Ben Salem, Néjib; Samaali, Taha Hilbert transforms associated with Dunkl-Hermite polynomials. (English) Zbl 1162.42002 SIGMA, Symmetry Integrability Geom. Methods Appl. 5, Paper 037, 17 p. (2009). Summary: We consider expansions of functions in \(L^p(\mathbb R,|x|^{2k}dx), 1\leq p<+\infty \) with respect to Dunkl-Hermite functions in the rank-one setting. We actually define the heat-diffusion and Poisson integrals in the one-dimensional Dunkl setting and study their properties. Next, we define and deal with Hilbert transforms and conjugate Poisson integrals in the same setting. The formers occur to be Calderón-Zygmund operators and hence their mapping properties follow from general results. Cited in 2 Documents MSC: 42A50 Conjugate functions, conjugate series, singular integrals 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) Keywords:Dunkl operator; Dunkl-Hermite functions; Hilbert transforms; conjugate Poisson integrals; Calderón-Zygmund operators PDFBibTeX XMLCite \textit{N. Ben Salem} and \textit{T. Samaali}, SIGMA, Symmetry Integrability Geom. Methods Appl. 5, Paper 037, 17 p. (2009; Zbl 1162.42002) Full Text: DOI arXiv EuDML