Norm estimates for \((C,\delta)\) means of Hermite expansions and bounds for \(\delta_{eff}\)
type: Item
instance of: scholarly article
title: Norm estimates for \((C,\delta)\) means of Hermite expansions and bounds for \(\delta_{eff}\)
zbMATH Open document ID: 0538.42012
DOI: 10.1007/BF01958018
published in: Acta Mathematica Hungarica
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 40G05
zbMATH DE Number: 3855703
zbMATH Keywords: mean convergence of Hermite expansions
zbMATH Keywords: weight
zbMATH Keywords: Cesàro means
zbMATH Keywords: Laguerre expansion
zbMATH Keywords: Christoffel-Darboux formula
author: Clemens Markett
MaRDI profile type: Publication
cites work: Norm inequalities for some orthogonal series
cites work: Mean convergence of orthogonal series and Lagrange interpolation
cites work: Mean Convergence of Expansions in Laguerre and Hermite Series
cites work: On summation processes of Fourier expansions in Banach spaces. I: Comparison theorems
cites work: Q5179890
cites work: On linear processes of approximation (II)
cites work: Q3878133
cites work: Q3244654
cites work: The Lebesgue Constants for Jacobi Series, I
cites work: Mean Cesaro summability of Laguerre expansions and norm estimates with shifted parameter
cites work: Mean Convergence of Hermite and Laguerre Series. I
cites work: Mean Cesaro Summability of Laguerre and Hermite Series
Recommended article: Two-weight norm inequalities for the Cesàro means of Hermite expansions
Recommended article: Mixed norm estimates for the Cesàro means associated with Dunkl–Hermite expansions
Recommended article: Two-weight norm inequalities for the Cesàro means of generalized Hermite expansions
Recommended article: On the \(C^ 1\)-norm of the Hermite interpolation operator
Recommended article: Q5699800
Recommended article: On Hardy's inequality for Hermite expansions
Recommended article: \(L^p\)-norms of Hermite polynomials and an extremal problem on Wiener chaos
Recommended article: The Hardy inequality for Hermite expansions
Recommended article: \(L^p\) eigenfunction bounds for the Hermite operator
Recommended article: Inequality for the norms of Hermite functions on a finite interval.
description: scientific article; zbMATH DE number 3855703