Density properties of Hankel translations of positive definite functions (Q5927599)

The space \(A_{\mu}\) of infinitely differentiable functions \(f\) with \((\frac 1{x}D)^k f\) bounded on \((0,1)\) is endowed with the topology generated by the family of seminorms \(w_{\mu}^{m,k}(\phi)=\sup_{0\leq x\leq m} |\Delta^k_{\mu}\phi (x)|\), where \(\Delta_{\mu}\) is Bessel operator \(x^{-2\mu-1}D x^{2\mu+1} D\). The article proves that, under an additional condition on the tempered growth of a Hankel positive definite function \(f\in A_{\mu}\) and a condition on the preimage of the Hankel transform for the function \(f\), the linear space generated by Hankel translations of the function \(f\) is dense in \(A_{\mu}\) in the sense of the above mentioned topology.