On Gegenbauer transformation on the half-line

Abstract

In this paper, the Gegenbauer transformation is constructed and some of its properties similar to the Fourier transformation are proved. An equation of Parseval–Plancherel type is obtained. The inversion theorem for the Gegenbauer transformation is proved.

Boundedness of Gegenbauer fractional maximal commutator on generalized Gegenbauer-Morrey spaces

“…Note that all result obtained in the paper are the future development of Gegenbauer harmonic analysis theory, foundations of which were laid in [32]. This theory was later developed in different directions: approximation and embedding theory, transformation theory, theory of singulars integrals, maximal functions theory, theory of potentials and its commutator.…”

Some aspects of harmonic analysis related to Gegenbauer expansions on the half-line

“…Introduction The Hardy-Lettlewood maximal function is an important tool of harmonic analysis. It was first introduced by Hardy and Littlewood in 1930 (see [21]) for 2π− periodical functions, and later it was extendet to the Euclidean spaces, some weighted measure spaces (see [4,28,29]) , symmetric spaces (see [5,26]), various Lie groups [9], for the Jacobitype hypergroups [6,7], for Chebli-Trimeche hypergroups [1], for the onedimensional Bessel-Kingman hypergroups [27], for the n− dimensional Bessel-Kingman hypergroups (n ≥ 1) [10,11,13], for Morrey-Bessel spaces [2,3,12,14], for Laguerre hypergroup [15,16,22,25]. The structure of the paper is as follows.…”