On Fourier transforms

Abstract:

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 \[ k_1 = \frac {1}{2\pi i} \int _{1/2 - i\infty }^{1/2 + i\infty } \frac {K(s)}{1 - s} x^{1-s} ds. \] 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.