnegative function then $\nu_{2k,\gamma}(f) = \mu_{2k,\gamma}(f)$ and $\nu_{2k+1,\gamma}(f) \leq \mu_{2k+2,\gamma}(f) + \mu_{2k,\gamma}(f)$ . (b) By (2.10) the q-Gaussian $E_{q^2}(-X^2)$ belongs to $\mathcal{I}_1^{\infty}$ and $$\mu_{2k,1}(E_{q^2}(-X^2)) = b_q \, (q;q^2)_k \, q^{2k^2+k}, \quad \mu_{2k+1,1}(E_{q^2}(-X^2)) = 0.$$ Hence $E_{q^2}(-X^2)$ is of left type 1 on L(1). It is also of strict left type 1 on L(1) by a similar argument as in (a). Remark 3.3. Let $g \in \mathcal{H}^{\mathbb{D}}_R$ with $g(x) = \sum_l c_l x^l$ and put $G(r) := \sum_l |c_l| r^l$ (we will repeatedly use this convention). Note that then also $\partial^k g \in \mathcal{H}^{\mathbb{D}}_R$ for every $k \in \mathbb{Z}_{\geq 0}$ . Since $|c_l| \leq \left(\sum_p |c_p| r^p\right) r^{-l}$ for every r < R and since $(q^l; q^{-1})_k \leq 1$ for $l \geq k$ we can also say that, for every |x| < r < R, $$|(\partial^k g)(x)| \le \frac{r}{r - |x|} \frac{1}{r^k (1 - q)^k} G(r).$$ It will be shown in the Appendix that condition (3.1) together with equality of left and right q-derivatives at 0 is equivalent to analyticity.

Definition 5.10. For every $c \in [0,1)$ we denote by $\mathcal{H}^{S}\mathcal{I}_{\gamma,>c}^{s\omega}$ the union of all spaces $\mathcal{H}^{S}\mathcal{I}_{\gamma,\alpha}^{s\omega}$ with $\alpha \in (c,1]$ .