Uncertainty inequalities and order of magnitude of Hankel transforms (Q2754938)

The introduction of the paper gives a very comprehensive review of the classical results for the Hankel transform. The transform is defined as, NEWLINE\[NEWLINE h_\mu(f)(x)=\int^\infty_0(xy)^{-\mu}J_\mu(xy)f(y)y^{2\mu+1}dy, NEWLINE\]NEWLINE where \(f\) is a function, which is transformable. NEWLINENEWLINENEWLINEThe paper studies the behaviour of the Hankel integral transform at infinity in the Cesaro sense. Several results surround this notion and one result is as follows: NEWLINENEWLINENEWLINEProposition. Assume that \(f\in L_{1,\mu}\cap L_{p,\mu}\), with \(1<p\leq 2\), and that \(0<\alpha+\frac{2(\mu+1)}p<2\). Then NEWLINE\[NEWLINE \lim_{\lambda \to \infty}\frac 1{\lambda^2}\int^\lambda_0 \left(1-\left(\frac t\lambda \right)^2\right)^{-1/p}t^\alpha h_\mu (f)(t)t^{2\mu+1}dt=0. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEThe proofs are elegant and complete.NEWLINENEWLINENEWLINEThe paper also establishes uncertainty inequalities for the transform and Laguerre expansions. A very nice Laguerre expansion result is as follows: NEWLINENEWLINENEWLINEProposition. Let \(f\in L_{2,\mu}\cap L_{2,\mu+2}\). Then NEWLINE\[NEWLINE \int^\infty_0x^6(|f(x)|^2+|h_\mu(f)(x)|^2)x^{2\mu+1}dx=2\sum^\infty_{n=0} |\langle x^2f,\phi^\mu_n\rangle_\mu|^2(\mu+2n+1). NEWLINE\]NEWLINENEWLINENEWLINENEWLINEThe paper concludes with an entropy inequality for the Hankel transform.