Real Paley–Wiener theorems for the Koornwinder–Swarttouw q-Hankel transform

Abstract

We derive two real Paley–Wiener theorems in the setting of quantum calculus. The first uses techniques due to Tuan and Zayed [V.K. Tuan, A.I. Zayed, Paley–Wiener-type theorems for a class of integral transforms, J. Math. Anal. Appl. 266 (1) (2002) 200–226] in order to describe the image of the space L q 2 ( 0 , R ) under Koornwinder and Swarttouw q-Hankel transform [T.H. Koornwinder, R.F. Swarttouw, On q-analogues of the Fourier and Hankel transforms, Trans. Amer. Math. Soc. 333 (1) (1992) 445–461] and contains as a special case a description of the domain of the q-sampling theorem associated with the q-Hankel transform [L.D. Abreu, A q-sampling theorem related to the q-Hankel transform, Proc. Amer. Math. Soc. 133 (4) (2005) 1197–1203]. The second characterizes the image of compactly supported q-smooth functions under a rescaled version of the q-Hankel transform and is a q-analogue of a recent result due to Andersen [N.B. Andersen, Real Paley–Wiener theorems for the Hankel transform, J. Fourier Anal. Appl. 12 (1) (2006) 17–25].