# Almost everywhere summability of Laguerre series

Studia Mathematica (1991)

• Volume: 100, Issue: 2, page 129-147
• ISSN: 0039-3223

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## Abstract

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We apply a construction of generalized twisted convolution to investigate almost everywhere summability of expansions with respect to the orthonormal system of functions ${\ell }_{n}^{a}\left(x\right)={\left(n!/\Gamma \left(n+a+1\right)\right)}^{1/2}{e}^{-x/2}{L}_{n}^{a}\left(x\right)$, n = 0,1,2,..., in ${L}^{2}\left({ℝ}_{+},{x}^{a}dx\right)$, a ≥ 0. We prove that the Cesàro means of order δ > a + 2/3 of any function $f\in {L}^{p}\left({x}^{a}dx\right)$, 1 ≤ p ≤ ∞, converge to f a.e. The main tool we use is a Hardy-Littlewood type maximal operator associated with a generalized Euclidean convolution.

## How to cite

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Stempak, Krzysztof. "Almost everywhere summability of Laguerre series." Studia Mathematica 100.2 (1991): 129-147. <http://eudml.org/doc/215878>.

@article{Stempak1991,
abstract = {We apply a construction of generalized twisted convolution to investigate almost everywhere summability of expansions with respect to the orthonormal system of functions $ℓ_n^a(x) = (n!/Γ(n+a+1))^\{1/2\} e^\{-x/2\} L_n^a(x)$, n = 0,1,2,..., in $L^2(ℝ_+, x^adx)$, a ≥ 0. We prove that the Cesàro means of order δ > a + 2/3 of any function $f ∈ L^p(x^adx)$, 1 ≤ p ≤ ∞, converge to f a.e. The main tool we use is a Hardy-Littlewood type maximal operator associated with a generalized Euclidean convolution.},
author = {Stempak, Krzysztof},
journal = {Studia Mathematica},
keywords = {Laguerre expansions; generalized twisted convolution; Riesz; Cesàro and Abel-Poisson means; Laguerre series; Watson's product formula; Laguerre polynomials; Riesz means; Abel-Poisson means; almost everywhere summability; Cesàro means; Hardy- Littlewood type maximal operator; generalized Euclidean convolution},
language = {eng},
number = {2},
pages = {129-147},
title = {Almost everywhere summability of Laguerre series},
url = {http://eudml.org/doc/215878},
volume = {100},
year = {1991},
}

TY - JOUR
AU - Stempak, Krzysztof
TI - Almost everywhere summability of Laguerre series
JO - Studia Mathematica
PY - 1991
VL - 100
IS - 2
SP - 129
EP - 147
AB - We apply a construction of generalized twisted convolution to investigate almost everywhere summability of expansions with respect to the orthonormal system of functions $ℓ_n^a(x) = (n!/Γ(n+a+1))^{1/2} e^{-x/2} L_n^a(x)$, n = 0,1,2,..., in $L^2(ℝ_+, x^adx)$, a ≥ 0. We prove that the Cesàro means of order δ > a + 2/3 of any function $f ∈ L^p(x^adx)$, 1 ≤ p ≤ ∞, converge to f a.e. The main tool we use is a Hardy-Littlewood type maximal operator associated with a generalized Euclidean convolution.
LA - eng
KW - Laguerre expansions; generalized twisted convolution; Riesz; Cesàro and Abel-Poisson means; Laguerre series; Watson's product formula; Laguerre polynomials; Riesz means; Abel-Poisson means; almost everywhere summability; Cesàro means; Hardy- Littlewood type maximal operator; generalized Euclidean convolution
UR - http://eudml.org/doc/215878
ER -

## References

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21. [21] G. N. Watson, Another note on Laguerre polynomials, J. London Math. Soc. 14 (1939), 19-22. Zbl0020.21805

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