2022/10/27 by Gorbachev, D. V., Ivanov, V. I., Tikhonov, S. Yu. · 1 citation
#33C45 #33C52 #42B10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2210.15730
For the kernel Bκ,a(x,y) of the (κ,a)-generalized Fourier transform Fκ,a, acting in L2(ℝd) with the weight |x|a-2vκ(x), where vκ is the Dunkl weight, we study the important question of when ‖Bκ,a‖∞=Bκ,a(0,0)=1. The positive answer was known for d≥ 2 and (2)/(a)∈ℕ. We investigate the case d=1 and (2)/(a)∈ℕ. Moreover, we give sufficient conditions on parameters for ‖Bκ,a‖∞>1 to hold with d≥ 1 and any a. We also study the image of the Schwartz space under the Fκ,a transform. In particular, we obtain that Fκ,a(S(ℝd))=S(ℝd) only if a=2. Finally, extending the Dunkl transform, we introduce non-deformed transforms generated by Fκ,a and study their main properties.