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On almost everywhere and mean convergence of Hermite and Laguerre expansions. (English) Zbl 0747.42014

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.

MSC:

42C15 General harmonic expansions, frames
42B99 Harmonic analysis in several variables
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