Beurling's theorem for the Clifford-Fourier transform
We give a generalization of Beurling's theorem for the Clifford-Fourier transform. Then, analogues of Hardy, Cowling-Price and Gelfand-Shilov theorems are obtained in Clifford analysis.
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arXiv subjects
Jamel El Kamel
Publications and source records attributed to Jamel El Kamel.
Uncertainty principles for the Clifford-Fourier transform
In this paper, we estabish an analogue of Hardy's theorem and Miyachi's theorem for the Clifford-Fourier transform.
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The uncertainty principle in Clifford analysis
In this paper, we provide the Heisenberg's inequality and the Hardy's theorem for the Clifford-Fourier transform on $\mathbb{R}^m$.
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Dunkl completely monotonic functions
We introduce the notion of Dunkl completely monotonic functions on $\left(-σ,σ\right), σ>0$. We establish a restrictive version of the analogue of Schoenberg's theorem in Dunkl setting.
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Turán Type Inequalities for Dunkl Kernel and $q$-Dunkl Kernel
We prove turán type inequalities for Dunkl kernel. We provide a $q$-integral representation for the $q$-Dunkl kernel. Using a $q$-version of Schwartz inequality, we get a turán type inequalities for $q$-Dunkl kernel.
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Dunkl positive definite functions
We introduce the notion of Dunkl positive definite and strictly positive definite functions on $\mathbb{R}^{d}$. This done by the use of the properties of Dunkl translation. We establish the analogue of Bochner's theorem in Dunkl setting. The case of radial functions is considered. We give a sufficient condition for a function to be Dunkl strictly positive definite on $\mathbb{R}^{d}.$
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A function Class of strictly positive definite and logarithmically completely monotonic functions related to the modified Bessel functions
In this paper we give some conditions for a class of functions related to Bessel functions to be positive definite or strictly positive definite . We present some properties and relationships involving logarithmically completely monotonic functions and strictly positive definite functions. In particular, we are interested with the modified Bessel functions.
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Almost Everywhere Convergence of Inverse Dunkl Transform on the Real Line
In this paper, we will first show that the maximal operator $S_*^α$ of spherical partial sums $S_R^α$, associated to Dunkl transform on $\mathbb{R}$ is bounded on $L^p(\mathbb{R}, |x|^{2α+1} dx)$ functions when $\frac{4(α+1)}{2α+3}<p<\frac{4(α+1)}{2α+1}$, and it implies that, for every $L^p(\mathbb{R}, |x|^{2α+1} dx)$ function $f(x)$, $S_R^αf(x)$ converges to $f(x)$ almost everywhere as $R\to \infty$. On the other hand we obtain a sharp version by showing that $S_*^α$ is bounded from the Lorentz space $L^{p_i,1}(\mathbb{R}, |x|^{2α+1})$ into $L^{p_i,\infty}(\mathbb{R}, |x|^{2α+1}),\quad i=0,1$ where $p_0=\frac{4(α+1)}{2α+3}$ and $p_1=\frac{4(α+1)}{2α+1}$.
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