Full Fourier-Bessel transform and the algebra of singular pseudodifferential operators

Abstract

The one-dimensional full Fourier-Bessel transform was introduced by I.A. Kipriyanov and V.V. Katrakhov on the basis of even and odd small (normalized) Bessel functions. We introduce a mixed full Fourier-Bessel transform and prove an inversion formula for it. Singular pseudodifferential operators are introduced on the basis of the mixed full Fourier-Bessel transform. This class of operators includes linear differential operators in which the singular Bessel operator and its (integer) powers or the derivative (only of the first order) of powers of the Bessel operator act in one of the directions. We suggest a method for constructing the asymptotic expansion of a product of such operators. We present the form of the adjoint singular pseudodifferential operator and show that the constructed algebra is, in a sense, a *-algebra.