On the index integral transformation with Nicholson's function as the kernel
This paper contains a new and interesting study of the Nicholson transformation of the form \[ g(x)= {\pi\over 4} \int^\infty_0 \bigl[ J^2_\alpha (\sqrt x)+ Y^2_\alpha (\sqrt x)\bigr] \tau f(\tau) d\tau, \] where \(\alpha= |{i \tau\over 2} |\) and integration is with respect to an index of the sum of squares of Bessel functions of the first and the second kind. Several properties of this index integral transformation including the composition properties, with other well-known integral transformation, the inversion theorem, Parseval's type relation are proved. Composition representation and relationships with the Meijer K-transform, the Mellin transform, the Kontorovich-Lebedev transform and the sine Fourier transform are discussed. It is shown that the Nicholson transformation is a bounded operator given by the formula \[ [JYf](x)=-{x\pi^{5/2}\over 2} \int^\infty_0 {d\over dx} \bigl[J^2_\alpha (\sqrt x)+ Y^2_\alpha(\sqrt x)\bigr] {\tau f(\tau) d\tau\over \cosh({\pi \tau\over 2})}. \] Although no particular application of the Nicholson transformation is given, it seems to be useful for solving problems in applied mathematics and physical sciences.
scientific article; zbMATH DE number 1797503
scientific article