On a family of differential-reflection operators: intertwining operators and Fourier transform of rapidly decreasing functions

We introduce a family of differential-reflection operators $\Lambda_{A, \varepsilon}$actingonsmoothfunctionsdefinedon$\mathbb R.$Here$A$ is a Strum-Liouville function with additional hypotheses and $\varepsilon\in \mathbb R.$Forspecialpairs$(A,\varepsilon),$ we recover Dunkl’s, Heckman’s and Cherednik’s operators (in one dimension). The spectral problem for the operators ${\mathrm{\Lambda }}_{A,\epsilon }$ is studied. In particular, we obtain suitable growth estimates for the eigenfunctions of ${\mathrm{\Lambda }}_{A,\epsilon }$. As the operators ${\mathrm{\Lambda }}_{A,\epsilon }$ are mixture of $d\mathrm{/}dx$ and reflection operators, we prove the existence of an intertwining operator $V_{A,\epsilon }$ between ${\mathrm{\Lambda }}_{A,\epsilon }$ and the usual derivative. The positivity of $V_{A,\epsilon }$ is also established. Via the eigenfunctions of ${\mathrm{\Lambda }}_{A,\epsilon },$ we introduce a generalized Fourier transform ${\mathcal{F}}_{A,\epsilon }\mathrm{.}$ An $L^p$-harmonic analysis for ${\mathcal{F}}_{A,\epsilon }$ is developed when $0<p\leq {2\over{1+\sqrt{1-\varepsilon^2}}}$and$-1\leq \varepsilon\leq 1.$ In particular, an $L^p$-Schwartz space isomorphism theorem for $\mathcal F_{A,\varepsilon}$ is proved.