Three-Dimensional Analog of the Integer-Order Hankel Transform

Mathematical notes. 2025. Vol. 118. No. 4. P. 794–810.

E. M. Novikova

For each integer nonnegative n, in some Hilbert space, we introduce an integral
transform H_n. It is similar to the well-known Hankel (Fourier–Bessel) transform of nth 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 H_n is intended for functions of three variables. For the constructed transform 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
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 H_n turn into a product.

Keywords: преобразование Фурье, convolution, Fourier transform, Hankel transform, integral transform, преобразование Ханкеля, интегральное преобразование, свертка

Publication based on the results of: Advanced materials and technologies (2025)