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

In 2012, Salem Ben Saǐd, Kobayashi, and Orsted defined the two-parametric $(k, a)$-generalized Fourier transform, acting in the space with weight $|x|^{a-2}v_k(x)$, $a>0$, where $v_k(x)$ 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 $\tau^y$. Its $L^p$-boundedness was recently established by Salem Ben Saǐd and Deleaval, but only on radial functions and for $1\le p\le 2$. Earlier, we proposed for $a=1$ a new positive generalized translation operator and proved that it is $L^p$ -bounded in $x$. In this paper, it is proved that it is $L^p$ -bounded in $t$. For the translation operator $\tau^y$, $L^p$-boundedness on radial functions is established for $2<p<\infty$. The operator $T^t$ is used to define a convolution and to prove Young's inequality. For $(k, 1)$-generalized means defined by convolution, sufficient conditions for $L^p$-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.