Norm estimates for \((C,\delta)\) means of Hermite expansions and bounds for \(\delta_{eff}\)
Let \(L^ p_{u(H)}\) be the space of functions f with norm \(\| f\|_ p=\| f(x)u(x)\|_{L^ p(-\infty,\infty)},\) where \(u(x)=\exp(-x^ 2/2),\) 1\(\leq p\leq \infty\). For \((C,\delta)\) means of Hermite expansions upper and lower norm estimates are established for any \(\delta\geq 0\), 1\(\leq p\leq \infty\). In particular, for \(p=1\), \(\delta =0\) it is shown that \(c_ 1n^{1/6}\| f\|_ 1\leq \|(C,0)_ nf\|_ 1\leq c_ 2n^{1/6}\| f\|_ 1\) for some constants \(c_ 1,c_ 2>0\). Results of \textit{R. Askey} and \textit{S. Wainger} [Am. J. Math. 87, 695-708 (1965; Zbl 0125.313)] and \textit{B. Muckenhoupt} [Trans. Am. Math. Soc. 147, 419-431, 433-460 (1970; Zbl 0191.076)] on partial sums and \textit{E. L. Poiani} [Trans. Am. Math. Soc. 173, 1-31 (1972; Zbl 0224.42014)] on (C,1) means are contained. Denoting by \(\delta_{eff}\) the ''effective'' \(\delta\), i.e. the largest \(\delta>0\) with the property that the Hermite expansions is not \((C,\delta)\) summable for at least one \(f\in L^ 1_{u(H)}\), it is shown that \(1/6\leq \delta_{eff}\leq 1/2\). Though the exact value of \(\delta_{eff}\) remains an open question, the present results imply that the Hermite expansion differs in a more essential way from the Laguerre expansion than could be expected in view of the known results. For the proof, recent results on Laguerre expansions with shifted parameter [the author, Anal. Math. 8, 19-37 (1982; Zbl 0515.42043)] are used.
scientific article; zbMATH DE number 3855703
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