We establish necessary and sufficient Boas-type criteria for generalized Lipschitz classes associated with the generalized Dunkl transform on the real line. The criteria are expressed through the growth of weighted spectral moments of \(\mathcal{F}_{D}(f)\) as the spectral radius tends to infinity. A symmetric generalized difference generated by the dual translation operators is used; its transform multiplier is \((1-j_{\alpha}(th))^{n}\), which gives the correct order \(2n\) at the origin and the required lower control away from the origin. Direct theorems are proved for the classes \(B_{\gamma}^{n}\) and \(b_{\gamma}^{n}\), while converse theorems are obtained under the standard one-sign condition on the transform. These results identify the precise connection between decay of the generalized Dunkl transform and \(Q\)-weighted smoothness measured by generalized translations.
Integral transforms and their inversion formulae are central tools in harmonic analysis, approximation theory, mathematical physics and the study of differential and integro-differential equations. Classical examples include the Fourier, Bessel, Hankel and Fourier–Bessel transforms, whose spectral representations convert smoothness and approximation properties of functions into quantitative estimates on transform coefficients or transform densities [1–6].
Boas-type theorems form a classical branch of this theory. In the trigonometric setting, they characterize Lipschitz regularity by the asymptotic size of weighted sums of Fourier coefficients. The original result of Boas for Fourier series with non-negative coefficients [7] was followed by refinements involving generalized moduli of smoothness and higher-order Lipschitz classes [8–10]. Related criteria have subsequently been developed for transforms and expansions adapted to non-Euclidean or weighted harmonic analysis, including Bessel, \(q\)-Bessel and Dunkl-type settings [11–14].
The generalized Dunkl transform considered here is associated with the first-order singular differential-difference operator \[D_{\alpha,q}f(x)=f'(x)+\left(\alpha+\frac12\right)\frac{f(x)-f(-x)}{x}+q(x)f(x), \qquad \alpha>-\frac12,\] where \(q\) is a real-valued odd \(C^{\infty}\) function. This operator contains the one-dimensional Dunkl operator as the special case \(q=0\), but the presence of \(q\) changes the natural measure of smoothness through the weight \[Q(x)=\exp\left(-\int_{0}^{x}q(t)\,dt\right).\]
The novelty of the present treatment lies in formulating Boas-type criteria for this generalized transform by using a symmetric difference adapted to the dual translation structure. This choice produces the multiplier \(1-j_{\alpha}(th)\) on the transform side, and therefore matches the \(2n\)-order spectral moment naturally associated with the \(n\)th generalized difference.
The main results show that, for \(0<\gamma<2n\), the estimate \[\int_{|t|\le a}|t|^{2n}|\mathcal{F}_{D}(f)(t)|\,d\mu_{\alpha}(t)=O(a^{2n-\gamma}),\] implies membership of \(f\) in the generalized Lipschitz class \(B_{\gamma}^{n}\); the corresponding little-oh estimate implies membership in \(b_{\gamma}^{n}\). Conversely, if \(\mathcal{F}_{D}(f)\) has one sign, membership in these classes yields the corresponding spectral moment estimates. Thus, within the stated hypotheses, the spectral growth of \(\mathcal{F}_{D}(f)\) and the generalized translation smoothness of \(f\) determine each other.
Let \(\alpha>-1/2\) and let \(q\) be a real-valued odd function of class \(C^{\infty}\) on \(\mathbb{R}\). Define \[\tag{1}\label{eq:Dalphaq} D_{\alpha,q}f(x)=f'(x)+\left(\alpha+\frac12\right)\frac{f(x)-f(-x)}{x}+q(x)f(x).\]
When \(q=0\), \(D_{\alpha,q}\) reduces to the classical one-dimensional Dunkl operator \[\tag{2}\label{eq:Dalpha} D_{\alpha}f(x)=f'(x)+\left(\alpha+\frac12\right)\frac{f(x)-f(-x)}{x}.\]
Set \[\tag{3}\label{eq:Qdef} Q(x)=\exp\left(-\int_{0}^{x}q(t)\,dt\right).\]
Since \(q\) is odd, \(Q\) is positive and even, and \(Q(0)=1\).
Let \[\,d\mu_{\alpha}(x)=|x|^{2\alpha+1}\,dx.\]
For \(1\le p<\infty\), define \[L_{\alpha}^{p}(\mathbb{R})=\left\{f:\mathbb{R}\to\mathbb{C}\;\hbox{measurable}:\; \|f\|_{p,\alpha}=\left(\int_{\mathbb{R}}|f(x)|^{p}\,d\mu_{\alpha}(x)\right)^{1/p}<\infty\right\},\] and \[L_{Q}^{p}(\mathbb{R})=\left\{f:\mathbb{R}\to\mathbb{C}\;\hbox{measurable}:\; Qf\in L_{\alpha}^{p}(\mathbb{R})\right\},\qquad \|f\|_{p,Q}=\|Qf\|_{p,\alpha}.\]
For each \(\lambda\in\mathbb{C}\), the equation \[D_{\alpha,q}u=i\lambda u,\qquad u(0)=1,\] has the unique solution \[\tag{4}\label{eq:psi} \psi_{\lambda}(x)=Q(x)e_{\alpha}(i\lambda x),\] where \(e_{\alpha}\) is the one-dimensional Dunkl kernel \[\tag{5}\label{eq:ealpha} e_{\alpha}(z)=j_{\alpha}(iz)+\frac{z}{2(\alpha+1)}j_{\alpha+1}(iz), \qquad z\in\mathbb{C},\] and \[\tag{6}\label{eq:jalpha} j_{\alpha}(z)=\Gamma(\alpha+1)\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!\,\Gamma(k+\alpha+1)}\left(\frac{z}{2}\right)^{2k}, \qquad z\in\mathbb{C}.\]
Moreover, for every \(x\in\mathbb{R}\), \(\lambda\in\mathbb{C}\) and \(k\in\mathbb{N}\), \[\tag{7}\label{eq:psiderivative} \left|\frac{\partial^{k}}{\partial\lambda^{k}}\psi_{\lambda}(x)\right| \le Q(x)|x|^{k}e^{|\operatorname{Im}\lambda|\,|x|}.\]
The following elementary estimates for the normalized Bessel function will be used repeatedly. For \(x\in\mathbb{R}\), \[\tag{8}\label{eq:jbound} |j_{\alpha}(x)|\le 1,\] and there exists a constant \(c_{\alpha}>0\) such that \[\tag{9}\label{eq:jloweraway} 1-j_{\alpha}(x)\ge c_{\alpha},\qquad |x|\ge 1.\]
In particular, \(j_{\alpha}\) is real-valued on \(\mathbb{R}\) and \(j_{\alpha}(x)\le 1\).
Definition 1. For \(f\in L_{Q}^{1}(\mathbb{R})\), the generalized Dunkl transform of \(f\) is defined by \[\tag{10}\label{eq:FDdef} \mathcal{F}_{D}(f)(\lambda)=\int_{\mathbb{R}}f(x)\psi_{-\lambda}(x)\,d\mu_{\alpha}(x),\qquad \lambda\in\mathbb{R}.\]
The Plancherel and inversion formulae take the following form. For \[\tag{11}\label{eq:malpha} m_{\alpha}=\frac{1}{2^{2\alpha+2}\Gamma(\alpha+1)^{2}},\] one has, initially on \(L_{Q}^{1}(\mathbb{R})\cap L_{Q}^{2}(\mathbb{R})\) and then by density, \[\tag{12}\label{eq:plancherel} \int_{\mathbb{R}}|f(x)|^{2}Q(x)^{2}\,d\mu_{\alpha}(x)=m_{\alpha}\int_{\mathbb{R}}|\mathcal{F}_{D}(f)(\lambda)|^{2}\,d\mu_{\alpha}(\lambda).\]
Thus \(m_{\alpha}^{1/2}\mathcal{F}_{D}\) is an isometric isomorphism from \(L_{Q}^{2}(\mathbb{R})\) onto \(L_{\alpha}^{2}(\mathbb{R})\). If \(f\in L_{Q}^{1}(\mathbb{R})\) and \(\mathcal{F}_{D}(f)\in L_{\alpha}^{1}(\mathbb{R})\), then \[\tag{13}\label{eq:inversion} Q(x)^{2}f(x)=m_{\alpha}\int_{\mathbb{R}}\mathcal{F}_{D}(f)(t)\psi_{t}(x)\,d\mu_{\alpha}(t),\qquad x\in\mathbb{R}.\]
Equivalently, this is the inversion formula for the ordinary Dunkl transform of the function \(Qf\).
For \(h\in\mathbb{R}\), the generalized dual translation operator is \[\tag{14}\label{eq:translation} T_{h}f(x)=\frac{Q(h)}{Q(x)}\tau_{-h}^{\alpha}(Qf)(x),\] where \(\tau_{h}^{\alpha}\) denotes the classical Dunkl translation. The latter can be represented by \[\tau_{-h}^{\alpha}g(x)=\int_{\mathbb{R}}g(z)\,d\mu_{h,x}^{\alpha}(z),\] where \(\mu_{h,x}^{\alpha}\) is a finite signed measure of total mass one, supported in \[[-|h|-|x|,-||h|-|x||]\cup[||h|-|x||,|h|+|x|],\] and satisfying \(|\mu_{h,x}^{\alpha}|\le 2\). The translation and the generalized transform are connected by \[\tag{15}\label{eq:translationtransform} \mathcal{F}_{D}(T_{h}f)(t)=\psi_{-t}(h)\mathcal{F}_{D}(f)(t).\]
Since \(Q\) is even and \[\frac{e_{\alpha}(-ith)+e_{\alpha}(ith)}{2}=j_{\alpha}(th),\] the symmetric averaging operator \[\tag{16}\label{eq:Ah} A_{h}f=\frac{T_{h}f+T_{-h}f}{2Q(h)},\] satisfies \[\tag{17}\label{eq:Ahtransform} \mathcal{F}_{D}(A_{h}f)(t)=j_{\alpha}(th)\mathcal{F}_{D}(f)(t).\]
The proofs of the Boas-type criteria use the following two-sided moment-tail estimates. They are stated for the measure \(\,d\mu_{\alpha}(t)=|t|^{2\alpha+1}\,dt\).
Lemma 1. Let \(H\) be a non-negative measurable function on \(\mathbb{R}\), let \(N>0\) and let \(0<\gamma<N\). If \[\tag{18}\label{eq:momentO} \int_{|t|\le a}|t|^{N}H(t)\,d\mu_{\alpha}(t)=O(a^{N-\gamma}),\qquad a\to\infty,\] then \[\tag{19}\label{eq:tailO} \int_{|t|>a}H(t)\,d\mu_{\alpha}(t)=O(a^{-\gamma}),\qquad a\to\infty.\]
Conversely, if \(H\in L_{\mathrm{loc}}^{1}(\mathbb{R},\,d\mu_{\alpha})\) and (19) holds, then (18) holds.
Proof. Assume (18). For large \(a\), \[\begin{aligned} \int_{|t|>a}H(t)\,d\mu_{\alpha}(t) &\le \sum_{k=0}^{\infty}\int_{2^{k}a<|t|\le 2^{k+1}a}H(t)\,d\mu_{\alpha}(t)\\ &\le \sum_{k=0}^{\infty}(2^{k}a)^{-N}\int_{|t|\le 2^{k+1}a}|t|^{N}H(t)\,d\mu_{\alpha}(t)\\ &\le C a^{-\gamma}\sum_{k=0}^{\infty}2^{-k\gamma}=O(a^{-\gamma}). \end{aligned}\]
Conversely, for large \(a\) the layer-cake representation gives \[\int_{|t|\le a}|t|^{N}H(t)\,d\mu_{\alpha}(t) \le C+N\int_{1}^{a}r^{N-1}\int_{|t|>r}H(t)\,d\mu_{\alpha}(t)\,dr.\]
Using (19), the last integral is \(O(a^{N-\gamma})\) because \(0<\gamma<N\). ◻
Lemma 2. Let \(H\) be a non-negative measurable function on \(\mathbb{R}\), let \(N>0\) and let \(0<\gamma<N\). If \[\tag{20}\label{eq:momento} \int_{|t|\le a}|t|^{N}H(t)\,d\mu_{\alpha}(t)=o(a^{N-\gamma}),\qquad a\to\infty,\] then \[\tag{21}\label{eq:tailo} \int_{|t|>a}H(t)\,d\mu_{\alpha}(t)=o(a^{-\gamma}),\qquad a\to\infty.\]
Conversely, if \(H\in L_{\mathrm{loc}}^{1}(\mathbb{R},\,d\mu_{\alpha})\) and (21) holds, then (20) holds.
Proof. The proof follows the preceding argument with the standard \(\varepsilon\)-splitting of the dyadic sums and of the layer-cake integral. The condition \(0<\gamma<N\) guarantees convergence of the dyadic remainder and yields the little-oh estimates. ◻
Lemma 3. There exist constants \(c_{\alpha},C_{\alpha}>0\) such that, for every \(x\in\mathbb{R}\), \[\tag{22}\label{eq:besselcomparison} c_{\alpha}\min\{1,x^{2}\}\le 1-j_{\alpha}(x)\le C_{\alpha}\min\{1,x^{2}\}.\]
Consequently, for every \(n\in\mathbb{N}^{*}\), \[\tag{23}\label{eq:besselcomparisonn} (1-j_{\alpha}(x))^{n}\le C_{\alpha,n}\min\{1,|x|^{2n}\}.\]
Proof. The series expansion (6) gives \[j_{\alpha}(x)=1-\frac{x^{2}}{4(\alpha+1)}+O(x^{4}),\qquad x\to0.\]
Hence \(1-j_{\alpha}(x)\) is comparable with \(x^{2}\) for \(|x|\le1\). For \(|x|\ge1\), the lower estimate follows from (9), while the upper estimate follows from (8). Combining the two ranges proves (22); raising the upper estimate to the \(n\)th power gives (23). ◻
For \(n\in\mathbb{N}^{*}\) and \(h\in\mathbb{R}\), define the generalized symmetric difference of order \(n\) by \[\tag{24}\label{eq:deltadef} \delta_{h}^{n}f=(I-A_{h})^{n}f,\] where \(A_{h}\) is given by (16). From (17), \[\tag{25}\label{eq:deltatransform} \mathcal{F}_{D}(\delta_{h}^{n}f)(t)=(1-j_{\alpha}(th))^{n}\mathcal{F}_{D}(f)(t).\]
This normalization is essential: the multiplier vanishes at \(h=0\), has order \(2n\) near the origin, and remains bounded away from zero on spectral regions where \(|th|\ge1\).
For a measurable function \(g\), set \[\tag{26}\label{eq:weightednorm} \|g\|_{\infty,Q}=\sup_{x\in\mathbb{R}}Q(x)|g(x)|.\]
Let \(0<\gamma<2n\). A function \(f\) belongs to \(B_{\gamma}^{n}\) if \[\tag{27}\label{eq:Bclass} \|\delta_{h}^{n}f\|_{\infty,Q}=O(h^{\gamma}),\qquad h\to0^{+},\] and belongs to \(b_{\gamma}^{n}\) if \[\tag{28}\label{eq:bclass} \|\delta_{h}^{n}f\|_{\infty,Q}=o(h^{\gamma}),\qquad h\to0^{+}.\]
Theorem 1. Let \(n\in\mathbb{N}^{*}\) and \(0<\gamma<2n\). Suppose that \(f\in L_{Q}^{1}(\mathbb{R})\cap C(\mathbb{R})\) and \[\tag{29}\label{eq:mainO} \int_{|t|\le a}|t|^{2n}|\mathcal{F}_{D}(f)(t)|\,d\mu_{\alpha}(t)=O(a^{2n-\gamma}),\qquad a\to\infty.\]
Then \(\mathcal{F}_{D}(f)\in L_{\alpha}^{1}(\mathbb{R})\) and \(f\in B_{\gamma}^{n}\).
Proof. Apply Lemma 1 with \(H(t)=|\mathcal{F}_{D}(f)(t)|\) and \(N=2n\). Since \(\mathcal{F}_{D}(f)\) is bounded near the origin for \(f\in L_{Q}^{1}(\mathbb{R})\), condition (29) implies \[\tag{30}\label{eq:tailmainO} \int_{|t|>a}|\mathcal{F}_{D}(f)(t)|\,d\mu_{\alpha}(t)=O(a^{-\gamma}),\qquad a\to\infty,\] and therefore \(\mathcal{F}_{D}(f)\in L_{\alpha}^{1}(\mathbb{R})\).
Using (13), (25) and \(|\psi_{t}(x)|\le Q(x)\) for \(t,x\in\mathbb{R}\), one obtains \[\tag{31}\label{eq:basicbound} Q(x)|\delta_{h}^{n}f(x)| \le m_{\alpha}\int_{\mathbb{R}}(1-j_{\alpha}(th))^{n}|\mathcal{F}_{D}(f)(t)|\,d\mu_{\alpha}(t).\]
Split the integral into \(|t|\le h^{-1}\) and \(|t|>h^{-1}\). By Lemma 3, \[\begin{aligned} \int_{|t|\le h^{-1}}(1-j_{\alpha}(th))^{n}|\mathcal{F}_{D}(f)(t)|\,d\mu_{\alpha}(t) &\le C h^{2n}\int_{|t|\le h^{-1}}|t|^{2n}|\mathcal{F}_{D}(f)(t)|\,d\mu_{\alpha}(t)\\ &=O(h^{\gamma}). \end{aligned}\]
For the second part, \[\int_{|t|>h^{-1}}(1-j_{\alpha}(th))^{n}|\mathcal{F}_{D}(f)(t)|\,d\mu_{\alpha}(t) \le C\int_{|t|>h^{-1}}|\mathcal{F}_{D}(f)(t)|\,d\mu_{\alpha}(t)=O(h^{\gamma}),\] by (30). Taking the supremum over \(x\) in (31) gives (27). ◻
Theorem 1 shows that the growth of the \(2n\)th weighted spectral moment controls the \(n\)th generalized translation smoothness of \(f\). The exponent \(2n-\gamma\) is dictated by the quadratic behavior of \(1-j_{\alpha}\) at the origin and by the \(n\)th power in (25).
Example 1. Consider the classical Dunkl case \(q=0\), so \(Q\equiv1\). Let \[f_{c}(x)=e^{-cx^{2}},\qquad c>0.\]
The Dunkl transform of \(f_{c}\) has the form \[\mathcal{F}_{D}(f_{c})(t)=C_{\alpha,c}e^{-t^{2}/(4c)},\qquad C_{\alpha,c}>0.\]
Hence, for every \(n\in\mathbb{N}^{*}\), \[\int_{|t|\le a}|t|^{2n}|\mathcal{F}_{D}(f_{c})(t)|\,d\mu_{\alpha}(t) \le C\int_{\mathbb{R}}|t|^{2n+2\alpha+1}e^{-t^{2}/(4c)}\,dt<\infty.\]
Thus (29) holds for each \(0<\gamma<2n\), and Theorem 1 yields \(f_{c}\in B_{\gamma}^{n}\).
Theorem 2. Let \(n\in\mathbb{N}^{*}\) and \(0<\gamma<2n\). Suppose that \(f\in L_{Q}^{1}(\mathbb{R})\cap L_{Q}^{2}(\mathbb{R})\cap C(\mathbb{R})\), \(\mathcal{F}_{D}(f)\in L_{\alpha}^{1}(\mathbb{R})\), and \(\mathcal{F}_{D}(f)\) does not change sign on \(\mathbb{R}\). If \(f\in B_{\gamma}^{n}\), then (29) holds.
Proof. It is enough to treat the case \(\mathcal{F}_{D}(f)\ge0\); the case \(\mathcal{F}_{D}(f)\le0\) follows by replacing \(\mathcal{F}_{D}(f)\) with \(-\mathcal{F}_{D}(f)\). For \(|t|\le h^{-1}\), Lemma 3 gives \[|t|^{2n}\le C h^{-2n}(1-j_{\alpha}(th))^{n}.\]
Consequently, \[\int_{|t|\le h^{-1}}|t|^{2n}\mathcal{F}_{D}(f)(t)\,d\mu_{\alpha}(t) \le C h^{-2n}\int_{\mathbb{R}}(1-j_{\alpha}(th))^{n}\mathcal{F}_{D}(f)(t)\,d\mu_{\alpha}(t).\]
Using (13) and (25) at \(x=0\), together with \(Q(0)=1\) and \(\psi_{t}(0)=1\), yields \[\int_{\mathbb{R}}(1-j_{\alpha}(th))^{n}\mathcal{F}_{D}(f)(t)\,d\mu_{\alpha}(t)=m_{\alpha}^{-1}\delta_{h}^{n}f(0).\]
Therefore, \[\int_{|t|\le h^{-1}}|t|^{2n}\mathcal{F}_{D}(f)(t)\,d\mu_{\alpha}(t) \le C h^{-2n}\|\delta_{h}^{n}f\|_{\infty,Q}=O(h^{\gamma-2n}).\]
Putting \(a=h^{-1}\) gives (29). ◻
The sign hypothesis in Theorem 2 is the mechanism that converts the value of the inverse transform at the origin into a positive spectral integral. Under this hypothesis, Theorems 1 and 2 give a two-sided characterization of \(B_{\gamma}^{n}\) by the growth of the moment in (29).
Example 2. In the case \(q=0\), the Gaussian \(f_{c}(x)=e^{-cx^{2}}\) satisfies \(\mathcal{F}_{D}(f_{c})(t)=C_{\alpha,c}e^{-t^{2}/(4c)}>0\) for all \(t\in\mathbb{R}\). It is continuous and belongs to \(L_{\alpha}^{1}(\mathbb{R})\cap L_{\alpha}^{2}(\mathbb{R})\). The preceding example gives \(f_{c}\in B_{\gamma}^{n}\) for \(0<\gamma<2n\), and Theorem 2 recovers the moment estimate (29).
Theorem 3. Let \(n\in\mathbb{N}^{*}\) and \(0<\gamma<2n\). Suppose that \(f\in L_{Q}^{1}(\mathbb{R})\cap C(\mathbb{R})\) and \[\tag{32}\label{eq:mainlittle} \int_{|t|\le a}|t|^{2n}|\mathcal{F}_{D}(f)(t)|\,d\mu_{\alpha}(t)=o(a^{2n-\gamma}),\qquad a\to\infty.\] Then \(\mathcal{F}_{D}(f)\in L_{\alpha}^{1}(\mathbb{R})\) and \(f\in b_{\gamma}^{n}\).
Proof. By Lemma 2, \[\int_{|t|>a}|\mathcal{F}_{D}(f)(t)|\,d\mu_{\alpha}(t)=o(a^{-\gamma}),\qquad a\to\infty.\]
The same decomposition used in the proof of Theorem 1 gives \[\begin{aligned} \int_{|t|\le h^{-1}}(1-j_{\alpha}(th))^{n}|\mathcal{F}_{D}(f)(t)|\,d\mu_{\alpha}(t)&=o(h^{\gamma}),\\ \int_{|t|>h^{-1}}(1-j_{\alpha}(th))^{n}|\mathcal{F}_{D}(f)(t)|\,d\mu_{\alpha}(t)&=o(h^{\gamma}). \end{aligned}\]
Together with (31), these estimates imply \[\|\delta_{h}^{n}f\|_{\infty,Q}=o(h^{\gamma}),\qquad h\to0^{+},\] which is precisely \(f\in b_{\gamma}^{n}\). ◻
The little-oh result distinguishes functions whose spectral moment grows strictly more slowly than \(a^{2n-\gamma}\). In terms of generalized differences, this sharper spectral decay is exactly reflected by the smaller smoothness scale \(b_{\gamma}^{n}\).
Example 3. Again take \(q=0\). If \(f\in\mathcal{S}(\mathbb{R})\), then the classical Dunkl transform also belongs to \(\mathcal{S}(\mathbb{R})\). Hence, for every \(n\in\mathbb{N}^{*}\), \[\int_{|t|\le a}|t|^{2n}|\mathcal{F}_{D}(f)(t)|\,d\mu_{\alpha}(t)=O(1),\qquad a\to\infty.\]
Since \(2n-\gamma>0\), this boundedness implies (32). Therefore Theorem 3 gives \(f\in b_{\gamma}^{n}\) for every \(0<\gamma<2n\).
Theorem 4. Let \(n\in\mathbb{N}^{*}\) and \(0<\gamma<2n\). Suppose that \(f\in L_{Q}^{1}(\mathbb{R})\cap L_{Q}^{2}(\mathbb{R})\cap C(\mathbb{R})\), \(\mathcal{F}_{D}(f)\in L_{\alpha}^{1}(\mathbb{R})\), and \(\mathcal{F}_{D}(f)\) does not change sign on \(\mathbb{R}\). If \(f\in b_{\gamma}^{n}\), then (32) holds.
Proof. Assume first that \(\mathcal{F}_{D}(f)\ge0\). As in the proof of Theorem 2, \[\int_{|t|\le h^{-1}}|t|^{2n}\mathcal{F}_{D}(f)(t)\,d\mu_{\alpha}(t) \le C h^{-2n}\int_{\mathbb{R}}(1-j_{\alpha}(th))^{n}\mathcal{F}_{D}(f)(t)\,d\mu_{\alpha}(t).\]
The inverse formula at the origin gives \[\int_{\mathbb{R}}(1-j_{\alpha}(th))^{n}\mathcal{F}_{D}(f)(t)\,d\mu_{\alpha}(t)=m_{\alpha}^{-1}\delta_{h}^{n}f(0).\]
Since \(f\in b_{\gamma}^{n}\), \[\int_{|t|\le h^{-1}}|t|^{2n}\mathcal{F}_{D}(f)(t)\,d\mu_{\alpha}(t)=o(h^{\gamma-2n}).\]
With \(a=h^{-1}\) this is (32). If \(\mathcal{F}_{D}(f)\le0\), the same argument is applied to \(-\mathcal{F}_{D}(f)\). ◻
Together, Theorems 3 and 4 provide the little-oh counterpart of the preceding characterization. The distinction between \(B_{\gamma}^{n}\) and \(b_{\gamma}^{n}\) is therefore encoded by the distinction between big-oh and little-oh growth of the same weighted spectral moment.
The generalized Dunkl transform associated with \(D_{\alpha,q}\) admits Boas-type smoothness criteria governed by weighted moments of \(\mathcal{F}_{D}(f)\). The symmetric difference \((I-A_{h})^{n}\) is the natural translation difference for this purpose because its transform multiplier is \((1-j_{\alpha}(th))^{n}\). This multiplier gives the exact order \(2n\) near zero and provides the spectral localization required for the direct and inverse estimates.
For \(0<\gamma<2n\), the moment condition (29) implies membership in \(B_{\gamma}^{n}\), while the sharper condition (32) implies membership in \(b_{\gamma}^{n}\). Under the one-sign condition on \(\mathcal{F}_{D}(f)\), these implications reverse. The results therefore characterize \(Q\)-weighted generalized Lipschitz smoothness through the asymptotic behavior of the generalized Dunkl transform and clarify the role of the deformation \(q\): it changes the weight and translations through \(Q\), while the decisive smoothness exponent is determined by the Bessel factor \(j_{\alpha}\).
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