Thangavelu, S. On almost everywhere and mean convergence of Hermite and Laguerre expansions. (English) Zbl 0747.42014 Colloq. Math. 60/61, No. 1, 21-34 (1990). The results of the author in Trans. Am. Math. Soc. 314, No. 1, 143-170 (1989; Zbl 0685.42016) are improved.Namely, for \(1\leq p\leq\infty\), \(\alpha>(n-1)/2\) the boundedness of the operator \(S^ \alpha_ R:L^ p\to L^ p\) is proved, where \(S_ R^ \alpha f\) are Riesz means of order \(\alpha\) of Hermite expansions of a function \(f\) (cf. loc.cit.). The a.e. convergence of \(S_ R^ \alpha f(x)\) to \(f(x)\) for \(f\in L^ p(R^ p)\), \(p\geq 2\), \(n\geq 2\) and \(\alpha>(n-1)(1/2-1/p)\) and the convergence of \(S_ R^ \alpha f(x)\) to \(f(x)\) at every Lebesgue point \(x\) of \(f\) if \(\alpha>(n-1)/2\) are proved.Moreover the a.e. convergence of Riesz means \(\sigma_ n^ \alpha f(r)\) of order \(\alpha>(2n-1)(1/2-1/p)\) of Laguerre expansions of a function \(f\in L^ p(R_ +,r^{2n-1}dr)\), \(2\leq p\leq\infty\) (the notations are from the author [Rev. Mat. Iberoam. 3, No. 1, 1-24 (1987; Zbl 0687.42015)]) is proved. Reviewer: A.Lukashov (Saratov) Cited in 7 Documents MSC: 42C15 General harmonic expansions, frames 42B99 Harmonic analysis in several variables Keywords:Laguerre expansion; Riesz means; Hermite expansion; Lebesgue point; a.e. convergence Citations:Zbl 0685.42016; Zbl 0687.42015 PDFBibTeX XMLCite \textit{S. Thangavelu}, Colloq. Math. 60/61, No. 1, 21--34 (1990; Zbl 0747.42014) Full Text: DOI