In a previous paper we introduced a family of differential-reflection operators ΛA , ϵ acting on smooth functions defined on R, where the spectral problem for the operators ΛA,ϵ has been studied. Here A is a Sturm-Liouville function with additional hypotheses and-1 ≤ ϵ ≤ 1. Via the eigenfunctions of ΛA,ϵ, we introduce in this paper a generalized Fourier transform fA,ϵ. An Lp-harmonic analysis for fA,ϵ is developed when 0 < p ≤ 2/1 + 1-ϵ2. In particular, an Lp-Schwartz space isomorphism theorem for fA,ϵ is proved.
- Differential-reflection operators
- L-harmonic analysis
- generalized Fourier transform
ASJC Scopus subject areas