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Uncertainty principles for the continuous Dunkl Gabor transform and the Dunkl continuous wavelet transform. (English) Zbl 1181.26036

Summary: We consider the Dunkl operators \(T_j\), \(j=1,\dots,d\), on \(\mathbb R^d\) and the harmonic analysis associated with these operators. We define a continuous Dunkl Gabor transform, involving the Dunkl translation operator, by proceeding as mentioned in [J. Fourier Anal. Appl. 9, No. 4, 321–339 (2003; Zbl 1037.42031)] by C. Wojciech and G. Gigante. We prove a Plancherel formula, an \(L^2_k\) inversion formula and a weak uncertainty principle for it. Then, we show that the portion of the continuous Dunkl Gabor transform lying outside some set of finite measure cannot be arbitrarily too small. Similarly, using the basic theory for the Dunkl continuous wavelet transform introduced by K. Trimèche in [Rocky Mt. J. Math. 32, No. 2, 889–917 (2002; Zbl 1034.44005)], an analogous of this result for the Dunkl continuous wavelet transform is given. Finally, an analogous of Heisenberg’s inequality for a continuous Dunkl Gabor transform (resp. Dunkl continuous wavelet transform) is proved.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
43A32 Other transforms and operators of Fourier type
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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