One-dimensional (k, a)-generalized Fourier transform

We study the two-parametric (k,a)-generalized Fourier transform F-k,F-a, k, a > 0, on the line. For a not equal 2 it has deformation properties and, in particular, for a function f from the Schwartz space S(R), F-k,F-a(f) may be not infinitely differentiable or rapidly decreasing at infinity. It is proved that the invariant set for the generalized Fourier transform F k,a and differential-difference operator |x|2-a Delta(k)f(x), where Delta(k) is the Dunkl Laplacian, is the class S-a(R) = {f(x) = F-1 (|x|(a/2)) + xF(2)(|x|(a/2)) : F-1, F-2 is an element of S(R), F-1, F-2 - are even}. For a = 1/r, r is an element of N, we consider two generalized translation operators Tau(y) and Tau(y) = (T-y+T-y) /2. Simple integral representations are proposed for them, which make it possible to prove their L-p-boundedness as 1 <= p <= infinity for lambda = r(2k - 1) > -1/2. For lambda >= 0 the generalized translation operator Ty is positive and its norm is equal to one. Two convolutions are defined and Young's theorem is proved for them. For generalized means defined using convolutions, a sufficient L-p-convergence condition is established. The generalized analogues of the Gauss-Weierstrass, Poisson, and Bochner-Riesz means are studied.