Abelian and Tauberian theorems for a class of integral transforms

Extending the Wiener-Ganelius method we give Abelian and precise Tauberian remainder results for a class of Fourier kernels which includes the Hankel transform $\text{?(x) = ∫}{}_0{}^{\mathrm{\infty }}\text{√xu J}{}_{\mathrm{r}}\text{(xu) ?(u) du, v ? ?}\text{1}\text{2}$. Further, we discuss applications to Fourier series and integrals.