Properties and application of a positive translation operator for (𝑘, 1)-Generalized Fourier transform

Abstract

In 2012, Salem Ben Saˇid, Kobayashi, and Orsted defined the two-parametric (𝑘, 𝑎)- generalized Fourier transform, acting in the space with weight |𝑥|𝑎−2𝑣𝑘(𝑥), 𝑎 > 0, where 𝑣𝑘(𝑥) is the Dunkl weight. The most interesting cases are 𝑎 = 2 and 𝑎 = 1. For 𝑎 = 2 the generalized Fourier transform coincides with the Dunkl transform and it is well studied. In case 𝑎 = 1 harmonic analysis, which is important, in particular, in problems of quantum mechanics, has not yet been sufficiently studied. One of the essential elements of harmonic analysis is the bounded translation operator, which allows one to determine the convolution and structural characteristics of functions. For 𝑎 = 1, there is a translation operator 𝜏 𝑦. Its 𝐿𝑝-boundedness
was recently established by Salem Ben Saˇid and Deleaval, but only on radial functions and for 1 ⩽ 𝑝 ⩽ 2. Earlier, we proposed for 𝑎 = 1 a new positive generalized translation operator and
proved that it is 𝐿𝑝 -bounded in 𝑥. In this paper, it is proved that it is 𝐿𝑝 -bounded in 𝑡. For the translation operator 𝜏 𝑦, 𝐿𝑝-boundedness on radial functions is established for 2 < 𝑝 < ∞. The operator 𝑇𝑡 is used to define a convolution and to prove Young’s inequality. For (𝑘, 1)-generalized means defined by convolution, sufficient conditions for 𝐿𝑝-convergence and convergence almost everywhere are established. The fulfillment of these conditions is verified for analogues of the classical summation methods of Gauss–Weierstrass, Poisson, Bochner–Riesz.