Opuscula Math. 29, no. 1 (2009), 93-110
Beurling's theorems and inversion formulas for certain index transforms
Semyon B. Yakubovich
Abstract. The familiar Beurling theorem (an uncertainty principle), which is known for the Fourier transform pairs, has recently been proved by the author for the Kontorovich-Lebedev transform. In this paper analogs of the Beurling theorem are established for certain index transforms with respect to a parameter of the modified Bessel functions. In particular, we treat the generalized Lebedev-Skalskaya transforms, the Lebedev type transforms involving products of the Macdonald functions of different arguments and an index transform with the Nicholson kernel function. We also find inversion formulas for the Lebedev-Skalskaya operators of an arbitrary index and the Nicholson kernel transform.
Keywords: Beurling theorem, Kontorovich-Lebedev transform, Lebedev-Skalskaya transforms, Fourier transform, Laplace transform, modified Bessel functions, uncertainty principle, the Nicholson function.
Mathematics Subject Classification: 44A10, 44A15, 33C10.