Mean summability methods for Laguerre series

Stempak, Krzysztof

doi:10.1090/S0002-9947-1990-0974528-8

Mean summability methods for Laguerre series
HTML articles powered by AMS MathViewer

by Krzysztof Stempak PDF

Trans. Amer. Math. Soc. 322 (1990), 671-690 Request permission

Abstract:

We apply a construction of generalized convolution in \[ {L^1}({\mathbb {R}_ + } \times \mathbb {R},{x^{2\alpha - 1}}dxdt),\qquad \alpha \geqslant 1,\] cf. [8], to investigate the mean convergence of expansions in Laguerre series. Following ideas of [4, 5] we construct a functional calculus for the operator \[ L = - \left ( {\frac {{{\partial ^2}}} {{\partial {x^2}}} + \frac {{2\alpha - 1}} {x}\frac {\partial } {{\partial x}} + {x^2}\frac {{{\partial ^2}}} {{\partial {t^2}}}} \right ),\qquad x > 0,\quad t \in \mathbb {R},\quad \alpha \geqslant 1.\] Then, arguing as in [3], we prove results concerning the mean convergence of some summability methods for Laguerre series. In particular, the classical Abel-Poisson and Bochner-Riesz summability methods are included.

References

Richard Askey and Stephen Wainger, Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math. 87 (1965), 695–708. MR 182834, DOI 10.2307/2373069
Jacek Cygan, Subadditivity of homogeneous norms on certain nilpotent Lie groups, Proc. Amer. Math. Soc. 83 (1981), no. 1, 69–70. MR 619983, DOI 10.1090/S0002-9939-1981-0619983-8
Jolanta Długosz, Almost everywhere convergence of some summability methods for Laguerre series, Studia Math. 82 (1985), no. 3, 199–209. MR 825478, DOI 10.4064/sm-82-3-199-209
A. Hulanicki, Subalgebra of $L_{1}(G)$ associated with Laplacian on a Lie group, Colloq. Math. 31 (1974), 259–287. MR 372536, DOI 10.4064/cm-31-2-259-287
Andrzej Hulanicki and Joe W. Jenkins, Almost everywhere summability on nilmanifolds, Trans. Amer. Math. Soc. 278 (1983), no. 2, 703–715. MR 701519, DOI 10.1090/S0002-9947-1983-0701519-0
Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0248482
Benjamin Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans. Amer. Math. Soc. 139 (1969), 231–242. MR 249917, DOI 10.1090/S0002-9947-1969-0249917-9
Krzysztof Stempak, An algebra associated with the generalized sub-Laplacian, Studia Math. 88 (1988), no. 3, 245–256. MR 932012, DOI 10.4064/sm-88-3-245-256
Krzysztof Stempak, A new proof of a Watson’s formula, Canad. Math. Bull. 31 (1988), no. 4, 414–418. MR 971567, DOI 10.4153/CMB-1988-060-8
David Jerison and Antonio Sánchez-Calle, Subelliptic, second order differential operators, Complex analysis, III (College Park, Md., 1985–86) Lecture Notes in Math., vol. 1277, Springer, Berlin, 1987, pp. 46–77. MR 922334, DOI 10.1007/BFb0078245