We introduce and study differential-reflection operators ΛA,ε acting on smooth functions defined on R. Here A is a Sturm-Liouville function with additional hypotheses and ε∈R. For special pairs (A, ε), we recover Dunkl's, Heckman's and Cherednik's operators (in one dimension). As, by construction, the operators ΛA,ε are mixture of d/dx and reflection operators, we prove the existence of an operator VA,ε so that ΛA,εVA,ε=VA,εd/dx. The positivity of the intertwining operator VA,ε is also established. Via the eigenfunctions of ΛA,ε, we introduce a generalized Fourier transform FA,ε. For -1≤ε≤1 and 0<p≤2/1+√1-ε2, we develop an Lp-Fourier analysis for FA,ε, and then we prove an Lp-Schwartz space isomorphism theorem for FA,ε. Details of this paper will be given in other articles  and .
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