  We introduce a family of differential-reflection operators $\Lambda_{A,
\varepsilon}$ acting on smooth functions defined on $\mathbb R.$ Here $A$ is a
Strum-Liouville function with additional hypotheses and $\varepsilon\in \mathbb
R.$ For special pairs $(A,\varepsilon),$ we recover Dunkl's, Heckman's and
Cherednik's operators (in one dimension). The spectral problem for the
operators $\Lambda_{A, \varepsilon}$ is studied. In particular, we obtain
suitable growth estimates for the eigenfunctions of $\Lambda_{A, \varepsilon}$.
  As the operators $\Lambda_{A, \varepsilon}$ are mixture of $d/dx$ and
reflection operators, we prove the existence of an intertwining operator
$V_{A,\varepsilon}$ between $\Lambda_{A, \varepsilon}$ and the usual
derivative. The positivity of $V_{A,\varepsilon}$ is also established.
  Via the eigenfunctions of $\Lambda_{A,\varepsilon},$ we introduce a
generalized Fourier transform $\mathcal F_{A,\varepsilon}.$ An $L^p$-harmonic
analysis for $\mathcal F_{A,\varepsilon}$ 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.
