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We introduce a family of differential-reflection operators $\Lambda_{A,\varepsilon}$ acting on smooth functions defined on $\mathbb R.$ Here $A$ is aStrum-Liouville function with additional hypotheses and $\varepsilon\in \mathbbR.$ For special pairs $(A,\varepsilon),$ we recover Dunkl's, Heckman's andCherednik's operators (in one dimension). The spectral problem for theoperators $\Lambda_{A, \varepsilon}$ is studied. In particular, we obtainsuitable growth estimates for the eigenfunctions of $\Lambda_{A, \varepsilon}$.  As the operators $\Lambda_{A, \varepsilon}$ are mixture of $d/dx$ andreflection operators, we prove the existence of an intertwining operator$V_{A,\varepsilon}$ between $\Lambda_{A, \varepsilon}$ and the usualderivative. The positivity of $V_{A,\varepsilon}$ is also established.  Via the eigenfunctions of $\Lambda_{A,\varepsilon},$ we introduce ageneralized Fourier transform $\mathcal F_{A,\varepsilon}.$ An $L^p$-harmonicanalysis for $\mathcal F_{A,\varepsilon}$ is developed when $0<p\leq{2\over{1+\sqrt{1-\varepsilon^2}}}$ and $-1\leq \varepsilon\leq 1.$ Inparticular, an $L^p$-Schwartz space isomorphism theorem for $\mathcalF_{A,\varepsilon}$ is proved.

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