On weighted generalized functions associated with quadratic forms

In this article we consider certain types of weighted generalized functions associated with nondegenerate quadratic forms. Such functions and their derivatives are used for constructing fundamental solutions of iterated ultra-hyperbolic equations with Bessel operator and for constructing negative real powers of ultra-hyperbolic operators with Bessel operator.Comment: 16 page

On fractional powers of the Bessel operator on a semiaxis

In this paper we study fractional powers of the Bessel differential operator defined on a semiaxis. Some important properties of such fractional powers of the Bessel differential operator are proved. They include connections with Legendre functions for kernel representations, fractional integral operators of Liouville and Saigo, Mellin transform and index laws. Possible applications are indicated to differential equations with fractional powers of the Bessel differential operator.Comment: English version (pp. 1--8) and Russian version (pp. 9--18

Fourier-Bessels transform of a generalized function Vanishing outside a bounded surface

The Fourier-Bessel transform of any generalized function f € S'ev vanishing outside a bounded surface for any test function ψ(х) €Se

Uniqueness of the solution of the cauchy problem for the general Euler-Poisson-Darboux equation

For the general Euler-Poisson-Darboux equation, we prove a theorem on the uniqueness of the solution of the Cauchy problem by the energy metho

Mean-value theorem for B-harmonic functions

We establish a mean value property for the functions which is satisfied to Laplace–Bessel equatio

On the theory of spaces of generalized Bessel potentials

Potential theory originates from the theory of electrostatic and gravitational potentials and the study of the Laplace, wave, Helmholtz, and Poisson equations. The celebrated Riesz potentials are the realizations of the real negative powers of the Laplace and wave operators. In the meantime, much attention in potential theory is paid to the Bessel potential generating the spaces of fractional smoothnes

On generalized Bessel potentials and perfect functional completions

In this paper, perfect completions are constructed using the norm associated with the kernel of the generalized Bessel potentia

Solving the Euler-Poisson-Darboux equation of fractional order

We consider the Cauchy problem for the one-dimensional, homogeneous Euler-Poisson-Darboux equation with a differential operator of fractional order in time being the left-sided fractional Bessel operator. At the same time, we use the ordinary differential operator in the space variable of the second orde