On Fourier transforms

If $f(x)$ and $g(x)$ satisfy the equations

$$g(x) = \frac{d}{{dx}}\int _0^\infty \frac{1}{t}f(t){k_1}(xt)dt,\quad f(x) = \frac{d}{{dx}}\int _0^\infty \frac{1}{t}g(t){k_1}(xt)dt,$$

then we call f and g a pair of ${k_1}$-transforms, where

$${k_1} = \frac{1}{{2\pi i}}\int _{{\raise0.5ex\hbox{$\scriptstyle ... ...r0.25ex\hbox{$\scriptstyle 2$}} + i\infty }\frac{{K(s)}}{{1 - s}}{x^{1 - s}}ds.$$

In this paper alternative sets of conditions are established for f and g to be ${k_1}$-transform provided $K(s)$ is decomposable in a special way. These conditions involve simpler functions, which replace the kernel ${k_1}(x)$. Results are proved for the function spaces ${L^2}$. The necessary and sufficient conditions are established for the two functions to be self-reciprocal. Conditions are given for generating pairs of transforms for a given kernel. Two examples are given at the end to illustrate the methods and the advantage of the results.