TY  - JOUR
ID  - Yakubovich2009
LB  - Yakubovich2009
AU  - Yakubovich, Semyon B.
TI  - A double index transform with a product of Macdonald's functions revisited
ST  - A double index transform with a product of Macdonald's functions revisited
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2009
VL  - 29
IS  - 3
SP  - 313
EP  - 329
DO  - http://dx.doi.org/10.7494/OpMath.2009.29.3.313
UR  - http://dx.doi.org/10.7494/OpMath.2009.29.3.313
AB  - We prove an inversion theorem for a double index transform, which is associated with the product of Macdonald's functions \(K_{i \tau}(\sqrt{x^2+y^2}-y) K_{i \tau}(\sqrt{x^2+y^2}+y)\), where \((x, y) \in \mathbb{R}_+ \times \mathbb{R}_+\) and \(i \tau, \tau \in \mathbb{R}_+\) is a pure imaginary index. The results obtained in the sequel are applied to find particular solutions of integral equations involving the square and the cube of the Macdonald function \(K_{i \tau}(t)\) as a kernel.
KW  - Macdonald function
KW  - index transform
KW  - Kontorovich-Lebedev transform
KW  - double Mellin transform
KW  - Plancherel theorem
KW  - Parseval equality
SN  - 1232-9274
ER  -