Three-Dimensional Analog of the Integer-Order Hankel Transform

For each integer nonnegative $$n$$ , in some Hilbert space, we introduce an integral transform $$\mathfrak{H}_n$$ . It is similar to the well-known Hankel (Fourier–Bessel) transform of $$n$$ th order due to the fact that it is related to the Fourier transform and its integral kernel is expressed in terms of the Bessel function $$J_n$$ . But, unlike the Hankel transform, intended for functions of one variable, the transform $$\mathfrak{H}_n$$ is intended for functions of three variables. For the constructed transform $$\mathfrak{H}_n$$ , we prove Plancherel’s theorem, the inversion formula, the similarity property and formulas of composition with some differential operators. In particular, we found second-order differential operators which under $$\mathfrak{H}_n$$ turn into operators of multiplication by certain functions, and we give an example of using the constructed transform to solve a differential equation for the Laplace operator with a Coulomb-type potential. In addition, we described a family of convolutions which under $$\mathfrak{H}_n$$ turn into a product.