Weighted norm inequalities for the finite Hankel transformation (Q1921962)

\textit{P. Heywood} and \textit{P. G. Rooney} [Proc. R. Soc. Edinb., Ser. A 45-50 (1984; Zbl 0564.44003)] obtained a weighted norm inequality for the Hankel transformation. The present article contains a corresponding result for the finite Hankel transformation on \([0,1]\). For \(f(t) \in L_1 [0,1]\), the finite Hankel transformation \(h_\nu f\), \(\nu \geq-1/2\) is defined as the sequence \[ (h_\nu f) (n) = {1 \over J_{\nu+1} (\lambda_n)} \int^1_0 f(t) J_\nu (\lambda_nt) \sqrt {2t} dt, \quad n= 1,2, \dots, \] where \(\lambda_n\) are the positive real zeros of the Bessel functions \(J_\nu\).