Beurling’s Theorem for the Q-Fourier-Dunkl Transform

The Q-Fourier-Dunkl transform satisfies some uncertainty principles in a similar way to the Euclidean Fourier transform. By using the heat kernel associated to the Q-Fourier-Dunkl operator, we establish an analogue of Beurling’s theorem for the Q-Fourier-Dunkl transform ℱQ on ℝ.

The Range of the Spectral Projection Associated with the Dunkl Laplacian

For s∈ℝ, denote by Pksf the “projection” of a function f in Dℝd into the eigenspaces of the Dunkl Laplacian Δk corresponding to the eigenvalue −s2. The parameter k comes from Dunkl’s theory of differential-difference operators. We shall characterize the range of Pks on the space of functions f∈Dℝd supported inside the closed ball BO,R¯. As an application, we provide a spectral version of the Paley-Wiener theorem for the Dunkl transform.

A variety of uncertainty principles for the Dunkl transform on ℝd

In this work, we prove Clarkson-type and Nash-type inequalities in the Dunkl setting on [Formula: see text] for [Formula: see text]-functions. By combining these inequalities, we show Heisenberg-type inequalities for the Dunkl transform on [Formula: see text], and we deduce local-type uncertainty inequalities for the Dunkl transform on [Formula: see text].